Simple harmonic motion is a special type of oscillation in which the particle oscillates on a straight line, the acceleration of the particle is always directed towards a fixed point on the line and its magnitude is proportional to the displacement of the particle from this point.
The fixed point is called centre of oscillation.
If we take centre of oscillation as the origin and the line of motion as the X axis, SHM can be defined by the equation
a = -ω²x ... (1)
Where ω² is a positive constant.
If x is positive, a is negative and if x is negative, a is positive. It means that the acceleration if always directed towards to the centre of oscillation.
As acceleration is; a = F/m
We can write SHM equation as
F/m = -ω²x
F = -mω²x
F = -kx ...(2)
Force constant or spring constant
The constant k = mω² is called the force constant or spring constant.
The resultant force on the particle is zero when it is at the centre of oscillation
Equation of motion of SHM
Terms Associated with SHM
a. Amplitude
b. Time period
c. Frequency and angular frequency
d. Phase
e. Phase constant
Thursday, April 24, 2008
Concept review Ch. 13 Fluid Mechanics
Fluid
Pressure in fluids
Pascal's law
Atmospheric pressure
Barometer
Archimedes' principle
Buoyancy
Floatation
Flow of fluids: Steady flow, turbulent flow
Incompressible fluid
nonviscous fluid
Equation of continuity
Bernoulli's equation
Ventury tube
Aspirator pump
Pressure in fluids
Pascal's law
Atmospheric pressure
Barometer
Archimedes' principle
Buoyancy
Floatation
Flow of fluids: Steady flow, turbulent flow
Incompressible fluid
nonviscous fluid
Equation of continuity
Bernoulli's equation
Ventury tube
Aspirator pump
Concept review Ch. 14 Mechanical Properties of Matter
Elasticity
macroscopic reason of elasticity
Stress
Volume stress
Strain
Shearing strain
Hooke's law
Modulii of elasticity
Relation between longitudinal stress and strain
Elastic potential energy of a strained body
Surface tension
Surface energy
Contact angle
Viscosity
Poiseuille's equation
Stokes' law
Critical velocity
Reynold's number
macroscopic reason of elasticity
Stress
Volume stress
Strain
Shearing strain
Hooke's law
Modulii of elasticity
Relation between longitudinal stress and strain
Elastic potential energy of a strained body
Surface tension
Surface energy
Contact angle
Viscosity
Poiseuille's equation
Stokes' law
Critical velocity
Reynold's number
Concept review Ch.15 Wave Motion and Waves on a String
Particles carry kinetic energy with themselves and transfer energy to other particles in collisions. In this instance, the particle travels for some distance in space and then collides with another particle (Borrowing the terminology from heat and mass transfer we may say there is both mass transfer and energy transfer).
Wave motion is another way of transporting energy. When we say hello to our friend, no material particle is ejected by us from our lips that falls on our friend's ear. We create some disturbance in the part of air close to our lips. Energy is transferred to these air particles by sometimes pushing them ahead or sometimes pulling them back. This affects the density of ther air near our mouth. This disturbance is transferred to the next layer of air and so on till the ear of the listener gets disturbed. Only the disturbance produced in the air travels and the air itself does not move This tope of motion of energy is called a wave motion.
Wave motion on a string.
Consider a long string with one end fixed to a wall and the other held by a person. If the string is held tight, and a small bump is made near the end held by the person, the disturbance travels down the string with a constant speed.
For an elastic and homogeneous string, the bump moves with constant speed to cover equal distances in equal time periods. The shape of the bump is not altered as it moves provided the bump is small.
Equation of a travelling wave
In the string the wave travels from the hand end to the wall end. Let us assue that hand held end is at the left side and the end fixed in the wall is in the right. Hence the X-axis is along the string towards right.
Let function f(t) represent the displacement y of the particle at x = 0 as a function of time
y(x=0,t) = f(t)
As the disturbance is travelling on the string towards right with a constant speed v, the displacement in the y direction produced at the left end at time t, reaches the point x at time t+(x/v)(it takes time equal to x/v). We can express the statement in a different way. The displacement of the particle at point x at time time was originated at the left hand at the time t-(x/v). If the left end is taken as x=0, the displacement at time t-(x/v) will be f(t-x/v) which is y(x=0,t-x/v).
Hence y(x,t) = y(x=0,t-x/v) = f(t-x/v)
y(x,t) represents the displacementof the particle at x at time t. It is generally abbreviated as y and the wave equation is written as
y = f(t-x/v) ...(1)
The equation represents a wave travelling in the positive x direction with a constant speed v.
Function f depends on how the source is providing disturbance.
If the wave is travelling in negative x-direction, with speed v it may be written as
y = f(t+x/v)
Alternative forms of wave equation
y = A sinω(t-x/v)
y = A sin(ωt-kx) …(i)
Where k = ω/v
Remember λ = 2 π v/ ω = 2 π/k or
π/k = λ/2
y = A sin k(vt-x) as kv = ω …(ii)
Interference of waves going in the same direction
Let the amplitudes of the two waves be A1 and A2 and the two waves differ in phase by an angle δ.
Their equations may be written as
y1 = A1 sin(kx-ωt)
y2 = A2 sin(kx-ωt+δ)
By trigonometric properties, the combination of waves (suuperposition of waves) is represented by
= A sin(kx-ωt+ε)
A² = A² cos² ε + A² sin² ε
=
A1² +A2² +2A1A2cosδ
where A1 +A1cosδ = A cosε
A2sinδ = A sinε
Resultant amplitude is maximum, when cosδ = +1 or δ = 2nπ. The maximum value is A1+A2.
Resultant amplitude is minimum, when cosδ = -1 or cosδ = (2n+1)π . The minimum value is A1-A2.
(These results are used in interference of light.)
Standing waves
Standing waves are produced when two sine waves of equal amplitude and frequency propagate on a long string in opposite directions.
The equations of two waves can be:
y1 = A sin(ωt-kx)
y2 = A sin(ωt+kx)
The equations of the resultant wave will be:
y = y1+y2
= A[sin(ωt-kx) + sin(ωt+kx)]
= 2Asin ωt cos kx
= (2A cos kx) sin ωt
Interpretation of the equation: The amplitude of the wave is |2A cos kx|, but the amplitude is not equal for all the particles.
There are some points where the amplitude |2A cos kx| = 0 all the time.
This will at points where
cos kx = 0
=> kx = (n + ½) π
=> x = (n + ½) λ/2
where n is an integer.
Although these points are not physically clamped, they remain fixed as the two waves pass them simultaneously. These points whose amplitude is zero always are called nodes.
At the points where |cos kx| = 1, the maximum amplitude will be present. These points at which the maximum amplitude is obtained are called antinodes.
When sin ωt = 0, the amplitude of all points is zero which means that all points are at their normal positions or mean positions.
When sin ωt = 1, all points for which cos kx is positive reach their positive maximum displacement. At the same time, all points for which cos kx is negative reach their negative maximum displacement.
From the equation is also clear that the separation between consecutive nodes or consecutive antinodes is λ/2.
As the particles at the nodes do not move at all, energy cannot be transmitted across them.
Wave motion is another way of transporting energy. When we say hello to our friend, no material particle is ejected by us from our lips that falls on our friend's ear. We create some disturbance in the part of air close to our lips. Energy is transferred to these air particles by sometimes pushing them ahead or sometimes pulling them back. This affects the density of ther air near our mouth. This disturbance is transferred to the next layer of air and so on till the ear of the listener gets disturbed. Only the disturbance produced in the air travels and the air itself does not move This tope of motion of energy is called a wave motion.
Wave motion on a string.
Consider a long string with one end fixed to a wall and the other held by a person. If the string is held tight, and a small bump is made near the end held by the person, the disturbance travels down the string with a constant speed.
For an elastic and homogeneous string, the bump moves with constant speed to cover equal distances in equal time periods. The shape of the bump is not altered as it moves provided the bump is small.
Equation of a travelling wave
In the string the wave travels from the hand end to the wall end. Let us assue that hand held end is at the left side and the end fixed in the wall is in the right. Hence the X-axis is along the string towards right.
Let function f(t) represent the displacement y of the particle at x = 0 as a function of time
y(x=0,t) = f(t)
As the disturbance is travelling on the string towards right with a constant speed v, the displacement in the y direction produced at the left end at time t, reaches the point x at time t+(x/v)(it takes time equal to x/v). We can express the statement in a different way. The displacement of the particle at point x at time time was originated at the left hand at the time t-(x/v). If the left end is taken as x=0, the displacement at time t-(x/v) will be f(t-x/v) which is y(x=0,t-x/v).
Hence y(x,t) = y(x=0,t-x/v) = f(t-x/v)
y(x,t) represents the displacementof the particle at x at time t. It is generally abbreviated as y and the wave equation is written as
y = f(t-x/v) ...(1)
The equation represents a wave travelling in the positive x direction with a constant speed v.
Function f depends on how the source is providing disturbance.
If the wave is travelling in negative x-direction, with speed v it may be written as
y = f(t+x/v)
Alternative forms of wave equation
y = A sinω(t-x/v)
y = A sin(ωt-kx) …(i)
Where k = ω/v
Remember λ = 2 π v/ ω = 2 π/k or
π/k = λ/2
y = A sin k(vt-x) as kv = ω …(ii)
Interference of waves going in the same direction
Let the amplitudes of the two waves be A1 and A2 and the two waves differ in phase by an angle δ.
Their equations may be written as
y1 = A1 sin(kx-ωt)
y2 = A2 sin(kx-ωt+δ)
By trigonometric properties, the combination of waves (suuperposition of waves) is represented by
= A sin(kx-ωt+ε)
A² = A² cos² ε + A² sin² ε
=
A1² +A2² +2A1A2cosδ
where A1 +A1cosδ = A cosε
A2sinδ = A sinε
Resultant amplitude is maximum, when cosδ = +1 or δ = 2nπ. The maximum value is A1+A2.
Resultant amplitude is minimum, when cosδ = -1 or cosδ = (2n+1)π . The minimum value is A1-A2.
(These results are used in interference of light.)
Standing waves
Standing waves are produced when two sine waves of equal amplitude and frequency propagate on a long string in opposite directions.
The equations of two waves can be:
y1 = A sin(ωt-kx)
y2 = A sin(ωt+kx)
The equations of the resultant wave will be:
y = y1+y2
= A[sin(ωt-kx) + sin(ωt+kx)]
= 2Asin ωt cos kx
= (2A cos kx) sin ωt
Interpretation of the equation: The amplitude of the wave is |2A cos kx|, but the amplitude is not equal for all the particles.
There are some points where the amplitude |2A cos kx| = 0 all the time.
This will at points where
cos kx = 0
=> kx = (n + ½) π
=> x = (n + ½) λ/2
where n is an integer.
Although these points are not physically clamped, they remain fixed as the two waves pass them simultaneously. These points whose amplitude is zero always are called nodes.
At the points where |cos kx| = 1, the maximum amplitude will be present. These points at which the maximum amplitude is obtained are called antinodes.
When sin ωt = 0, the amplitude of all points is zero which means that all points are at their normal positions or mean positions.
When sin ωt = 1, all points for which cos kx is positive reach their positive maximum displacement. At the same time, all points for which cos kx is negative reach their negative maximum displacement.
From the equation is also clear that the separation between consecutive nodes or consecutive antinodes is λ/2.
As the particles at the nodes do not move at all, energy cannot be transmitted across them.
Concept review Ch. 16 Sound waves
Nature and propagation
Sound is produced in a material medium by a vibrating source.
Sound waves constitute alternate compression and rarefaction pulses traveling in the medium.
Sound is audible only if the frequency of alternation of pressure is between 20 Hz to 20,000 Hz.
Displacement wave and Pressure Wave
A longitudinal wave in a fluid can be described either in terms of the longitudinal displacement suffered by the particles of the medium or in terms of the excess pressure generated due to the compression or rarefaction.
Speed of a sound wave
v = √(B/ ρ)
where
v = velocity
B = Bulk modulus of the material.
ρ = normal density of the fluid
Hence the velocity of a longitudinal wave in a medium depends on its elastic properties and inertial properties of the medium.
Newton’s formula for speed of sound in a gas
v = √(P/ρ)
The density of air at temperature 0°C and pressure 76 cm of mercury column is ρ = 1.293 kg/m³
So P = .76m*(13.6*10^3 kg/ m³)*(9.8 m/s²) = 101292.8
Hence P/ ρ = 78339.37
√(P/ρ) = 279.8917 m/s
The velocity of sound in air comes as 280 m/s.
But the measured value of speed of sound in air is 332 m/s
Laplace's correction
Laplace suggested a correction. With Laplace’s correction the formula is
v = √( γ P/ρ)
where γ = Cp/Cv (Cp and Cv are molar heat capacities at constant pressure and constant volume respectively)
With this new formula the value comes out to be 331.1723 closer to 332 m/s.
Effect of pressure, temperature and humidity on speed of a sound wave
The speed of sound is not affected by the change in pressure provided the temperature is kept constant. If pressure is changed but the temperature is kept constant, the density varies proportionately and P/ρ remains constant.
Speed of sound increases with increasing humidity. The density of water vapour is less than dry air at the same pressure. Thus, the density of moist air is less than that of dry air.
Intensity of sound waves
The intensity of a sound wave is defined as the average energy crossing a unit cross sectional area perpendicular to the direction of propagation of the wave in unit time.
The loudness of sound that we feel is mainly related to the intensity of sound. It also depends on the frequency to some extent.
Appearance of sound to human ear
The appearance of sound to human ear is characterised by three parameters.
1. pitch
2. loudness
3. quality
1. Pitch: Higher the frequency, higher will be the pitch.
2. Loudness: Loudness that we sense is related to the intensity of sound though it is not proportional to it.
3. A sound generated by a source contains a number of frequency components in it. Certain sounds have well defined frequencies which have considerable amplitude. Such sounds are particularly pleasant to the ear.
Interference of sound waves
Resultant change in pressure due to superposition of two sound waves.
p1 = p01 sin(kx- ωt)
p2 = p02 sin[k(x + ∆x)- ωt]
= p02 sin{(kx- ωt)+ δ]
Where δ = k∆x = 2 π ∆x/ λ
P = p0sin[(kx- ωt)+ ε]
where
p0² = p01² + p02² + 2 p01 p02 cos δ
tan ε = p02 sin δ /( p01 + p02cos δ)
the resultant amplitude is maximum when δ = 2n π and is minimum when δ = (2n+1) π.
Hence when δ = 2n π there is constructive interference
When δ = (2n+1) π there is destructive interference.
Beats
The phenomenon of periodic variation of intensity of sound when two sound waves of slightly different frequencies interfere, is called beats.
Bending of waves from an obstacle or an opening is called diffraction.
Diffraction effects are appreciable when the dimensions of openings or the obstacles are comparable or smaller than the wave length of the wave.
Doppler effect
If the source of sound or the observer or both, move with respect to the medium, the frequency observed may be different from the frequency of the source. This apparent change in frequency of the wave due to motion of the source or the observer is called Doppler effect.
Mach number
Mach Number = µs/v
µs = speed of source creating the sound wave
v = velocity of sound wave
Sound is produced in a material medium by a vibrating source.
Sound waves constitute alternate compression and rarefaction pulses traveling in the medium.
Sound is audible only if the frequency of alternation of pressure is between 20 Hz to 20,000 Hz.
Displacement wave and Pressure Wave
A longitudinal wave in a fluid can be described either in terms of the longitudinal displacement suffered by the particles of the medium or in terms of the excess pressure generated due to the compression or rarefaction.
Speed of a sound wave
v = √(B/ ρ)
where
v = velocity
B = Bulk modulus of the material.
ρ = normal density of the fluid
Hence the velocity of a longitudinal wave in a medium depends on its elastic properties and inertial properties of the medium.
Newton’s formula for speed of sound in a gas
v = √(P/ρ)
The density of air at temperature 0°C and pressure 76 cm of mercury column is ρ = 1.293 kg/m³
So P = .76m*(13.6*10^3 kg/ m³)*(9.8 m/s²) = 101292.8
Hence P/ ρ = 78339.37
√(P/ρ) = 279.8917 m/s
The velocity of sound in air comes as 280 m/s.
But the measured value of speed of sound in air is 332 m/s
Laplace's correction
Laplace suggested a correction. With Laplace’s correction the formula is
v = √( γ P/ρ)
where γ = Cp/Cv (Cp and Cv are molar heat capacities at constant pressure and constant volume respectively)
With this new formula the value comes out to be 331.1723 closer to 332 m/s.
Effect of pressure, temperature and humidity on speed of a sound wave
The speed of sound is not affected by the change in pressure provided the temperature is kept constant. If pressure is changed but the temperature is kept constant, the density varies proportionately and P/ρ remains constant.
Speed of sound increases with increasing humidity. The density of water vapour is less than dry air at the same pressure. Thus, the density of moist air is less than that of dry air.
Intensity of sound waves
The intensity of a sound wave is defined as the average energy crossing a unit cross sectional area perpendicular to the direction of propagation of the wave in unit time.
The loudness of sound that we feel is mainly related to the intensity of sound. It also depends on the frequency to some extent.
Appearance of sound to human ear
The appearance of sound to human ear is characterised by three parameters.
1. pitch
2. loudness
3. quality
1. Pitch: Higher the frequency, higher will be the pitch.
2. Loudness: Loudness that we sense is related to the intensity of sound though it is not proportional to it.
3. A sound generated by a source contains a number of frequency components in it. Certain sounds have well defined frequencies which have considerable amplitude. Such sounds are particularly pleasant to the ear.
Interference of sound waves
Resultant change in pressure due to superposition of two sound waves.
p1 = p01 sin(kx- ωt)
p2 = p02 sin[k(x + ∆x)- ωt]
= p02 sin{(kx- ωt)+ δ]
Where δ = k∆x = 2 π ∆x/ λ
P = p0sin[(kx- ωt)+ ε]
where
p0² = p01² + p02² + 2 p01 p02 cos δ
tan ε = p02 sin δ /( p01 + p02cos δ)
the resultant amplitude is maximum when δ = 2n π and is minimum when δ = (2n+1) π.
Hence when δ = 2n π there is constructive interference
When δ = (2n+1) π there is destructive interference.
Beats
The phenomenon of periodic variation of intensity of sound when two sound waves of slightly different frequencies interfere, is called beats.
Bending of waves from an obstacle or an opening is called diffraction.
Diffraction effects are appreciable when the dimensions of openings or the obstacles are comparable or smaller than the wave length of the wave.
Doppler effect
If the source of sound or the observer or both, move with respect to the medium, the frequency observed may be different from the frequency of the source. This apparent change in frequency of the wave due to motion of the source or the observer is called Doppler effect.
Mach number
Mach Number = µs/v
µs = speed of source creating the sound wave
v = velocity of sound wave
Concept review Ch. 17 Light waves
Nature of light waves
Huygen's principle
Young's double hole experiment
Young's double slit experiment
Alternate bright and dark rings observed.
Fringe width = w = Dλ/d
D = distance betwee slits and the screen
λ = wave length of light wave
d = distance between slits.
visit for past JEE objective questions on Young's double slit experiment.
(This is an important topic on which many questions were asked.)
http://iit-jee-physics-ps.blogspot.com/2008/05/past-jee-oq-wave-motion-light-waves-1.html
Optical path
Interference
Fresnel's biprism
Coherent light source
There will be a constant phase difference between the two light waves in young's double experiment
Incoherent light source
the phase differene will be random - No interference will be there
Diffraction of light
Fraunhoffer diffraction by a single slit
Fresnel diffraction
Limit of resolution
Scattering of light
Polarization of light
Polaroids
Huygen's principle
Young's double hole experiment
Young's double slit experiment
Alternate bright and dark rings observed.
Fringe width = w = Dλ/d
D = distance betwee slits and the screen
λ = wave length of light wave
d = distance between slits.
visit for past JEE objective questions on Young's double slit experiment.
(This is an important topic on which many questions were asked.)
http://iit-jee-physics-ps.blogspot.com/2008/05/past-jee-oq-wave-motion-light-waves-1.html
Optical path
Interference
Fresnel's biprism
Coherent light source
There will be a constant phase difference between the two light waves in young's double experiment
Incoherent light source
the phase differene will be random - No interference will be there
Diffraction of light
Fraunhoffer diffraction by a single slit
Fresnel diffraction
Limit of resolution
Scattering of light
Polarization of light
Polaroids
Concept review Ch. 18 Geometrical Optics
Reflection at smooth surfaces
Spherical mirrors
Refraction at plane surfaces
Critical angle
Optical fibre
Prism
Angle of minimum deviation
Refraction at spherical surfaces
Refractoin through thin lenses
Len maker's formula
Lens formula
Power of a lens
Defects of images
a. spherical aberration
b. Coma
c. Astigmatism
d. Curvature
e. Distortion
Chromatic aberrations
Spherical mirrors
Refraction at plane surfaces
Critical angle
Optical fibre
Prism
Angle of minimum deviation
Refraction at spherical surfaces
Refractoin through thin lenses
Len maker's formula
Lens formula
Power of a lens
Defects of images
a. spherical aberration
b. Coma
c. Astigmatism
d. Curvature
e. Distortion
Chromatic aberrations
Concept review Ch. 19 Optical Instrument
Eye
Simple microscope
Compound microscope
Telescope
Defects of vision
Nearsightedness
Farsightedness
Astigmatism
Simple microscope
Compound microscope
Telescope
Defects of vision
Nearsightedness
Farsightedness
Astigmatism
Concept review Ch. 20 Dispersion and Spectra
Dispersion
Dispersive power
Spectrum
Emission spectra
Absorption spectra
Ultraviolet spectrum
Infrared specturm
Spectrometer
Rainbow
Dispersive power
Spectrum
Emission spectra
Absorption spectra
Ultraviolet spectrum
Infrared specturm
Spectrometer
Rainbow
Concept review Ch. 22 Photometry
Total radiant flux
Luminsoity of radiant flux
Luminous flux
Relative luminosity
Luminous efficiency
Luminous intensity or Illuminating power
Illuminance
Inverse square law
Lambert's cosine law
Photometer
Luminsoity of radiant flux
Luminous flux
Relative luminosity
Luminous efficiency
Luminous intensity or Illuminating power
Illuminance
Inverse square law
Lambert's cosine law
Photometer
Concept review Ch. 23 Heat and Temperature
Hot and cold bodies
A hot body has more internal energy than an otherwise identical cold body.
When a hot body and cold body are kept in contact, energy is transferred from hot body to cold body and the cold body warms up and hot body cools down.
The transfer to energy from a hot body to a cold body is nonmechanical process. This energy that is transferred from one body to the other, without any mechanical work involved, is called heat.
Zeroth law of thermodynamics
If two bodies A and B are in thermal equilibrium and A and C are also in thermal equilibrium then B and C are in also in thermal equilibrium
All bodies in thermal equilibrium are assigned equal temperature.
Heat flows from the body at higher temperature to the body at lower temperature
* To measure temperature, we can choose a substance and look for a measurable property of the substance which monotonically changes with temperature.
* Length of mercury in long capillary, resistance of a wire, pressure of gas when volume is kept constant are properties which can be used for temperature measurement.
Mercury Thermometer
A change in one degree of temperature results in a change of (l2–l1)/(t2-t1) in the length of mercury column. Hence the length of the capillary can be graduated in degrees.
Centigrade system assumes ice point at 0°C and the steam point at 100°C.
Another popular system known as Fahrenheit system assumes 32°F for the ice point and 212°F for the steam point
Hence conversion formula is F = 32 + 9C/5
Platinum resistance thermometer
Electric resistance of a metal wire increases monotonically with temperature and may be used to define a temperature scale. A platinum wire is often used to construct a thermometer based on this scale. Thermometer using platinum wire is called platinum resistance thermometer.
This thermometer also uses the principle of Wheatstone bridge.
Constant volume gas thermometer
The temperature of triple point of water is assigned a value of 273.16 K
The temperature of ice point on the ideal gas scale is 273.15 K and of the steam point is T = 373.15 K. The interval between the two is 100 K.
Centigrade scale or Celsius scale is defined to have ice point at 0°C and steam point at 100°C. The interval is 100°C.
Hence if θ represents the Celsius or centigrade temperature
V = T – 273.15 K
Ideal gas equation
pV = nRT
n = the amount of gas in moles
R = universal constant = 8.314 J/mol-K
Callendar’s Compensated Constant Pressure Thermometer is also there.
An adiabatic wall does not allow heat flow even between two bodies at different temperatures.
Diathermic wall allows heat transfer between two bodies at different temperatures through it rapidly.
Thermal expansion
Average coefficient of linear expansion
Average (α) = (1/L)*∆L/∆T
Coefficient of linear expansion at temperature T
α = Lim (∆T->0)(1/L)*∆L/∆T = (1/L)dL/dT
Suppose the length of a rod is L0 at 0° C and Lθ at temperature θ measured in Celsius. If α is small and constant over the given temperature interval,
α = Lθ-L0/L0*θ or
Lθ = L0(1+αθ)
The coefficient of volume expansion is defined in a similar way.
γ = (1/V)dV/dT
It is also known as coefficient of cubical expansion.
Vθ = V0(1+γθ)
γ = 3α
A hot body has more internal energy than an otherwise identical cold body.
When a hot body and cold body are kept in contact, energy is transferred from hot body to cold body and the cold body warms up and hot body cools down.
The transfer to energy from a hot body to a cold body is nonmechanical process. This energy that is transferred from one body to the other, without any mechanical work involved, is called heat.
Zeroth law of thermodynamics
If two bodies A and B are in thermal equilibrium and A and C are also in thermal equilibrium then B and C are in also in thermal equilibrium
All bodies in thermal equilibrium are assigned equal temperature.
Heat flows from the body at higher temperature to the body at lower temperature
* To measure temperature, we can choose a substance and look for a measurable property of the substance which monotonically changes with temperature.
* Length of mercury in long capillary, resistance of a wire, pressure of gas when volume is kept constant are properties which can be used for temperature measurement.
Mercury Thermometer
A change in one degree of temperature results in a change of (l2–l1)/(t2-t1) in the length of mercury column. Hence the length of the capillary can be graduated in degrees.
Centigrade system assumes ice point at 0°C and the steam point at 100°C.
Another popular system known as Fahrenheit system assumes 32°F for the ice point and 212°F for the steam point
Hence conversion formula is F = 32 + 9C/5
Platinum resistance thermometer
Electric resistance of a metal wire increases monotonically with temperature and may be used to define a temperature scale. A platinum wire is often used to construct a thermometer based on this scale. Thermometer using platinum wire is called platinum resistance thermometer.
This thermometer also uses the principle of Wheatstone bridge.
Constant volume gas thermometer
The temperature of triple point of water is assigned a value of 273.16 K
The temperature of ice point on the ideal gas scale is 273.15 K and of the steam point is T = 373.15 K. The interval between the two is 100 K.
Centigrade scale or Celsius scale is defined to have ice point at 0°C and steam point at 100°C. The interval is 100°C.
Hence if θ represents the Celsius or centigrade temperature
V = T – 273.15 K
Ideal gas equation
pV = nRT
n = the amount of gas in moles
R = universal constant = 8.314 J/mol-K
Callendar’s Compensated Constant Pressure Thermometer is also there.
An adiabatic wall does not allow heat flow even between two bodies at different temperatures.
Diathermic wall allows heat transfer between two bodies at different temperatures through it rapidly.
Thermal expansion
Average coefficient of linear expansion
Average (α) = (1/L)*∆L/∆T
Coefficient of linear expansion at temperature T
α = Lim (∆T->0)(1/L)*∆L/∆T = (1/L)dL/dT
Suppose the length of a rod is L0 at 0° C and Lθ at temperature θ measured in Celsius. If α is small and constant over the given temperature interval,
α = Lθ-L0/L0*θ or
Lθ = L0(1+αθ)
The coefficient of volume expansion is defined in a similar way.
γ = (1/V)dV/dT
It is also known as coefficient of cubical expansion.
Vθ = V0(1+γθ)
γ = 3α
Concept review Ch. 24 Kinetic Theory of Gases
Any sample of gas is made of molecules.
The observed behaviour of gas results from the behaviour of its large number of molecules.
Kinetic theory of gases attempts to develop a model of the molecular behaviour which should result in the observed behaviour an ideal gas.
Assumptions of kinetic theory of gases
1. All gases are made of molecules moving randomly in all directions
2. The size of molecule is much smaller than the average separation between the molecules.
3. The molecules exert no force on each other or on the walls of the container except during collision (no atraction force or repulsion force).
4. All collisions between two molecules or between a molecule and a wall are perfectly elastic. Also the time spent during a collision is negligibly small.
5. the molecules obey Newton's laws of motion.
6. When a gas is left for sufficient time in a closed container, it comes to a steady state. The density and the distribution of molecules with different velocities are independent of position, direction and time.
The assumptions are close to the real situations at low densities.
The molecular size is roughly 100 times smaller than the average separation between the molecules at 0.1 atm and room temperature.
The real molecules do exert electric forces on each other but these forces can be neglected as the average separation between molecules is large as compared to their size.
Pressure of an ideal gas
p = (1/3)ρ*Avg(v²) ........ (1)
where
ρ = density of gas = mass per unit area
Avg(v²) = average of the speeds of molecules squared
pV = (1/3)M*Avg(v²) ....... (2)
M = Mass of gas in the closed container
pV = (1/3)nm*Avg(v²) ........ (3)
n = number of molecules of gas in the container
m = mass of each molecule
RMS Speed: The square root of mean square speed is called root-mean-square speed or rms speed.
It is denoted by the symbol vrms
Avg(v²) = (vrms)²
The equation (1) can be written as
p = (1/3)ρ*(vrms)²
Then
vrms) = √[3p/ρ] = √[3pV/M]
Total translational kinetic energy of all the molecules of the gas is
K = Σ (1/2mv² = (1/2)M(vrms)² ... (4)
The average kinetic energy of a molecule = (1/2)m(vrms)²
Then from equation (2)
K = (3/2)pV
according to the kinetic theory of gases, the internal energy of an ideal gas is the same as the total translational kinetic energy of its molecules.
For different kinds of gases, it is not the rms speed but average kinetic energy of individual molecules that has a fixed value at a given temperature.
The heavier molecules move with smaller rms speed and the lighter molecules move with larger rms speed.
All gas laws can be deduced from kinetic theory of gases.
Ideal gas equation
pV = nRT
R = universal gas constant = 8.314 J/mol-L
The average speed of molecules is somewhat less than the rms speed.
Average speed = (Σv)/n = √[8kT/πm]= √[8RT/πM0]
Due to the random motion of molecules of gas, the centre of mass of the gas does not change.
Maxwell’s speed distribution law
The rms speed of an oxygen molecule in a sample at 300 K is about 480 m/s. As it is a square root of average of squared values of the individual velocities of various molecules, many molecules will have speed more than 480 m/s and many will have speed less than 480 m/s. Maxwell derived an equation giving the speed distribution of molecules.
If dN represents the number of molecules with speeds between v and v + dv then
dN = 4 πN[m/2πkT]3/2v²e-(mv²/2kT)dv
The speed vp at which dN/Dv is maximum is called the most probable speed.
vp = SQRT(2kT/m)
Thermodynamic State
A thermodynamic state of a given sample of an ideal gas is completely described if its pressure and its volume are given.
Equation of state
An equation describing the relation between pressure, volume and temperature of a given sample of a substance is called the equation of state for that substance
For an ideal gas it is pV = nRT
For a real gas the equation is [p + a/V²][V-b] = nRT
Where
a and b are small positive constants.
a is related to the average force of attraction between the molecules.
b is related to the total volume of the molecules.
This equation is given by van der Waals.
Brownian motion
Brownian motion was random motion of molecules in liquid observed by Robert Brown. It is similar to random motion of molecules in gases. To observe the Brownian motion in liquid we need light suspended particles. The motion increases at higher temperatures. Liquids with smaller viscosity and smaller density will show more intense Brownian motion.
Vapour
In general a gas can be liquefied either by increasing the pressure or by decreasing the temperature. However, it the temperature is sufficiently high, no amount of pressure can liquefy the gas. The temperature above which this behavior occurs is called the critical temperature of the substance. A gas below its critical temperature is called vapour.
Critical temperature of a substance: Above this temperature, the substance cannot be converted into liquid by increasing the pressure i.e., by compressing.
Vapour: Vapour is gas below the critical temperature of the substance.
Water cannot be liquefied at a temperature greater than374.1°C by increasing the pressure. 374.1°C is the critical temperature of water. Hence below 374.1° water in gas form is called water vapour and above 374.1° it is called water gas.
Vapours obey Dalton’s law of partial pressure.
Evaporation
Evaporation is a process in which molecules escape slowly from the surface of a liquid. Only those molecules whose kinetic energy is more than the average escape from the surface. Because of this, as more and more molecules evaporate, kinetic energy of the remaining liquid decreases and temperature goes down. This effect is observed in water in pots.
Saturated vapour pressure
When we place an open flask with a liquid in a closed jar, after sufficient time, volume of the liquid becomes constant. We know that liquid evaporates, but at this point in time, when volume of liquid is constant, we have to interpret that rate of transformation from liquid to vapour equals the rate of transformation from vapour to liquid.
If some vapour from outside is injected into the space above the liquid in the jar, we observe that the volume of liquid will increase. That means more vapour is getting transformed into liquid.
When a space actually contains the maximum possible amount of vapour, the vapour is said to be saturated or it is called saturated vapour. If the amount of vapour in a space is less than the maximum possible, the vapour is called unsaturated vapour.
Saturated vapour increases with temperature. At higher temperature, the space contains more vapour. This is because more liquid molecules escape from liquid surface at higher temperatures.
The pressure exerted by a saturated vapour is called saturated vapour pressure. As a higher temperature more amount of vapour is there of a liquid, saturated vapour pressure is also higher at higher temperatures for a substance. Even into an empty jar, as vapour increases more and more, and pressure increases above the SVP, vapour starts condensing and liquid forms. This is as per definition of vapour. A vapour can be transformed into a liquid at a constant temperature by increasing pressure.
In the atmosphere around us air which is mixture of nitrogen and oxygen and water vapour are mixed with each other. If a given volume air contains maximum amount of vapour possible, the air called saturated with water vapour.
Boiling
As we heat a liquid, the kinetic energy of the entire liquid increases, energy of many molecules in various places becomes sufficient to break the molecular attraction. Hence vapour bubbles form, float to the surface and escape from the liquid surface. This phenomenon is called boiling and the temperature at which it occurs is called boiling point.
Boiling point of a liquid depends on the external pressure over its surface. Boiling occurs at a temperature where the SVP equals the external pressure. Hence in a pressure cooker boiling of water occurs at a higher temperature. At a higher temperature only SVP of water vapour equals the higher pressure in the cooker.
Dew point
The temperature at which the saturation vapour pressure is equal to the present vapour pressure is called the dew point.
Example: Air at temperature 15°C has a pressure of 8.94 mm Hg. This air unsaturated at SVP of air at 15°C is 12.67 Hg mm. At 10°C, SVP of air is 8.94 mm. Therefore, dew point of this air is 10°C.
If the temperature is of an unsaturated air is decreased below dew point, some vapour will condense.
Humidity
The amount of water vapour present in a unit volume of air is called the absolute humility. Its general units are g/m³
The ratio of amount of water vapour present in a given volume to the amount of water vapour required to saturate the volume at the same temperature is called the relative humidity.
Relative humidity = RH =
Amount of water vapour present in a given volume of air
at a given temperature
-------------------------------------------------------------------------------
Amount of water vapour required to saturate the same volume of air at the same temperature
RH may also be defined as
Vapour pressure of air/SVP at the same temperature
SVP = Saturation vapour pressure: The pressure exerted by a saturated vapour is called saturation vapour pressure.
The RH may also be defined as
SVP at the dew point/SVP at the air temperature
As the vapour pressure of air at the actual temperature is equal to the SVP at the dew point.
Determination of relative humidity
Regnault’s hygrometer has two test tubes whose outer surfaces are silvered with thermometers. In one of the test tubes some ether is taken and there is an arrangement for sucking outside air through the ether. As the outside air is sucked through the ether, it evaporates and temperature in the test tube falls. As the temperature is decreasing at a particular temperature, vapour in the air outside the test tube starts condensing on the test tube surface and the silver surface becomes dull or hazy. This temperature, which is dew point, is noted along with the temperature of the other test tube. Then sucking of air through the test tube having ether is stopped and temperature of the tube slowly starts increasing. As the temperature crosses the dew point, the silver surface starts shining again. This temperature is noted again. The average of these two temperatures gives the dew point (average is taken to minimize experimental error). So the dew point (temperature) and air temperature outside are known. If f and F are the saturation vapour pressures of air at the dew point and the air temperature respectively, the relative humidity is f/F*100%.
Phase diagrams
It is a diagram that shows the relation between saturated vapour pressure and temperature in the case of gas, liquid and solid. The three phases solid to liquid, liquid to gas as well as solid to gas (if feasible in the case of a substance) are shown in this diagram. There is a point in this diagram where all three phases can exist simultaneously in equilibrium. It is called triple point.
For carbon dioxide, the triple point is 5.11 atm pressure and 216.55 K temperature. At atmospheric pressure, the diagram shows that carbon can be in solid state at low temperatures or in vapour state at high temperatures. Solid carbon dioxide when heated at atmospheric pressure becomes gas directly. Hence it is called dry ice.
Dew and fog
In winters, as temperature falls, surfaces of window-panes, flowers, grass etc. become still colder due to radiation. The air near to them becomes saturated and water vapour condenses and droplets are formed on them. It is known as dew. At still lower temperature entire air in the atmosphere gets saturated, water vapour condenses around or on the dust particles in the air. These dust particles floating in air with condensed water vapour on them form a thick mist, which restricts visibility. This thick mist is called fog.
The observed behaviour of gas results from the behaviour of its large number of molecules.
Kinetic theory of gases attempts to develop a model of the molecular behaviour which should result in the observed behaviour an ideal gas.
Assumptions of kinetic theory of gases
1. All gases are made of molecules moving randomly in all directions
2. The size of molecule is much smaller than the average separation between the molecules.
3. The molecules exert no force on each other or on the walls of the container except during collision (no atraction force or repulsion force).
4. All collisions between two molecules or between a molecule and a wall are perfectly elastic. Also the time spent during a collision is negligibly small.
5. the molecules obey Newton's laws of motion.
6. When a gas is left for sufficient time in a closed container, it comes to a steady state. The density and the distribution of molecules with different velocities are independent of position, direction and time.
The assumptions are close to the real situations at low densities.
The molecular size is roughly 100 times smaller than the average separation between the molecules at 0.1 atm and room temperature.
The real molecules do exert electric forces on each other but these forces can be neglected as the average separation between molecules is large as compared to their size.
Pressure of an ideal gas
p = (1/3)ρ*Avg(v²) ........ (1)
where
ρ = density of gas = mass per unit area
Avg(v²) = average of the speeds of molecules squared
pV = (1/3)M*Avg(v²) ....... (2)
M = Mass of gas in the closed container
pV = (1/3)nm*Avg(v²) ........ (3)
n = number of molecules of gas in the container
m = mass of each molecule
RMS Speed: The square root of mean square speed is called root-mean-square speed or rms speed.
It is denoted by the symbol vrms
Avg(v²) = (vrms)²
The equation (1) can be written as
p = (1/3)ρ*(vrms)²
Then
vrms) = √[3p/ρ] = √[3pV/M]
Total translational kinetic energy of all the molecules of the gas is
K = Σ (1/2mv² = (1/2)M(vrms)² ... (4)
The average kinetic energy of a molecule = (1/2)m(vrms)²
Then from equation (2)
K = (3/2)pV
according to the kinetic theory of gases, the internal energy of an ideal gas is the same as the total translational kinetic energy of its molecules.
For different kinds of gases, it is not the rms speed but average kinetic energy of individual molecules that has a fixed value at a given temperature.
The heavier molecules move with smaller rms speed and the lighter molecules move with larger rms speed.
All gas laws can be deduced from kinetic theory of gases.
Ideal gas equation
pV = nRT
R = universal gas constant = 8.314 J/mol-L
The average speed of molecules is somewhat less than the rms speed.
Average speed = (Σv)/n = √[8kT/πm]= √[8RT/πM0]
Due to the random motion of molecules of gas, the centre of mass of the gas does not change.
Maxwell’s speed distribution law
The rms speed of an oxygen molecule in a sample at 300 K is about 480 m/s. As it is a square root of average of squared values of the individual velocities of various molecules, many molecules will have speed more than 480 m/s and many will have speed less than 480 m/s. Maxwell derived an equation giving the speed distribution of molecules.
If dN represents the number of molecules with speeds between v and v + dv then
dN = 4 πN[m/2πkT]3/2v²e-(mv²/2kT)dv
The speed vp at which dN/Dv is maximum is called the most probable speed.
vp = SQRT(2kT/m)
Thermodynamic State
A thermodynamic state of a given sample of an ideal gas is completely described if its pressure and its volume are given.
Equation of state
An equation describing the relation between pressure, volume and temperature of a given sample of a substance is called the equation of state for that substance
For an ideal gas it is pV = nRT
For a real gas the equation is [p + a/V²][V-b] = nRT
Where
a and b are small positive constants.
a is related to the average force of attraction between the molecules.
b is related to the total volume of the molecules.
This equation is given by van der Waals.
Brownian motion
Brownian motion was random motion of molecules in liquid observed by Robert Brown. It is similar to random motion of molecules in gases. To observe the Brownian motion in liquid we need light suspended particles. The motion increases at higher temperatures. Liquids with smaller viscosity and smaller density will show more intense Brownian motion.
Vapour
In general a gas can be liquefied either by increasing the pressure or by decreasing the temperature. However, it the temperature is sufficiently high, no amount of pressure can liquefy the gas. The temperature above which this behavior occurs is called the critical temperature of the substance. A gas below its critical temperature is called vapour.
Critical temperature of a substance: Above this temperature, the substance cannot be converted into liquid by increasing the pressure i.e., by compressing.
Vapour: Vapour is gas below the critical temperature of the substance.
Water cannot be liquefied at a temperature greater than374.1°C by increasing the pressure. 374.1°C is the critical temperature of water. Hence below 374.1° water in gas form is called water vapour and above 374.1° it is called water gas.
Vapours obey Dalton’s law of partial pressure.
Evaporation
Evaporation is a process in which molecules escape slowly from the surface of a liquid. Only those molecules whose kinetic energy is more than the average escape from the surface. Because of this, as more and more molecules evaporate, kinetic energy of the remaining liquid decreases and temperature goes down. This effect is observed in water in pots.
Saturated vapour pressure
When we place an open flask with a liquid in a closed jar, after sufficient time, volume of the liquid becomes constant. We know that liquid evaporates, but at this point in time, when volume of liquid is constant, we have to interpret that rate of transformation from liquid to vapour equals the rate of transformation from vapour to liquid.
If some vapour from outside is injected into the space above the liquid in the jar, we observe that the volume of liquid will increase. That means more vapour is getting transformed into liquid.
When a space actually contains the maximum possible amount of vapour, the vapour is said to be saturated or it is called saturated vapour. If the amount of vapour in a space is less than the maximum possible, the vapour is called unsaturated vapour.
Saturated vapour increases with temperature. At higher temperature, the space contains more vapour. This is because more liquid molecules escape from liquid surface at higher temperatures.
The pressure exerted by a saturated vapour is called saturated vapour pressure. As a higher temperature more amount of vapour is there of a liquid, saturated vapour pressure is also higher at higher temperatures for a substance. Even into an empty jar, as vapour increases more and more, and pressure increases above the SVP, vapour starts condensing and liquid forms. This is as per definition of vapour. A vapour can be transformed into a liquid at a constant temperature by increasing pressure.
In the atmosphere around us air which is mixture of nitrogen and oxygen and water vapour are mixed with each other. If a given volume air contains maximum amount of vapour possible, the air called saturated with water vapour.
Boiling
As we heat a liquid, the kinetic energy of the entire liquid increases, energy of many molecules in various places becomes sufficient to break the molecular attraction. Hence vapour bubbles form, float to the surface and escape from the liquid surface. This phenomenon is called boiling and the temperature at which it occurs is called boiling point.
Boiling point of a liquid depends on the external pressure over its surface. Boiling occurs at a temperature where the SVP equals the external pressure. Hence in a pressure cooker boiling of water occurs at a higher temperature. At a higher temperature only SVP of water vapour equals the higher pressure in the cooker.
Dew point
The temperature at which the saturation vapour pressure is equal to the present vapour pressure is called the dew point.
Example: Air at temperature 15°C has a pressure of 8.94 mm Hg. This air unsaturated at SVP of air at 15°C is 12.67 Hg mm. At 10°C, SVP of air is 8.94 mm. Therefore, dew point of this air is 10°C.
If the temperature is of an unsaturated air is decreased below dew point, some vapour will condense.
Humidity
The amount of water vapour present in a unit volume of air is called the absolute humility. Its general units are g/m³
The ratio of amount of water vapour present in a given volume to the amount of water vapour required to saturate the volume at the same temperature is called the relative humidity.
Relative humidity = RH =
Amount of water vapour present in a given volume of air
at a given temperature
-------------------------------------------------------------------------------
Amount of water vapour required to saturate the same volume of air at the same temperature
RH may also be defined as
Vapour pressure of air/SVP at the same temperature
SVP = Saturation vapour pressure: The pressure exerted by a saturated vapour is called saturation vapour pressure.
The RH may also be defined as
SVP at the dew point/SVP at the air temperature
As the vapour pressure of air at the actual temperature is equal to the SVP at the dew point.
Determination of relative humidity
Regnault’s hygrometer has two test tubes whose outer surfaces are silvered with thermometers. In one of the test tubes some ether is taken and there is an arrangement for sucking outside air through the ether. As the outside air is sucked through the ether, it evaporates and temperature in the test tube falls. As the temperature is decreasing at a particular temperature, vapour in the air outside the test tube starts condensing on the test tube surface and the silver surface becomes dull or hazy. This temperature, which is dew point, is noted along with the temperature of the other test tube. Then sucking of air through the test tube having ether is stopped and temperature of the tube slowly starts increasing. As the temperature crosses the dew point, the silver surface starts shining again. This temperature is noted again. The average of these two temperatures gives the dew point (average is taken to minimize experimental error). So the dew point (temperature) and air temperature outside are known. If f and F are the saturation vapour pressures of air at the dew point and the air temperature respectively, the relative humidity is f/F*100%.
Phase diagrams
It is a diagram that shows the relation between saturated vapour pressure and temperature in the case of gas, liquid and solid. The three phases solid to liquid, liquid to gas as well as solid to gas (if feasible in the case of a substance) are shown in this diagram. There is a point in this diagram where all three phases can exist simultaneously in equilibrium. It is called triple point.
For carbon dioxide, the triple point is 5.11 atm pressure and 216.55 K temperature. At atmospheric pressure, the diagram shows that carbon can be in solid state at low temperatures or in vapour state at high temperatures. Solid carbon dioxide when heated at atmospheric pressure becomes gas directly. Hence it is called dry ice.
Dew and fog
In winters, as temperature falls, surfaces of window-panes, flowers, grass etc. become still colder due to radiation. The air near to them becomes saturated and water vapour condenses and droplets are formed on them. It is known as dew. At still lower temperature entire air in the atmosphere gets saturated, water vapour condenses around or on the dust particles in the air. These dust particles floating in air with condensed water vapour on them form a thick mist, which restricts visibility. This thick mist is called fog.
Concept review Ch. 25 Calorimetry
Heat
Heat is a form of energy.
It is energy in transit whenever temperature differences exist.
Once it is transferred it becomes the internal energy of the receiving body.
Calorie: The amount of heat needed to increase the temperature of 1 g of water from 14.5°C to 15.5°C at a pressure of 1 atm is called 1 calorie.
1 cal = 4.186 joule
Principle of calorimetry
The total heat given by the hot objects equals the total heat received by the cold objects.
Specific heat capacity
Q = ms∆θ
m = mass of the body
s = Specific heat capacity
∆θ = temperature change in celsius
Q = nC∆θ
n = number of moles
C = molar heat capacity
∆θ = temperature change in celsius
Determination of specific heat capacity in laboratory
Regnault’s apparatus is used to determine the specific heat capacity of solids heavier than water and insoluble in it.
Calculation of specific heat from observations
m1 = mass of the solid
m2 = mass of the calorimeter and the stirrer
m3 = mass of the water
s1 = specific heat capacity of the solid
s2 = specific heat capacity of material of the calorimeter and the stirrer
s3 = specific heat capacity of water
θ1 = initial temperature of the solid
θ2 = initial temperature of the calorimeter, stirrer and water
θ = final temperature of the mixture
We get s1 = (m2s2 + m3s3)( θ – θ2)/[m1(θ1 – θ)]
Specific heat capacity of a liquid can also be measured with the Regnault apparatus. Here a solid of known specific heat capacity is taken and the experimental liquid is taken in the calorimeter in place of water.
Specific latent heat of fusion and vaporization
Heat is to be supplied to a body during a phase change [from solid to liquid (melting) and liquid to gas (vaporization)]. During the processes of phase change, melting or vaporization, temperature remains constant. The amount of heat supplied is written as
Q = mL
m = mass of substance
L = specific latent heat of fusion
Specific latent heat of fusion is also called as latent heat of fusion. During vaporization also similar relation holds. But L during vaporization (latent heat of vaporization or specific latent heat of vaporization in this case) is higher than L during melting.
When vapour condenses or liquid solidifies, heat is released to the surrounding by the substance.
As heat is supplied as latent heat, internal energy of a body is larger in liquid phase than in solid phase. Similarly, internal energy of a body is larger in gaseous phase than in liquid phase.
Measurement of specific latent heat of fusion of ice.
A calorie meter if filled with water. A piece of ice is taken and as it is melting, it is dried with a blotting paper and put into the calorimeter. The initial temperature of the water is observed and after all the ice has melted the temperature of the water is taken once again. The weight of the empty calorimeter and weight including water are taken as usual.
Observations:
m1 = mass of the calorimeter and the stirrer
m2 = mass of the water
m3 = mass of the ice
s1 = specific heat capacity of material of the calorimeter and the stirrer
s2 = specific heat capacity of water
θ1 = initial temperature of the water in calorimeter
θ2 = final temperature of the water
L = (m1s1 + m2s2)( θ1 - θ2)/m3 - s2θ2
Measurement of specific latent heat of vaporization of water
The experimental set up helps us to send steam with a measured temperature into a calorimeter with known mass of water. The mass of the steam condensed is equal to the increase in the mass of water in the calorimeter after the steam is sent into it.
Observations:
m1 = mass of the calorimeter and the stirrer
m2 = mass of the water
m3 = mass of the steam condensed
s1 = specific heat capacity of material of the calorimeter and the stirrer
s2 = specific heat capacity of water
θ1 = temperature of the steam
θ2 = initial temperature of the water in calorimeter
θ3 = final temperature of the water in the calorimeter after the condensation of steam
L = (m1s1 + m2s2)( θ3 - θ2)/m3 - s2( θ1 - θ2)
Mechanical equivalent of heat
W = JH
When
W is measured in joule
H is measured in calorie
J is expressed in joule/ calorie = 4.186 joule/calorie
The value of J gives how many joules of mechanical work is needed to raise the temperature of water by 1°C.
Measurement of the mechanical equivalent of heat
Searle’s Cone method is used for it.
There is an outer vessel which is rotated using a spindle and an inner vessel which does not move. As the outer vessel is rotated due to friction heat is generated and the water in the inner vessel absorbs it. The temperature of the water in the inner vessel is observed first and then the outer vessel is rotated till water temperature in the inner vessel increases by 5°C. the number of revolution made by the outer vessel is counted.
Observations
m1 = mass of the water in inner vessel
m2 = mass of the two vessels taken together
M = mass of the pan and weights in it
s1 = specific heat capacity of water
s2 = specific heat capacity of material of the vessels
θ1 = initial temperature of the water in calorimeter
θ2 = final temperature of the water
n = number revolutions of the outer vessel
r = radius of the disc
J = 2 πMgr/(m1s1 + m2s2)( θ2 – θ1)
Heat is a form of energy.
It is energy in transit whenever temperature differences exist.
Once it is transferred it becomes the internal energy of the receiving body.
Calorie: The amount of heat needed to increase the temperature of 1 g of water from 14.5°C to 15.5°C at a pressure of 1 atm is called 1 calorie.
1 cal = 4.186 joule
Principle of calorimetry
The total heat given by the hot objects equals the total heat received by the cold objects.
Specific heat capacity
Q = ms∆θ
m = mass of the body
s = Specific heat capacity
∆θ = temperature change in celsius
Q = nC∆θ
n = number of moles
C = molar heat capacity
∆θ = temperature change in celsius
Determination of specific heat capacity in laboratory
Regnault’s apparatus is used to determine the specific heat capacity of solids heavier than water and insoluble in it.
Calculation of specific heat from observations
m1 = mass of the solid
m2 = mass of the calorimeter and the stirrer
m3 = mass of the water
s1 = specific heat capacity of the solid
s2 = specific heat capacity of material of the calorimeter and the stirrer
s3 = specific heat capacity of water
θ1 = initial temperature of the solid
θ2 = initial temperature of the calorimeter, stirrer and water
θ = final temperature of the mixture
We get s1 = (m2s2 + m3s3)( θ – θ2)/[m1(θ1 – θ)]
Specific heat capacity of a liquid can also be measured with the Regnault apparatus. Here a solid of known specific heat capacity is taken and the experimental liquid is taken in the calorimeter in place of water.
Specific latent heat of fusion and vaporization
Heat is to be supplied to a body during a phase change [from solid to liquid (melting) and liquid to gas (vaporization)]. During the processes of phase change, melting or vaporization, temperature remains constant. The amount of heat supplied is written as
Q = mL
m = mass of substance
L = specific latent heat of fusion
Specific latent heat of fusion is also called as latent heat of fusion. During vaporization also similar relation holds. But L during vaporization (latent heat of vaporization or specific latent heat of vaporization in this case) is higher than L during melting.
When vapour condenses or liquid solidifies, heat is released to the surrounding by the substance.
As heat is supplied as latent heat, internal energy of a body is larger in liquid phase than in solid phase. Similarly, internal energy of a body is larger in gaseous phase than in liquid phase.
Measurement of specific latent heat of fusion of ice.
A calorie meter if filled with water. A piece of ice is taken and as it is melting, it is dried with a blotting paper and put into the calorimeter. The initial temperature of the water is observed and after all the ice has melted the temperature of the water is taken once again. The weight of the empty calorimeter and weight including water are taken as usual.
Observations:
m1 = mass of the calorimeter and the stirrer
m2 = mass of the water
m3 = mass of the ice
s1 = specific heat capacity of material of the calorimeter and the stirrer
s2 = specific heat capacity of water
θ1 = initial temperature of the water in calorimeter
θ2 = final temperature of the water
L = (m1s1 + m2s2)( θ1 - θ2)/m3 - s2θ2
Measurement of specific latent heat of vaporization of water
The experimental set up helps us to send steam with a measured temperature into a calorimeter with known mass of water. The mass of the steam condensed is equal to the increase in the mass of water in the calorimeter after the steam is sent into it.
Observations:
m1 = mass of the calorimeter and the stirrer
m2 = mass of the water
m3 = mass of the steam condensed
s1 = specific heat capacity of material of the calorimeter and the stirrer
s2 = specific heat capacity of water
θ1 = temperature of the steam
θ2 = initial temperature of the water in calorimeter
θ3 = final temperature of the water in the calorimeter after the condensation of steam
L = (m1s1 + m2s2)( θ3 - θ2)/m3 - s2( θ1 - θ2)
Mechanical equivalent of heat
W = JH
When
W is measured in joule
H is measured in calorie
J is expressed in joule/ calorie = 4.186 joule/calorie
The value of J gives how many joules of mechanical work is needed to raise the temperature of water by 1°C.
Measurement of the mechanical equivalent of heat
Searle’s Cone method is used for it.
There is an outer vessel which is rotated using a spindle and an inner vessel which does not move. As the outer vessel is rotated due to friction heat is generated and the water in the inner vessel absorbs it. The temperature of the water in the inner vessel is observed first and then the outer vessel is rotated till water temperature in the inner vessel increases by 5°C. the number of revolution made by the outer vessel is counted.
Observations
m1 = mass of the water in inner vessel
m2 = mass of the two vessels taken together
M = mass of the pan and weights in it
s1 = specific heat capacity of water
s2 = specific heat capacity of material of the vessels
θ1 = initial temperature of the water in calorimeter
θ2 = final temperature of the water
n = number revolutions of the outer vessel
r = radius of the disc
J = 2 πMgr/(m1s1 + m2s2)( θ2 – θ1)
Concept review Ch. 26 Laws of Thermodynamics
First law of thermodynamics
∆U = ∆Q -∆W
or ∆Q = ∆U + ∆W
In an ideal monatomic gas, the internal energy of the gas is simply translational kinetic energy of all its molecules.
the first law may be taken as a statement that there exists an internal energy function U that has a fixed value in a given state.
Remember that when work is done by the system,∆W is positive, If work is done on the system ∆W is negative.
When heat is given to the system ∆Q is positive. If heat is given by the system ∆Q is negative.
A positive ∆W decreases internal energy and a positive ∆Q increases internal energy.
Work done in an isothermal process on an ideal gas
W = nRTln(V2/V1)
Work done an isobaric process
W = p(V2-V1)
Work done an in isochoric process(volume of gas is constant)
zero
Second law of thermodynamics
Kelvin-Planck statement
It is not possible to design a heat engine which works in cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work.
∆U = ∆Q -∆W
or ∆Q = ∆U + ∆W
In an ideal monatomic gas, the internal energy of the gas is simply translational kinetic energy of all its molecules.
the first law may be taken as a statement that there exists an internal energy function U that has a fixed value in a given state.
Remember that when work is done by the system,∆W is positive, If work is done on the system ∆W is negative.
When heat is given to the system ∆Q is positive. If heat is given by the system ∆Q is negative.
A positive ∆W decreases internal energy and a positive ∆Q increases internal energy.
Work done in an isothermal process on an ideal gas
W = nRTln(V2/V1)
Work done an isobaric process
W = p(V2-V1)
Work done an in isochoric process(volume of gas is constant)
zero
Second law of thermodynamics
Kelvin-Planck statement
It is not possible to design a heat engine which works in cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work.
Concept review Ch. 27 Specific Heat Capacities of Gases
Specific heat capacity of a substance
The specific heat capacity of a substance is defined as the heat supplied per unit mass of the substance per unit rise in the temperature.
This definition applies to any mode of the substance, solid, liquid or gas.
In the case of gases, to define the specific heat capacity, the process should also be specified.
There can be many processes, but two processes are more important and correspondingly two specific heat capacities are defined for gases. They are constant pressure process and constant volume process.
Molar heat capacity at constant pressure (Cp) and molar heat capacity at constant volume (Cv).
For a given amount of heat, the rise in temperature of a gas at constant pressure is smaller than the rise in temperature at constant volume. (At constant volume work done by the gas zero. At constant pressure work is done in expansion)
Determination of Cp
Hence Cp>Cv
Cp-Cv = R
and Cp/Cv is termed as γ
Determination of Cp of a gas
Regnault’s apparatus is used to measure Cp of a gas.
It has an arrangement to send gas through a pipe where in two manometers measure the pressure at two point and in between there is an adjusting screw to change the rate of flow. As the gas is flowing through this arrangement, adjusting screw is changed to maintain a constant difference of pressure between two manometers and this ensures that gas is at constant pressure when it is flowing through the calorimeter. This gas flows through an oil bath tank wherein it is given heat and then flows through the calorimeter wherein it gives out heat.
Observations
W = the water equivalent of the calorimeter with the coil
M = mass of water in the calorimeter
θ1 = temperature of the oil bath which gives heat to the gas.
θ2 = initial temperature of water in calorimeter
θ3 = Final temperature of water in calorimeter
n = amount of the gas (in moles) passed through the water
s = specific heat capacity of water
Cp of a gas. = (W + m) s (θ3 - θ2)/[n (θ1 – (θ2+ θ3)/2]
Determination of n = amount of the gas (in moles) passed through the water
It depends on the levels of mercury in the manometer attached to gas tank. If the difference in levels of the manometer is h and the atmospheric pressure (separately measured) is equal to a height H of mercury. The difference h is noted at the start of the experiment and at the end of the experiment. The pressure of the gas varies between p1 = H+h at the beginning to p2 = H+h at the end. Under the assumption of ideal gas
p1V = n1RT and p2V = n2RT
n is equal to n1 – n2 = (p1 - p2)V/RT
Determination of Cv of a gas
Joly’s differential steam calorimeter is used to measure Cv of a gas. The arrangement has two hollow copper spheres attached to two pans of a sensitive balance. In one of the spheres the gas for which Cv is to be measured is filled. At the start the temperature of the steam chamber without any steam is noted. It is the temperature of the gas at the beginning (θ1). Steam is sent through the steam chamber in which these two hollow spheres are there. Steam condenses on the hollow spheres and it collected in the pans attached to the hollow spheres. More steam condenses on the sphere having gas in it. After steady state conditions are reached temperature measurement is taken. This the final temperature of the gas.
Observations
m1 = the mass of gas taken or filled in the hollow sphere
m2 = the mass of extra steam condensed on the pan of the sphere having gas
θ1 = the temperature of the gas at the beginning
θ2 = the temperature of the gas at the end
L = Specific latent heat of vaporization of water.
M = molecular weight of the gas
Cv = Mm2L/[m1(θ1 – θ2)]
Isothermal process
A thermal process on a system is called isothermal if the temperature of the system remains constant during the process. The internal energy of ideal gas used in the system remains constant in an isothermal process. The gas obeys Boyle’s law.
As the temperature remains constant in the process, the molar heat capacity is infinity.
Cisothermal = ΔQ/nΔT = infinity
An isothermal process can be achieved by immersing the system in large reservoir and performing the process very slowly. Any heat produced by the system is absorbed by the reservoir without any appreciable change in temperature. Similar any heat required is supplied by the reservoir to the system with out any appreciable change in temperature. An process done in an open air can also be an isothermal process as the outside air acts as reservoir.
Adiabatic process
A process on a system is termed adiabatic if no heat is supplied t it or extracted from it. In such a system temperature may change but no heat is added or removed from the system.
If work is done by the gas, then the internal energy of the system decreases and temperature falls.
In a reversible adiabatic process pVγ remains constant
Relation between p and T in an adiabatic process
T γ /p γ -1 = constant
Relation between V and t in an adiabatic process
TV p γ -1 = constant
Work done in an adiabatic process
W = [p1V1 – p2V2]/( γ -1)
Where
W = work in the adiabatic process
p1,V1 = initial pressure and volume
p2,V2 = final pressure and volume
Equipartition of Energy
Equipartition of energy states that the average energy of a molecule in a gas associated with each degree of freedom is ½ kT where k is the Boltzmann constant and T is its absolute temperature.
Monatomic gas molecule has 3 degrees of freedom. For diatomic molecules, the degree of freedom is 5 if the molecule does not vibrate and is 7 if it vibrates.
For a sample of diatomic ga, internal energy U = nNA ((5/2) kT) = n(5/2)RT if the molecules do not vibrate.
n = number of moles in the sample
nNA = Avogadro’s number
For a sample of diatomic gas, Cv = 5R/2
Cp = Cv +R = 7R/2
According to equipartition theorem, the molar heat capacities should be independent of temperature. It is valid for most of the temperatures, but at very high temperatures, this theorem may not hold.
The specific heat capacity of a substance is defined as the heat supplied per unit mass of the substance per unit rise in the temperature.
This definition applies to any mode of the substance, solid, liquid or gas.
In the case of gases, to define the specific heat capacity, the process should also be specified.
There can be many processes, but two processes are more important and correspondingly two specific heat capacities are defined for gases. They are constant pressure process and constant volume process.
Molar heat capacity at constant pressure (Cp) and molar heat capacity at constant volume (Cv).
For a given amount of heat, the rise in temperature of a gas at constant pressure is smaller than the rise in temperature at constant volume. (At constant volume work done by the gas zero. At constant pressure work is done in expansion)
Determination of Cp
Hence Cp>Cv
Cp-Cv = R
and Cp/Cv is termed as γ
Determination of Cp of a gas
Regnault’s apparatus is used to measure Cp of a gas.
It has an arrangement to send gas through a pipe where in two manometers measure the pressure at two point and in between there is an adjusting screw to change the rate of flow. As the gas is flowing through this arrangement, adjusting screw is changed to maintain a constant difference of pressure between two manometers and this ensures that gas is at constant pressure when it is flowing through the calorimeter. This gas flows through an oil bath tank wherein it is given heat and then flows through the calorimeter wherein it gives out heat.
Observations
W = the water equivalent of the calorimeter with the coil
M = mass of water in the calorimeter
θ1 = temperature of the oil bath which gives heat to the gas.
θ2 = initial temperature of water in calorimeter
θ3 = Final temperature of water in calorimeter
n = amount of the gas (in moles) passed through the water
s = specific heat capacity of water
Cp of a gas. = (W + m) s (θ3 - θ2)/[n (θ1 – (θ2+ θ3)/2]
Determination of n = amount of the gas (in moles) passed through the water
It depends on the levels of mercury in the manometer attached to gas tank. If the difference in levels of the manometer is h and the atmospheric pressure (separately measured) is equal to a height H of mercury. The difference h is noted at the start of the experiment and at the end of the experiment. The pressure of the gas varies between p1 = H+h at the beginning to p2 = H+h at the end. Under the assumption of ideal gas
p1V = n1RT and p2V = n2RT
n is equal to n1 – n2 = (p1 - p2)V/RT
Determination of Cv of a gas
Joly’s differential steam calorimeter is used to measure Cv of a gas. The arrangement has two hollow copper spheres attached to two pans of a sensitive balance. In one of the spheres the gas for which Cv is to be measured is filled. At the start the temperature of the steam chamber without any steam is noted. It is the temperature of the gas at the beginning (θ1). Steam is sent through the steam chamber in which these two hollow spheres are there. Steam condenses on the hollow spheres and it collected in the pans attached to the hollow spheres. More steam condenses on the sphere having gas in it. After steady state conditions are reached temperature measurement is taken. This the final temperature of the gas.
Observations
m1 = the mass of gas taken or filled in the hollow sphere
m2 = the mass of extra steam condensed on the pan of the sphere having gas
θ1 = the temperature of the gas at the beginning
θ2 = the temperature of the gas at the end
L = Specific latent heat of vaporization of water.
M = molecular weight of the gas
Cv = Mm2L/[m1(θ1 – θ2)]
Isothermal process
A thermal process on a system is called isothermal if the temperature of the system remains constant during the process. The internal energy of ideal gas used in the system remains constant in an isothermal process. The gas obeys Boyle’s law.
As the temperature remains constant in the process, the molar heat capacity is infinity.
Cisothermal = ΔQ/nΔT = infinity
An isothermal process can be achieved by immersing the system in large reservoir and performing the process very slowly. Any heat produced by the system is absorbed by the reservoir without any appreciable change in temperature. Similar any heat required is supplied by the reservoir to the system with out any appreciable change in temperature. An process done in an open air can also be an isothermal process as the outside air acts as reservoir.
Adiabatic process
A process on a system is termed adiabatic if no heat is supplied t it or extracted from it. In such a system temperature may change but no heat is added or removed from the system.
If work is done by the gas, then the internal energy of the system decreases and temperature falls.
In a reversible adiabatic process pVγ remains constant
Relation between p and T in an adiabatic process
T γ /p γ -1 = constant
Relation between V and t in an adiabatic process
TV p γ -1 = constant
Work done in an adiabatic process
W = [p1V1 – p2V2]/( γ -1)
Where
W = work in the adiabatic process
p1,V1 = initial pressure and volume
p2,V2 = final pressure and volume
Equipartition of Energy
Equipartition of energy states that the average energy of a molecule in a gas associated with each degree of freedom is ½ kT where k is the Boltzmann constant and T is its absolute temperature.
Monatomic gas molecule has 3 degrees of freedom. For diatomic molecules, the degree of freedom is 5 if the molecule does not vibrate and is 7 if it vibrates.
For a sample of diatomic ga, internal energy U = nNA ((5/2) kT) = n(5/2)RT if the molecules do not vibrate.
n = number of moles in the sample
nNA = Avogadro’s number
For a sample of diatomic gas, Cv = 5R/2
Cp = Cv +R = 7R/2
According to equipartition theorem, the molar heat capacities should be independent of temperature. It is valid for most of the temperatures, but at very high temperatures, this theorem may not hold.
Concept review Ch. 28 Heat transfer
Revision points
Heat can be transferred from one place to another by three different methods.
Conduction
Convection
Radiation
Conduction takes place in metals.
Convection takes place liquids and gases.
No medium is required for radiation
Thermal conduction
In a metal rod, if one end is heated, the molecules at that end gain heat and their average kinetic energy increases. They collide with neighbouring molecules which have less kinetic energy and the energy sharing takes place. The molecules which now gained some additional energy in turn collide with their neighbour at lower energy and share energy with them. This way, heat is passed along the rod from one end to the other end from molecule to molecule.
The average position of any molecule does not change. Hence there is no mass movement in thermal conduction.
Thermal conductivity is a measure of the ability of a material to conduct heat.
Heat current: In steady state, if ΔQ amount of heat crosses through any cross section material in time Δt, ΔQ/Δt is called the heat current.
In steady state, heat current is proportional to the area of cross section A, proportional to the temperature differences between the ends of body (for example metal rod) and inversely proportional to the length if area of cross section is uniform throughout the body.
Heat current between two ends = ΔQ/Δt = KA(T1 – T2)/x
Where
K = constant for the material of the body and is called the thermal conductivity
Units: J/s-m-K or W/m-K we can also write °C in place of K.
A = area of cross section
T1, T2) = higher and lower temperatures of ends.
x = distance between ends
If the area of cross section is not uniform, the formula is applied to small thicknesses only.
ΔQ/Δt = - KAdT/dx
The quantity dT/dx is called the temperature gradient. The minus sign indicates that in the direction of heat flow there has to be drop in temperature.
In solids metals are better conductors than nonmetals. Metals have free electrons that move in the body of the metal freely and help in carrying thermal energy from one place to another.
Thermal conductivities
Aluminium 209 K(W/m-K)
Glass 0.669 K(W/m-K)
When a rug is placed in bright sun on a tiled floor, one can easily stand on rug. But one cannot stand on the bare floor. Reason: Both rug and floor are at the same temperature. No heat transfer takes place between surfaces at the same temperature. But the rug has very low conductivity and therefore heat current going into the foot is small when one is standing on rug.
Thermal resistance: The quantity x/KA in the equation for ΔQ/Δt is termed the thermal resistance R. So we can write
ΔQ/Δt = KA(T1 – T2)/x = (T1 – T2)/R
We can write ΔQ/Δt as i heat current and then write i = (T1 – T2)/R.
This formula is equivalent to the Ohm’s law formula of electric current. Many results derive based on the Ohm’s law for electric will be applicable to the heat current formula also.
When thermal resistances of same cross section are joined in series, just as in an electrical circuit current in same in the circuit and in both resistances, in the thermal resistances also the heat current will be same.
Hence the equivalent thermal resistance when thermal restances R1 and R2 are connected in series is R1+R2.
Heat current in the system will be i = T1-T2/R1+R2
When thermal resistances are connected in parallel, that is both the left ends of the two metal rods are a temperature T1 and connected to the same heat source and both right ends of the two metal rods are at the same temperature T2
i1 = T1-T2/R1
i2 = T1-T2/R2
i = i1 + i2 = T1-T2[1/R1 + 1/R2]
Hence equivalent resistance R = 1/R1 + 1/R2
If many thermal resistances are connected in parallel, equivalent resistance
R = 1/R1 + 1/R2 + 1/R3 ...
Measurement of Thermal Conductivity of a Solid
Searle’s apparatus is used for measuring thermal conductivity of a solid. One end of the cylindrical rod of a metal whose thermal conductivity is to be measured is placed in steam chamber. At the other end water circulates through a copper tube whose inlet and outlets have thermometers to measure inlet (θ4) and outlet (θ3) temperatures. In between, on the rod there are two drilled holes filled with mercury and thermometers are placed in these holes to measure temperatures (θ1 and θ2, θ1 is greater than θ2). After the steam is passed into the steam chamber, sufficient time is allowed for the system to stabilize, that is all temperatures are constant. Then for a measured t water is collected in a flask and weighed.
Heat flow between holes having thermometers = KA(θ1 – θ2)t/x
Where
K = thermal conductivity of the material
A = area of cross section of cylindrical rod
x = distance between the holes (thermometers)
The heat flow into water collected = ms(θ3 – θ4)
m = mass of water collected
s = specific heat capacity of water
Both the heat flows are same and hence KA(θ1 – θ2)t/x = ms(θ3 – θ4)
K = [x ms(θ3 – θ4)]/ [A(θ1 – θ2)t]
Convection
In convection heat transferred from one place to the other side by the actual motion of heated material. When water is kept in a vessel and heated on a stove, the water at the bottom gets heat due to conduction through the vessels bottom. Its density decreases and consequently it rises. Hot molecules keep going up and cold molecules keep coming down. Mass transfer accomplishes heat transfer from part to the other part of the liquid.
If the heated material is forced to move using a pump, it is called forced convection.
The movement of liquid in heat transfer and anomalous expansion of water (it expands as temperature falls from 4°C to 0°C.) saves the lives of acquatic animal during severe winters. During winters, as the surface of water gets cool, the water at the surface becomes dense and goes inside and hot water from inside comes up. But after 4°C, water at the surface does not become dense, it expands and hence water at the surface stays at the surface and freezes into ice. Heat transfer through ice is a very slow process. Therefore water is there under freezed ice at the surface acquatic animals survive in the water.
Radiation
Radiation word has two meanings. One, the energy transmitted by radiation process is called radiation. The other is the radiation process. In radiation process, heat transfer takes place without the involvement of any medium. Energy emitted by a body travels in the space and when it falls on material body, a part of it is absorbed and the thermal energy of the receiving body increases.
Prevost Theory of Exchange
According to this theory all bodies radiate at all temperatures. The amount of thermal radiation radiated per unit time depends on the nature of the emitting surface, its area and its temperature.
Every body absorbs part of the thermal radiation emitted by the surrounding bodies when this radiation falls on it.
If a body radiates more than what it absorbs, its temperature falls.
Kirchoff’s law
The ratio of emissive power to absorptive power is the same for all bodies at a given temperature and is equal to the emissive power of a blackbody at that temperature.
Thus
E(body)/a(body) = E(blackbody)
Stefan-Boltzmann Law
The energy of thermal radiation emitted by per unit time by a black body of surface area A is given by
U = σAT4
Where
σ = Stefan Boltzmann constant = 5.67*10-8 W/m²-K4
Newton’s law of cooling
dT/dt = -bA(T-T0)
b = a constant depends on the nature of the surface involved
a = surface area exposed of the body
T- T0) = temperature difference between the body and surrounding
Detection and Measurement of Radiation
Bolometer and thermopile are used to detect and measure thermal radiation.
Heat can be transferred from one place to another by three different methods.
Conduction
Convection
Radiation
Conduction takes place in metals.
Convection takes place liquids and gases.
No medium is required for radiation
Thermal conduction
In a metal rod, if one end is heated, the molecules at that end gain heat and their average kinetic energy increases. They collide with neighbouring molecules which have less kinetic energy and the energy sharing takes place. The molecules which now gained some additional energy in turn collide with their neighbour at lower energy and share energy with them. This way, heat is passed along the rod from one end to the other end from molecule to molecule.
The average position of any molecule does not change. Hence there is no mass movement in thermal conduction.
Thermal conductivity is a measure of the ability of a material to conduct heat.
Heat current: In steady state, if ΔQ amount of heat crosses through any cross section material in time Δt, ΔQ/Δt is called the heat current.
In steady state, heat current is proportional to the area of cross section A, proportional to the temperature differences between the ends of body (for example metal rod) and inversely proportional to the length if area of cross section is uniform throughout the body.
Heat current between two ends = ΔQ/Δt = KA(T1 – T2)/x
Where
K = constant for the material of the body and is called the thermal conductivity
Units: J/s-m-K or W/m-K we can also write °C in place of K.
A = area of cross section
T1, T2) = higher and lower temperatures of ends.
x = distance between ends
If the area of cross section is not uniform, the formula is applied to small thicknesses only.
ΔQ/Δt = - KAdT/dx
The quantity dT/dx is called the temperature gradient. The minus sign indicates that in the direction of heat flow there has to be drop in temperature.
In solids metals are better conductors than nonmetals. Metals have free electrons that move in the body of the metal freely and help in carrying thermal energy from one place to another.
Thermal conductivities
Aluminium 209 K(W/m-K)
Glass 0.669 K(W/m-K)
When a rug is placed in bright sun on a tiled floor, one can easily stand on rug. But one cannot stand on the bare floor. Reason: Both rug and floor are at the same temperature. No heat transfer takes place between surfaces at the same temperature. But the rug has very low conductivity and therefore heat current going into the foot is small when one is standing on rug.
Thermal resistance: The quantity x/KA in the equation for ΔQ/Δt is termed the thermal resistance R. So we can write
ΔQ/Δt = KA(T1 – T2)/x = (T1 – T2)/R
We can write ΔQ/Δt as i heat current and then write i = (T1 – T2)/R.
This formula is equivalent to the Ohm’s law formula of electric current. Many results derive based on the Ohm’s law for electric will be applicable to the heat current formula also.
When thermal resistances of same cross section are joined in series, just as in an electrical circuit current in same in the circuit and in both resistances, in the thermal resistances also the heat current will be same.
Hence the equivalent thermal resistance when thermal restances R1 and R2 are connected in series is R1+R2.
Heat current in the system will be i = T1-T2/R1+R2
When thermal resistances are connected in parallel, that is both the left ends of the two metal rods are a temperature T1 and connected to the same heat source and both right ends of the two metal rods are at the same temperature T2
i1 = T1-T2/R1
i2 = T1-T2/R2
i = i1 + i2 = T1-T2[1/R1 + 1/R2]
Hence equivalent resistance R = 1/R1 + 1/R2
If many thermal resistances are connected in parallel, equivalent resistance
R = 1/R1 + 1/R2 + 1/R3 ...
Measurement of Thermal Conductivity of a Solid
Searle’s apparatus is used for measuring thermal conductivity of a solid. One end of the cylindrical rod of a metal whose thermal conductivity is to be measured is placed in steam chamber. At the other end water circulates through a copper tube whose inlet and outlets have thermometers to measure inlet (θ4) and outlet (θ3) temperatures. In between, on the rod there are two drilled holes filled with mercury and thermometers are placed in these holes to measure temperatures (θ1 and θ2, θ1 is greater than θ2). After the steam is passed into the steam chamber, sufficient time is allowed for the system to stabilize, that is all temperatures are constant. Then for a measured t water is collected in a flask and weighed.
Heat flow between holes having thermometers = KA(θ1 – θ2)t/x
Where
K = thermal conductivity of the material
A = area of cross section of cylindrical rod
x = distance between the holes (thermometers)
The heat flow into water collected = ms(θ3 – θ4)
m = mass of water collected
s = specific heat capacity of water
Both the heat flows are same and hence KA(θ1 – θ2)t/x = ms(θ3 – θ4)
K = [x ms(θ3 – θ4)]/ [A(θ1 – θ2)t]
Convection
In convection heat transferred from one place to the other side by the actual motion of heated material. When water is kept in a vessel and heated on a stove, the water at the bottom gets heat due to conduction through the vessels bottom. Its density decreases and consequently it rises. Hot molecules keep going up and cold molecules keep coming down. Mass transfer accomplishes heat transfer from part to the other part of the liquid.
If the heated material is forced to move using a pump, it is called forced convection.
The movement of liquid in heat transfer and anomalous expansion of water (it expands as temperature falls from 4°C to 0°C.) saves the lives of acquatic animal during severe winters. During winters, as the surface of water gets cool, the water at the surface becomes dense and goes inside and hot water from inside comes up. But after 4°C, water at the surface does not become dense, it expands and hence water at the surface stays at the surface and freezes into ice. Heat transfer through ice is a very slow process. Therefore water is there under freezed ice at the surface acquatic animals survive in the water.
Radiation
Radiation word has two meanings. One, the energy transmitted by radiation process is called radiation. The other is the radiation process. In radiation process, heat transfer takes place without the involvement of any medium. Energy emitted by a body travels in the space and when it falls on material body, a part of it is absorbed and the thermal energy of the receiving body increases.
Prevost Theory of Exchange
According to this theory all bodies radiate at all temperatures. The amount of thermal radiation radiated per unit time depends on the nature of the emitting surface, its area and its temperature.
Every body absorbs part of the thermal radiation emitted by the surrounding bodies when this radiation falls on it.
If a body radiates more than what it absorbs, its temperature falls.
Kirchoff’s law
The ratio of emissive power to absorptive power is the same for all bodies at a given temperature and is equal to the emissive power of a blackbody at that temperature.
Thus
E(body)/a(body) = E(blackbody)
Stefan-Boltzmann Law
The energy of thermal radiation emitted by per unit time by a black body of surface area A is given by
U = σAT4
Where
σ = Stefan Boltzmann constant = 5.67*10-8 W/m²-K4
Newton’s law of cooling
dT/dt = -bA(T-T0)
b = a constant depends on the nature of the surface involved
a = surface area exposed of the body
T- T0) = temperature difference between the body and surrounding
Detection and Measurement of Radiation
Bolometer and thermopile are used to detect and measure thermal radiation.
Monday, April 14, 2008
Some JEE 2008 Questions - Paper I
1. Figure shows three resistor configurations R1, R2 and R3 connected to 3V battery. If the power dissipated
by the configuration R1, R2 and R3 is P1, P2 and P3, respectively, then
Figure : not included
(A) P1 > P2 > P3 (B) P1 > P3 > P2
(B) P2 > P1 > P3 (D) P3 > P2 > P1.
Resistances and power
Principle involved
P = V square/R equivalent
2. Which one of the following statement is WRONG in the context of X–rays generated from X–ray tube ?
(A) wavelength of characteristic X–rays decreases when the atomic number of the target increases
(B) cut–off wavelength of the continuous X–rays depends on the atomic number of the target
(C) intensity of the characteristic X–rays depends on the electrical power given to the X–ray tube
(D) cut–off wavelength of the continuous X–rays depends on the energy of the electrons in the X–ray tube.
3. Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is 60º).
In the position of minimum deviation, the angle of refraction will be
(A) 30º for both the colours (B) greater for the violet colour
(C) greater for the red colour (D) equal but not 30º for both the colours.
4. An ideal gas is expanding such that PT^2
= constant. The coefficient of volume expansion of the gas is
(A)1/T
(B)2/T
(C) 3/T
(D) 4/T
Multiple correct answer questions
1. In a Young’s double slit experiment, the separation between the two slits is d and the wavelength of the
light is λ. The intensity of light falling on slit 1 is four times the intensity of light falling on slit 2. Choose
the correct choice(s),
(A) if d = λ, the screen will contain only one maximum
(B) if λ< d < 2λ, at least one more maximum (besides the central maximum) will be observed on the screen
(C) if the intensity of light falling on slit 1 is reduced so that it becomes equal to that of slit 2, the
intensities of the observed dark and bright fringes will increase
(D) if the intensity of light falling on slit 2 is reduced so that it becomes equal to that of slit 1, the intensities of the observed dark and bright fringes will increase.
Assertion reason type
1. STATEMENT – 1
Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions
are simultaneously allowed to roll without slipping down an inclined plane from the same height. The
hollow cylinder will reach the bottom of the inclined plane first.
and
STATEMENT – 2
By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when
they reach the bottom of the incline.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
2. STATEMENT – 1
The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when
held vertically up, but tends to narrow down when held vertically down.
and
STATEMENT – 2
In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
3. STATEMENT – 1
In a Meter Bridge experiment, null point for an unknown resistance is measured. Now, the unknown
resistance is put inside an enclosure maintained at a higher temperature. The null point can be obtained at
the same point as before by decreasing the value of the standard resistance.
and
STATEMENT – 2
Resistance of a metal increases with increase in temperature.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
4. STATEMENT – 1
An astronaut in an orbiting space station above the Earth experiences weightlessness.
and
STATEMENT – 2
An object moving around the Earth under the influence of Earth’s gravitational force is in a state of free–
fall.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
Linked comprehension
In a mixture of H – He+ gas (He+ is singly ionized He atom), H atoms and He
+ ions are excited to their respective first excited states. Subsequently, H atoms transfer their total excitation energy to He+ ions (by collisions). Assume that the Bohr model of atom is exactly valid.
1. The quantum number n of the state finally populated in He+ ions is
(A) 2 (B) 3
(C) 4 (D) 5.
2. 45. The wavelength of light emitted in the visible region by He+ ions after collisions with H atoms is
(A) 6.5 × 10^–7 m
(B) 5.6 × 10^–7 m
(C) 4.8 × 10^–7 m
(D) 4.0 × 10^–7 m.
3. The ratio of the kinetic energy of the n = 2 electron for the H atom to that of He+ ion is
(A)1/4
(B) 1/2
(C) 1
(D) 2.
.
by the configuration R1, R2 and R3 is P1, P2 and P3, respectively, then
Figure : not included
(A) P1 > P2 > P3 (B) P1 > P3 > P2
(B) P2 > P1 > P3 (D) P3 > P2 > P1.
Resistances and power
Principle involved
P = V square/R equivalent
2. Which one of the following statement is WRONG in the context of X–rays generated from X–ray tube ?
(A) wavelength of characteristic X–rays decreases when the atomic number of the target increases
(B) cut–off wavelength of the continuous X–rays depends on the atomic number of the target
(C) intensity of the characteristic X–rays depends on the electrical power given to the X–ray tube
(D) cut–off wavelength of the continuous X–rays depends on the energy of the electrons in the X–ray tube.
3. Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is 60º).
In the position of minimum deviation, the angle of refraction will be
(A) 30º for both the colours (B) greater for the violet colour
(C) greater for the red colour (D) equal but not 30º for both the colours.
4. An ideal gas is expanding such that PT^2
= constant. The coefficient of volume expansion of the gas is
(A)1/T
(B)2/T
(C) 3/T
(D) 4/T
Multiple correct answer questions
1. In a Young’s double slit experiment, the separation between the two slits is d and the wavelength of the
light is λ. The intensity of light falling on slit 1 is four times the intensity of light falling on slit 2. Choose
the correct choice(s),
(A) if d = λ, the screen will contain only one maximum
(B) if λ< d < 2λ, at least one more maximum (besides the central maximum) will be observed on the screen
(C) if the intensity of light falling on slit 1 is reduced so that it becomes equal to that of slit 2, the
intensities of the observed dark and bright fringes will increase
(D) if the intensity of light falling on slit 2 is reduced so that it becomes equal to that of slit 1, the intensities of the observed dark and bright fringes will increase.
Assertion reason type
1. STATEMENT – 1
Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions
are simultaneously allowed to roll without slipping down an inclined plane from the same height. The
hollow cylinder will reach the bottom of the inclined plane first.
and
STATEMENT – 2
By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when
they reach the bottom of the incline.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
2. STATEMENT – 1
The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when
held vertically up, but tends to narrow down when held vertically down.
and
STATEMENT – 2
In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
3. STATEMENT – 1
In a Meter Bridge experiment, null point for an unknown resistance is measured. Now, the unknown
resistance is put inside an enclosure maintained at a higher temperature. The null point can be obtained at
the same point as before by decreasing the value of the standard resistance.
and
STATEMENT – 2
Resistance of a metal increases with increase in temperature.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
4. STATEMENT – 1
An astronaut in an orbiting space station above the Earth experiences weightlessness.
and
STATEMENT – 2
An object moving around the Earth under the influence of Earth’s gravitational force is in a state of free–
fall.
(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for
Statement– 1.
(C) Statement – 1 is True, Statement – 2 is False.
(D) Statement – 1 is False, Statement – 2 is True.
Linked comprehension
In a mixture of H – He+ gas (He+ is singly ionized He atom), H atoms and He
+ ions are excited to their respective first excited states. Subsequently, H atoms transfer their total excitation energy to He+ ions (by collisions). Assume that the Bohr model of atom is exactly valid.
1. The quantum number n of the state finally populated in He+ ions is
(A) 2 (B) 3
(C) 4 (D) 5.
2. 45. The wavelength of light emitted in the visible region by He+ ions after collisions with H atoms is
(A) 6.5 × 10^–7 m
(B) 5.6 × 10^–7 m
(C) 4.8 × 10^–7 m
(D) 4.0 × 10^–7 m.
3. The ratio of the kinetic energy of the n = 2 electron for the H atom to that of He+ ion is
(A)1/4
(B) 1/2
(C) 1
(D) 2.
.
Wednesday, April 9, 2008
Study Guides/Chapters having Links to Audiovisual Lectures
Motion in a Straight Line and Plane
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-3.html
Newton's Laws
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-5.html
Work and Energy
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-8.html
Center of Mass, Linear Momentum
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-9.html
Torque and Rotational Statics
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-10.html
Simple Harmonic Motion
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-12.html
Fluid Mechanics
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-13.html
Wave Motion
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-15.html
Sound Waves
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-16.html
Geometrical Optics
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-18.html
Kinetic Theory of Gases
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-24.html
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-3.html
Newton's Laws
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-5.html
Work and Energy
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-8.html
Center of Mass, Linear Momentum
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-9.html
Torque and Rotational Statics
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-10.html
Simple Harmonic Motion
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-12.html
Fluid Mechanics
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-13.html
Wave Motion
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-15.html
Sound Waves
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-16.html
Geometrical Optics
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-18.html
Kinetic Theory of Gases
http://iit-jee-physics.blogspot.com/2007/10/study-guide-h-c-verma-jee-physics-ch-24.html
Sunday, April 6, 2008
Online material AV
AP Physics B Semester 1
Unit 1: Newtonian Mechanics
Chapter 1: Kinematics
Lesson 1: Motion in One Dimension
Lesson 2: Motion in Two Dimensions
Chapter 2: Newton's Laws of Motion
Lesson 3: Newton's First Law
Lesson 4: Newton's Second Law
Lesson 5: Newton's Third Law
Lesson 6: Applications of Newton's Laws
Chapter 3: Work, Energy, and Power
Lesson 7: Work and Work-Energy Theorem
Lesson 8: Conservative Forces and Potential Energy
Lesson 9: Conservation of Energy
Lesson 10: Power
Chapter 4: Systems of Particles, Linear Momentum
Lesson 11: Center of Mass
Lesson 12: Impulse and Momentum
Lesson 13: Conservation of Linear Momentum, Collisions
Chapter 5: Circular Motion and Rotation
Lesson 14: Uniform Circular Motion
Lesson 15: Torque and Rotational Statics
Chapter 6: Oscillations and Gravitation
Lesson 16: Simple Harmonic Motion
Lesson 17: Mass on a Spring
Lesson 18: Pendulum and Other Oscillations
Lesson 19: Newton's Law of Gravity
Lesson 20: Orbits of Planets and Satellites
Unit 2: Fluid Mechanics and Thermal Physics
Chapter 7: Fluid Mechanics
Lesson 21: Hydrostatic Pressure
Lesson 22: Buoyancy
Lesson 23: Fluid Flow Continuity
Lesson 24: Bernoulli's Equation
Chapter 8: Temperature and Heat
Lesson 25: Mechanical Equivalent of Heat
Lesson 26: Specific and Latent Heat
Lesson 27: Heat Transfer and Thermal Expansion
Chapter 9: Kinetic Theory and Thermodynamics
Lesson 28: Ideal Gases
Lesson 29: Laws of Thermodynamics
http://www.curriki.org/nroc/Introductory_Physics_1/
AP Physics B Semester II
Unit 3: Electricity and Magnetism
Chapter 10: Electrostatics
Lesson 30: Electric Charges and Coulomb's Law
Lesson 31: Electric Fields
Lesson 32: Electric Potential
Chapter 11: Conductors and Capacitors
Lesson 33: Electrostatics with Conductors
Lesson 34: Capacitors
Chapter 12: Electric Circuits
Lesson 35: Current, Resistance, Power
Lesson 36: DC Circuits with Batteries and Resistors
Lesson 37: Capacitors in Circuits
Chapter 13: Magnetostatics
Lesson 38: Forces on Moving Charges
Lesson 39: Forces on Current-carrying Wires in Magnetic Fields
Lesson 40: Fields of Long, Current-carrying Wires
Chapter 14: Electromagnetism
Lesson 41: Electromagnetic Induction
Unit 4: Waves and Optics
Chapter 15: Wave Motion
Lesson 42: Wave Basics
Lesson 43: Properties of Traveling Waves
Lesson 44: Properties of Standing Waves
Lesson 45: Sound Waves and Doppler Shift
Chapter 16: Physical Optics
Lesson 46: Interference and Diffraction
Lesson 47: Dispersion of Light and the Electromagnetic Spectrum
Chapter 17: Geometric Optics
Lesson 48: Reflection and Refraction
Lesson 49: Mirrors
Lesson 50: Lenses
Unit 5: Atomic and Nuclear Physics
Chapter 18: Atomic Physics and Quantum Effect
Lesson 51: Photons and the Photoelectric Effect
Lesson 52: Atomic Energy Levels
Lesson 53: Wave-particle Duality
Chapter 19: Nuclear Physics
Lesson 54: Nuclear Reactions
Lesson 55: Mass-Energy Equivalence
http://www.curriki.org/nroc/Introductory_Physics_2/
Unit 1: Newtonian Mechanics
Chapter 1: Kinematics
Lesson 1: Motion in One Dimension
Lesson 2: Motion in Two Dimensions
Chapter 2: Newton's Laws of Motion
Lesson 3: Newton's First Law
Lesson 4: Newton's Second Law
Lesson 5: Newton's Third Law
Lesson 6: Applications of Newton's Laws
Chapter 3: Work, Energy, and Power
Lesson 7: Work and Work-Energy Theorem
Lesson 8: Conservative Forces and Potential Energy
Lesson 9: Conservation of Energy
Lesson 10: Power
Chapter 4: Systems of Particles, Linear Momentum
Lesson 11: Center of Mass
Lesson 12: Impulse and Momentum
Lesson 13: Conservation of Linear Momentum, Collisions
Chapter 5: Circular Motion and Rotation
Lesson 14: Uniform Circular Motion
Lesson 15: Torque and Rotational Statics
Chapter 6: Oscillations and Gravitation
Lesson 16: Simple Harmonic Motion
Lesson 17: Mass on a Spring
Lesson 18: Pendulum and Other Oscillations
Lesson 19: Newton's Law of Gravity
Lesson 20: Orbits of Planets and Satellites
Unit 2: Fluid Mechanics and Thermal Physics
Chapter 7: Fluid Mechanics
Lesson 21: Hydrostatic Pressure
Lesson 22: Buoyancy
Lesson 23: Fluid Flow Continuity
Lesson 24: Bernoulli's Equation
Chapter 8: Temperature and Heat
Lesson 25: Mechanical Equivalent of Heat
Lesson 26: Specific and Latent Heat
Lesson 27: Heat Transfer and Thermal Expansion
Chapter 9: Kinetic Theory and Thermodynamics
Lesson 28: Ideal Gases
Lesson 29: Laws of Thermodynamics
http://www.curriki.org/nroc/Introductory_Physics_1/
AP Physics B Semester II
Unit 3: Electricity and Magnetism
Chapter 10: Electrostatics
Lesson 30: Electric Charges and Coulomb's Law
Lesson 31: Electric Fields
Lesson 32: Electric Potential
Chapter 11: Conductors and Capacitors
Lesson 33: Electrostatics with Conductors
Lesson 34: Capacitors
Chapter 12: Electric Circuits
Lesson 35: Current, Resistance, Power
Lesson 36: DC Circuits with Batteries and Resistors
Lesson 37: Capacitors in Circuits
Chapter 13: Magnetostatics
Lesson 38: Forces on Moving Charges
Lesson 39: Forces on Current-carrying Wires in Magnetic Fields
Lesson 40: Fields of Long, Current-carrying Wires
Chapter 14: Electromagnetism
Lesson 41: Electromagnetic Induction
Unit 4: Waves and Optics
Chapter 15: Wave Motion
Lesson 42: Wave Basics
Lesson 43: Properties of Traveling Waves
Lesson 44: Properties of Standing Waves
Lesson 45: Sound Waves and Doppler Shift
Chapter 16: Physical Optics
Lesson 46: Interference and Diffraction
Lesson 47: Dispersion of Light and the Electromagnetic Spectrum
Chapter 17: Geometric Optics
Lesson 48: Reflection and Refraction
Lesson 49: Mirrors
Lesson 50: Lenses
Unit 5: Atomic and Nuclear Physics
Chapter 18: Atomic Physics and Quantum Effect
Lesson 51: Photons and the Photoelectric Effect
Lesson 52: Atomic Energy Levels
Lesson 53: Wave-particle Duality
Chapter 19: Nuclear Physics
Lesson 54: Nuclear Reactions
Lesson 55: Mass-Energy Equivalence
http://www.curriki.org/nroc/Introductory_Physics_2/
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