## Monday, July 23, 2007

### H C Verma Ch. 24. Kinetic Theory of Gases – Study Guide

This chapter contains the assumptions for developing kinetic theory of gases, derivation of an expression for pressure exerted by gas on a container, developing a concept of average of the speeds of the molecules, root-mean-square (rms) speed of molecules, derivation of translational kinetic energy of a gas, deriving the functional relation between Temperature of a gas and its rms speed of molecules, and using the kinetic theory to prove Boyle’s law, Charles’ law, Charles’ law of Pressure, Avogadro’s law, Graham’s law of diffusion, and Dalton’s law of partial pressures. The other concepts discussed are Boltzmann constant, Universal gas constant, Maxwell’s speed distribution law, description of thermodynamic state, equation of state, Brownian motion, vapour, evaporation, saturated and unsaturated vapour pressure, dew point, humidity, phase diagrams, dew and fog.

Key Points

Any sample of gas is made of molecules.

Assumptions of kinetic theory of gases :
• All gases are made up of molecules moving in al directions
• The size of the molecule is much smaller than the average separation between the molecules
• The molecules exert no force on each other or the walls of the container except during collisions
• All collisions between two molecules or the walls of the container are perfectly elastic. also the time spent during a collision is negligibly small
• The molecules obey newtons laws of motion.
• When a gas is left for a sufficient time it comes to a steady state, the density and the distribution of molecules with different velocities are independent of position ,direction ,and time. This assumption may be justified if the number of molecules is very large.

The molecular size is roughly 100 times smaller than the average separation between the molecules at 0.1atm and the room temperature

 According to the kinetic theory the internal energy of the ideal gas is the same as the total translational kinetic energy of its molecules .
 v-tr is the rms speed of the molecules at 273.16k and hence is a constant for the given gas .
 The absolute temperature of the given gas is proportional to the square of the rms speed of its molecules
 The absolute temperature of the given sample of the gas is proportional to the total translational kinetic energy of its molecules .
 We find that for a different kinds of gases, it is not the rms speed but the average kinetic energy of individual molecules that has a fixed value at a given temperature. The heavier molecules move with a smaller rms speed and the lighter molecules move with larger speed
 Grahams law of diffusion: when two gases at same pressure and temperature are allowed to diffuse into each other,the rate of diffusion of each gas is inversely proportional to the square of the root of the density of the gas. This is known as grahams law of diffusion.
 Dalton’s law of partial pressure: Daltons law of partial pressure says that the pressure exerted by a mixture of several gases equals the sum of the pressure exerted by each gas occupying the same volume as that of the mixture.
 The boltzmann constant: the universal constant k is known as the boltzmann constant and its value is k=1.38*10 ^-23 J/K
 The universal gas constant : R=N(a)k [n(a) =avagadros number] is another universal constant known as the universal gas constant its value is r=8.314J/mol-K.
 The average speed v-bar somewhat smaller then the rms speed.
 Maxwell derived an equation giving the distribution of the molecules in different speeds.
 The speed v (p) at which dn/dv is maximum is called the most probable speed.
 A thermodynamic state of a given sample of an ideal gas is completely described if its pressure and its volume are given
 The equation relating pressure, volume and temperature of a given sample of gaseous substance is called the equation of state for that gaseous substance. For ideal gas it is pV = nRT. For a real gas, van der Waals derived the following equation: [p+(a/V^2)][V-b] = nRT.
 In liquids also, molecules are in constant random motion. Such a phenomenon and motion is called Brownian motion. To observe brownian motion using normal microscope, we need to have light suspended particles in liquids.
 If the temperature is sufficiently high, no amount of pressure can liquify the gas. The temperature above which this behaviour occurs is called the critical temperature of the substance.
 A gas below its critical temperature is called vapour.
 For water critical temperature is 374.1 degrees Celsius.
 Evaporation is a process in which molecules escape slowly from the surface of a liquid due to their random motion.
 The temperature at which the saturation vapour pressure is equal to the present vapour pressure is called the dew point.
 Dew point is measured using Reganault’s hygrometer.
 Triple points is the point at which all three phases of a substance exist in equilibrium.
 Water vapour condensing on flowers, grass is termed dew.
 Water vapour condensing on dust particles in air is forms thick mist called fog.

JEE questions from this chapter:

there is question from this chapter in JEE 2007 in paper II

1.
Statement-1
The total translational kinetic energy of all the molecules of an ideal gas is 1.5 times of the product of its pressure and its volume.
because
Statement-2
the molecules of a gas collide with each other and the velocities of molecules change due to the collision.

(A) Statement-1 is true and Statement-2 is true and Statement-2 is a correct explation for Statement-1.
(B) Statement-1 is true and Statement-2 is true and Statement-2 is a not correct explation for Statement-1.
(C) Statement-1 is true and Statement-2 is false.
(D) Statement-1 is false and Statement-2 is true