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
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.
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.
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)]
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.
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)
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.