Molar specific heat capacity of a gas

Specific heat capacity

Specific heat capacity of a substance is defined as the quantity of heat required to raise the temperature of 1 kg of the substance through 1K. Its unit is J kg^?1K^?1.

Molar specific heat capacity of a gas

Molar specific heat capacity of a gas is defined as the quantity of heat required to raise the temperature of 1 mole of the gas through 1K. Its unit is J mol^?1 K^?1.

Specific heat capacity of a gas may have any value between ?∞ and +∞ depending upon the way in which heat energy is given.

Let m be the mass of a gas and C its specific heat capacity. Then ∆Q = m ? C ? ∆T where ∆Q is the amount of heat absorbed and ∆T is the corresponding rise in temperature.

(i.e) C = ∆ Q / m ∆T

Case (i)

If the gas is insulated from its surroundings and is suddenly compressed, it will be heated up and there is rise in temperature, even though no heat is supplied from outside

(i.e) ∆Q = 0

C=0

Case (ii)

If the gas is allowed to expand slowly, in order to keep the temperature constant, an amount of heat ∆Q is supplied from outside,

Then C = ∆Q / (m x ∆T ) = ∆Q/0 = +infinity

(∵ ∆Q is +ve as heat is supplied from outside)

Case (iii)

If the gas is compressed gradually and the heat generated ∆Q is conducted away so that temperature remains constant, then

Then C = ∆Q / (m x ∆T ) = -∆Q/0 = -infinity

Thus we find that if the external conditions are not controlled, the value of the specific heat capacity of a gas may vary from +∞ to -∞

Hence, in order to find the value of specific heat capacity of a gas, either the pressure or the volume of the gas should be kept constant.

Consequently a gas has two specific heat capacities (i) Specific heat capacity at constant volume (ii) Specific heat capacity at constant pressure.

Molar specific heat capacity of a gas at constant volume

Molar specific heat capacity of a gas at constant volume C_V is defined as the quantity of heat required to raise the temperature of one mole of a gas through 1 K, keeping its volume constant.

Molar specific heat capacity of a gas at constant pressure

Molar specific heat capacity of a gas at constant pressure C_p is defined as the quantity of heat to raise the temperature of one mole of a gas through 1 K keeping its pressure constant.

Specific heat capacity of monoatomic, diatomic and triatomic gases

Monoatomic gases like argon, helium etc. have three degrees of freedom.

We know, kinetic energy per molecule, per degree of freedom is ? kT.

Kinetic energy per molecule with three degrees of freedom is 3/2 kT.

Total kinetic energy of one mole of the monoatomic gas is given by E = 3/2. kT x N = 3/2 . RT, where N is the Avogadro number.

dE/ dT = 3/2  R

If dE is a small amount of heat required to raise the temperature of 1 mole of the gas at constant volume, through a temperature dT,

dE = 1 ? C_V ? dT

C_V = dE/dT = 3/2 .  R

As R = 8.31 J mol^?1 K^?1

CV = 3/2 ? 8.31=12.465 J mol^?1 K^?1

Then C_P ? C_V = R

C_P = C_V + R

=3R /2 + R

= 5R/2 = (5/2) x 8.31

 C_p = 20.775 J mol^?1 K^?1

In diatomic gases like hydrogen, oxygen, nitrogen etc., a molecule has five degrees of freedom. Hence the total energy associated with one mole of diatomic gas is

E = 5 ? 1/2 kT ? N = 5/2 RT

Also, Cv = dE/dT =d/dT (5/2 . RT) = 5/2 . R

C_v = ( 5/2 ) . 8.31  = 20.775 J mol^?1 K^?1

But C_p = C_v + R

=5/2 . R +  R = (7/2)R

C_p =(7/2)  8.31

= 29.085 J mol^?1 K^?1

similarly, Cp and Cv can be calculated for triatomic gases.

Internal energy

Internal energy U of a system is the energy possessed by the system due to molecular motion and molecular configuration. The internal kinetic energy U_K of the molecules is due to the molecular motion and the internal potential energy U_P is due to molecular configuration. Thus

U = U_K + U_P

It depends only on the initial and final states of the system. In case of an ideal gas, it is assumed that the intermolecular forces are zero. Therefore, no work is done, although there is change in the intermolecular distance. Thus U_P = O. Hence, internal energy of an ideal gas has only internal kinetic energy, which depends only on the temperature of the gas.

In a real gas, intermolecular forces are not zero. Therefore, a definite amount of work has to be done in changing the distance between the molecules. Thus the internal energy of a real gas is the sum of internal kinetic energy and internal potential energy. Hence, it would depend upon both the temperature and the volume of the gas.