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 1 K. Its unit is J kg^-1 K^-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 -∞ depending upon the way in which heat energy is given.

Let m be the mass of a gas and C it specific heat capacity. Then ΔQ = m x C x ΔT where ΔQ is the amount of heat absorbed and ΔT is the corresponding rise in temperature.

Case (i)

            If  the gas in 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)      ΔQ=0

            ∴    C=0

Case (ii)

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

                         Then

(because  ΔQ is + ve as heat is supplied from outside)

Case (iii)

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

 (Because ΔQ is –ve as heat is supplied by the system)

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 degree of freedom.

We know, kinetic energy per molecule, per degree of freedom is 1/2 kT.

Therefore 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

where N is the A vogadro number.

Therefore

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 x C_v x dT

AS R = 8.31 J mol^-1 K^-1

Then C_P - C_V = R

C_P = C_V + R

=3/2R + R = 5/2R = 5/2 x 8.31

Therefore 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 x 1/2 kT x N = 5/2RT

Also,

C_V = 5/2 x 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 x 8.31

= 29.085 J mol^-1 K^-1

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