Two Questions on Internal Energy of a Gas

The molar heat capacity (molar specific heat) of a gas at constant volume is given by

C_v = (n/2)R where ‘n’ is the number of degrees of freedom of the gas molecule and ‘R’ is the universal gas constant.

The molar heat capacity at constant pressure is given by

C_p = C_v + R = (n/2)R + R = [(n+2)/2]R, in accordance with Meyer’s relation.

The number of degrees of freedom of a mono atomic gas molecule is 3. Therefore, in the case of a mono atomic gas, C_v = (3/2)R and C_p = (5/2)R.

The number of degrees of freedom of diatomic gas molecules is 5 and that of triatomic and polyatomic gas molecules is 6 if we do not consider the possible vibrational modes. Therefore, in the case of a diatomic gas, C_v = (5/2)R and C_p = (7/2)R where as in the case of triatomic and polyatomic gases, C_v = (6/2)R = 3R and C_p = 4R.

Now, consider the following MCQ:

When an ideal diatomic gas without vibrational modes is heated at constant pressure, the fraction of heat energy supplied which increases the internal energy of the gas is

(a) 5/7 (b) 3/5 (c) 2/5 (d) 2/3 (e) 1/3

In the case of a diatomic gas, the molar specific heats at constant volume and constant pressure are respectively (5/2)R and (7/2)R. This means that the increase in internal energy of one mole of the gas when we supply (7/2)R joule of heat at constant pressure is (5/2)R joule. The remaining R joule of heat is spent on doing work during the expansion of the gas.

The fraction of the total energy supplied which increases the internal energy of the gas is therefore (5/2)R/(7/2)R = 5/7 [Option(a)].

[Note that this fraction in the case of a mono atomic gas will be 3/5].

Let us consider another question:

Two moles of an ideal mono atomic gas is heated from 20°C to 120°C. The increase in internal energy of the gas is nearly (Universal gas constant = 8.3 J mol^–1 K^–1).

(a) 1660 J (b) 2490 J (c) 3320 (d) 4150 J (e) 4980 J

Some of you may be inclined to wonder which molar specific heat you have to use here (C_p or C_v?). Don’t have any doubt. The increase in internal energy depends on the temperature rise and not on the conditions of constant volume or constant pressure. When you heat a gas at constant volume, the entire heat supplied is used to increase the internal energy. When you heat the gas at constant pressure, a part of the heat supplied is used for doing external work since the gas has to expand to keep the pressure constant. In both cases, the increase in the internal energy of the gas is the same if the temperature rise is the same. You have to use the molar specific heat at constant volume, which is equal to (3/2)R in the case of a mono atomic gas, for calculating the increase in internal energy.

So, increase in internal energy = No. of moles×C_v×Temperature rise = 2× (3/2)R(120–20) = 2490 J, on substituting the value 8.3 for the universal gas constant R.