Question

2. Heat added to a substance goes partly into the translational kinetic energy of its molecules, which directly elevates temperature. For some substances, large proportions of heat also go into vibrations and rotations of the molecules. Discuss whether you’d expect materials in which a lot of energy goes into non-translational molecular motions to have a high or low specific heat capacity?

Answer 1

The materials in which lot of energy goes into non-translational molecular motions should have a high specific heat.

Consider the following two points:

• Specific heat is the amount of heat required to raise a temperature of 1kg of a substance by 1°C.
• Temperature is a measure of translational kinetic energy of the gas molecules

From these two points we can infer that specific heat is the amount of heat required to increase kinetic energy of the molecules by an amount that corresponds to 1°C rise in temperature of 1kg of substance.

If there are non-translational molecular motion, all the heat is not used to increase only translational kinetic energy. Some of the heat is used to increase energies of the non-translational motion. So lesser rise in translational kinetic energy with the same amount of heat, hence more heat is required to increase the temperature by 1°C. Thus, the specific is high in such gases.

By equipartition theorem, every degree of freedom of a molecule has energy 0.5kT, where k is the Boltzmann constant. Consider a molecule of two atoms (like O_{​​​​​​2}), there are three translational degree of freedoms (translational motion along x, y and z directions) and two rotational degrees of freedom. Total there are 5 degrees of freedom, hence the average energy of the molecule is 5×0.5kT=2.5kT

The heat required to raise the temperature of the molecule by 1°C is 2.5k.

Compare this energy with the energy required to raise the temperature of monoatomic gas molecule but 1°C. Monoatomic atom has only 3 translational degrees of freedom. The average energy of the molecule at temperature T is 3×0.5kT=1.5kT

The energy required to raise the temperature of the molecule by 1°C is 1.5k

The energy required to raise the temperature of the diatomic gas by 1°C is higher than that required for a monoatomic gas, therefore specific heat of diatomic gas is higher.