Degrees of Freedom and Molar Specific Heats — Question 9

Question 9

A classical solid stores potential energy that a monatomic ideal gas does not

Model a classical solid as atoms independently oscillating harmonically in three directions. A monatomic ideal gas has only three free translational kinetic terms. One mole of each is brought into thermal contact in a rigid insulated apparatus. Initially the gas is at 600 K and the solid at 200 K. Assume both models remain classical and their capacities constant; neglect expansion and apparatus heat.

1. Derive the molar $C_V$ of each model and the final common temperature.

2. At that temperature, compare the gas mean kinetic energy per atom with the solid mean kinetic and potential energies per atom. Explain the factor of two in their heat capacities.

Question 9 - Solution

A classical solid stores potential energy that a monatomic ideal gas does not

(a) Method and calculation

$$f_{\mathrm{gas}}=3:\ C_{V,g}=\tfrac32R,\quad f_{\mathrm{solid}}=3+3=6:\ C_{V,s}=3R.$$$$\tfrac32R(600-T_f)=3R(T_f-200)\ \Rightarrow\ T_f=\frac{1000}{3}\,\mathrm K.$$

(b) Interpretation and further calculation

$$\langle K\rangle_g=\langle K\rangle_s=\tfrac32k_BT_f,\quad\langle V\rangle_s=\tfrac32k_BT_f.$$$$\langle E\rangle_s=3k_BT_f=2\langle E\rangle_g.$$Confining harmonic forces add three potential-energy terms. They do not double the kinetic energy at a common temperature.

Physical check

The classical solid result is a model limit; it is not asserted to hold at arbitrarily low temperature.