Question 2.163: A system consists of N very weakly interacting particles at ...
A system consists of N very weakly interacting particles at a temperature sufficiently high such that classical statistics are applicable. Each particle has mass m and oscillates in one direction about its equilibrium position. Calculate the heat capacity at temperature T in each of the following cases:
(a) The restoring force is proportional to the displacement x from the equilibrium position.
(b) The restoring force is proportional to x^{3}.
The results may be obtained without explicitly evaluating integrals.
According to the virial theorem, if the potential energy of each particle is V \propto x^{n} \text {, then the average kinetic energy } \bar{T} and the average potential energy \bar{V} \text { satisfy the relation } 2 \bar{T}=n \bar{V}. According to the theorem of equipartition of energy. \bar{T}=\frac{1}{2} k T for a one-dimensional motion. Hence we can state the following:
\text { (a) As } f \propto x, V \propto x^{2} \text {, and } n=2 \text {. Then } \bar{V}=\bar{T}=\frac{1}{2} k T, E=\bar{V}+\bar{T}= kT. Thus the heat capacity per particle is k and C_{v}=N k.
\text { (b) As } f \propto x^{3}, V \propto x^{4} \text { and } n=4 \text {. Then } \bar{V}=\frac{1}{2} \bar{T}=\frac{1}{4} k T, E=\frac{3}{4} k T .
Thus the heat capacity per particle is \frac{3}{4} k \text { and } C_{v}=\frac{3}{4} N k for the whole system.