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

Question

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 [latex]x^{3}[/latex].
The results may be obtained without explicitly evaluating integrals.

Accepted Answer

According to the virial theorem, if the potential energy of each particle is [latex]V \propto x^{n} ext {, then the average kinetic energy } \bar{T}[/latex] and the average potential energy [latex]\bar{V} ext { satisfy the relation } 2 \bar{T}=n \bar{V}[/latex]. According to the theorem of equipartition of energy. [latex]\bar{T}=\frac{1}{2} k T[/latex] for a one-dimensional motion. Hence we can state the following:

[latex] ext { (a) As } f \propto x, V \propto x^{2} ext {, and } n=2 ext {. Then } \bar{V}=\bar{T}=\frac{1}{2} k T, E=\bar{V}+\bar{T}=[/latex] kT. Thus the heat capacity per particle is k and [latex]C_{v}=N k[/latex].

[latex] ext { (b) As } f \propto x^{3}, V \propto x^{4} ext { and } n=4 ext {. Then } \bar{V}=\frac{1}{2} \bar{T}=\frac{1}{4} k T, E=\frac{3}{4} k T [/latex].

Thus the heat capacity per particle is [latex]\frac{3}{4} k ext { and } C_{v}=\frac{3}{4} N k[/latex] for the whole system.

Question 2.163: A system consists of N very weakly interacting particles at ...

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