Q 7.46 - An Introduction to Thermal Physics

Question

 Sometimes it is useful to know the free energy of a photon gas.

(a) Calculate the (Helmholtz) free energy directly from the definition

 (Express the answer in terms of T' and V.)

(b) Check the formula $S=-(\partial F/\partial T{)}_V$ for this system.

(c) Differentiate F with respect to V to obtain the pressure of a photon gas. Check that your result agrees with that of the previous problem.

(d) A more interesting way to calculate F is to apply the formula $F=-kT\ln Z$ separately to each mode (that is, each effective oscillator), then sum over all modes. Carry out this calculation, to obtain

$F=8\pi V\frac{(kT{)}^4}{(hc{)}^3}{\int }_0^{\infty }{x}^2\ln \left(1-e^{-x}\right)dx$

Integrate by parts, and check that your answer agrees with part (a).

Step-by-Step Solution

Answer

Hence, the free energy is $F=-\frac{1}{3}U$

Step 1: Given information

 Sometimes it is useful to know the free energy of a photon gas.

 F= U-TS.

Step 2: Explanation

The Helmholtz free energy is:

$F=U-T S \left(1\right)$

The total energy is:

$U=\frac{8{\pi }^5(kT{)}^4}{15(hc{)}^3}V$

The entropy is:

$S=\frac{32{\pi }^5}{45}V{\left(\frac{kT}{hc}\right)}^{3}k$

Let,

$A=\frac{8{\pi }^2{k}^4}{15(hc{)}^3}$

The total energy and entropy is:

$U=AV{T}^4 \text{ and } S=\frac{4}{3}AV{T}^3$

Substitute into (1)

$$\begin{gathered} F=AV{T}^4-\frac{4}{3}AV{T}^4 \\ F=-\frac{1}{3}AV{T}^4 \\ F=-\frac{1}{3}U \end{gathered}$$

Step 3: Explanation

(b) At constant volume, the entropy equals the negative partial derivative of the Helmholtz energy with respect to temperature.

$S=-{\left(\frac{\partial F}{\partial T}\right)}_V$

Substitute F,

$$\begin{gathered} S=-\frac{\partial }{\partial T}\left(-\frac{1}{3}AV{T}^4\right) \\ S=\frac{4}{3}AV{T}^3 \end{gathered}$$

(c) At constant temperature, the pressure equals the negative partial derivative of the Helmholtz energy with respect to the volume, i.e.

$P=-{\left(\frac{\partial F}{\partial V}\right)}_T$

Substitute with F

$$\begin{gathered} P=-\frac{\partial }{\partial V}\left(-\frac{1}{3}AV{T}^4\right) \\ P=\frac{1}{3}A{T}^4 \\ P=\frac{1}{3}\frac{U}{V} \end{gathered}$$

(d) The Helmholtz free energy for one mode is given by:

$F=-kT\ln \left(Z\right)$

Where Z is partition function and is given as:

$Z=\frac{1}{1-e^{-ϵ/kT}}$

Substitute in the above equation:

$$\begin{gathered} F=-kT\ln \left(\frac{1}{1-e^{-ϵ/kT}}\right) \\ F=kT\ln \left(1-e^{-ϵ/kT}\right) \end{gathered}$$

Because we have two polarisation modes, the total Helmholtz free energy equals the sum of F over all modes increased by a factor of two.

$F=2\sum _{n_x}\sum _{n_y}\sum _{n_z}kT\ln \left(1-e^{-ϵ/kT}\right)=2\sum _{n_x,n_y,n_z}kT\ln \left(1-e^{-ϵ/kT}\right)$

Step 4: Explanation

Consider we have a cubic box with volume of V and side width of L, the allowed energy for the photon is:

$ϵ=\frac{hcn}{2L}$

Hence

$F=2\sum _{n_x,n_y,n_z}kT\ln \left(1-e^{-hcn/2LkT}\right)$

Now we need to convert the total to an integral in spherical coordinates, which we can do by multiplying it by the spherical integration factor $n^2\sin \left(\theta \right)$, which gives us:

$$\begin{gathered} F=2kT{\int }_0^{\pi /2}d\Phi {\int }_0^{\pi /2}\sin \left(\theta \right)d\theta {\int }_0^{\infty }{n}^2\ln \left(1-e^{-hcn/2LkT}\right)dn \\ F=\pi kT{\int }_0^{\infty }{n}^2\ln \left(1-e^{-hcn/2LkT}\right)dn \end{gathered}$$

Let

$x=\frac{hcn}{2LkT} \to dx=\frac{hc}{2LkT}dn$

Thus,

$F=\pi kT{\left(\frac{2LkT}{hc}\right)}^3{\int }_0^{\infty }{x}^2\ln \left(1-e^{-x}\right)dx$

Volume of the box is:

$F=8\pi kTV{\left(\frac{kT}{hc}\right)}^3{\int }_0^{\infty }{x}^2\ln \left(1-e^{-x}\right)dx$

Integrate by parts:

$F=8\pi kTV{\left(\frac{kT}{hc}\right)}^3\left[{\left.\frac{x^3}{3}\ln \left(1-e^{-x}\right)\right|}_0^{\infty }-{\int }_0^{\infty }\frac{x^3}{3}\frac{e^{-x}}{1-e^{-x}}\right]$

Substituting values

$$\begin{gathered} F=-8\pi kTV{\left(\frac{kT}{hc}\right)}^3{\int }_0^{\infty }\frac{x^3}{3}\frac{e^{-x}}{1-e^{-x}} \\ F=-\frac{8\pi }{3}V\frac{(kT{)}^4}{(hc{)}^3}{\int }_0^{\infty }\frac{x^3{e}^{-x}}{1-e^{-x}} \\ F=-\frac{8\pi }{3}V\frac{(kT{)}^4}{(hc{)}^3}{\int }_0^{\infty }\frac{x^3}{e^x-1} \end{gathered}$$

Comparing with the equation,

$F=-\frac{1}{3}U$