Q. 7.69 - An Introduction to Thermal Physics

Question

If you have a computer system that can do numerical integrals, it's not particularly difficult to evaluate $\mu$ for $T>T_c$.

(a)  As usual when solving a problem on a computer, it's best to start by putting everything in terms dimensionless                      variables. So define $t=T/T_c, c=\mu /k{T}_c,\text{ and }x=ϵ/k{T}_c$. Express the integral that defines $\mu$, equation                                $N={\int }_0^{\infty }g\left(ϵ\right)\frac{1}{e^{(ϵ-\mu )/kT}-1}dϵ$, in terms of these variables, you should obtain the equation

                                                            $2.315={\int }_0^{\infty }\frac{\sqrt{x}dx}{e^{(x-c)/t}-1}$

(b) According to given figure , the correct value of $c$ when $T=2T_c$ , is approximately $-0.8$. Plug in these values and        check that the equation above is approximately satisfied.

(c) Now vary $\mu$, holding $T$ fixed, to find the precise value of $\mu$ for $T=2T_c$ . Repeat for values of$T/T_c$ ranging from $1.2$ up       to $3.0$, in increments of $0.2$. Plot a graph of $\mu$ as a function of temperature.

Step-by-Step Solution

Answer

k

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d