B$.$ Explidit calculation of $N^{**}$

We position ourselves at the thermodynamiclimit. Justify the transformation

${\displaystyle \underset{p}{\overset{?}{}}}\frac{V}{h^2}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{d}^{a}p$

$2$ In this limit$,$ explain the calculation of $N^{**}$ without actually carrying it out. In particular, show that it is perfectly

legitimate to consider integration from $p=||vec(p)||=0$ for this alculation.

$[$This justifies in netropoct the separais treabueut of the fuodawes tal level$].$

For zin$0,1[,$ we define:

$g_{\frac{a}{2}}(z)=\frac{2}{\sqrt[\phantom{2}]{}}{\displaystyle \underset{0}{\overset{}{}}}\frac{x^{\frac{1}{2}}}{z^{-1}{e}^x-1}dx$

We will assume that

$g_{\frac{3}{2}}(z)={\displaystyle \underset{n=1}{\overset{}{}}}\frac{z^n}{n^{\frac{2}{2}}}$ and ${\displaystyle \underset{n=1}{\overset{}{}}}{n}^{-\frac{3}{2}}=2612$

Provide the expression for $N^{**}$ and $g_{\frac{a}{2}}(z)$ where $z$ will be spedified $a$ a function of $k,T$ and $$

C$.$ Condensation condition

Study the variation of $g_{\frac{2}{2}}(z)$ within its domain of definition.

$2$ Deduce that at any temperature $T,N^{**}$ is incresed by $N_{\text{m}\text{a}\text{x}}^{\text{'}}(V,{}_{nh})$ where $N_{\text{m}\text{a}\text{x}}^{\text{'}}$ will be expressed as a function of $V$ and

the thermal wavdength ${}_{th}$ :

${}_{sh}=\frac{h}{\sqrt[\phantom{2}]{2mon{k}_{R}T}}$

What happens when the temperature $T$ decreases suffidently? Spedify the temperature $T_R$ at which this phenomenon

begins.

For $T>T_{}$ show that the approximation $N=N$ is realistic. What strict inequality does this imply for $?$

D$.$ Assume conditions whare $E=\frac{V}{{}_{dh}^3}(k_{D}T)\frac{2}{\sqrt[\phantom{2}]{}}{\displaystyle \underset{0}{\overset{}{}}}\frac{x^{\frac{a}{2}}dx}{e^x-1}{\displaystyle \underset{0}{\overset{}{}}}\frac{x^{\frac{3}{2}}dx}{e^x-1}=\frac{3}{2}\frac{\sqrt[\phantom{2}]{}}{2}1\mathrm{.}341EN,T{T}_E{T}_{2}T$

Show that the averageenergy $of$ the boson gas $is$ given $by$ the expression:

$E=\frac{V}{{}_{dh}^3}(k_{D}T)\frac{2}{\sqrt[\phantom{2}]{}}{\displaystyle \underset{0}{\overset{}{}}}\frac{x^{\frac{a}{2}}dx}{e^x-1}$

The mathematiol formula tabks provide the integral

${\displaystyle \underset{0}{\overset{}{}}}\frac{x^{\frac{3}{2}}dx}{e^x-1}=\frac{3}{2}\frac{\sqrt[\phantom{2}]{}}{2}1\mathrm{.}341$

$2$ Express theenergy $Eof$ the system $as$ a function $ofN,T$ and $T_E-$

Calculate the constant$-$volume heat capadity $of$ the gas $as$ a function $ofN,T$ and $T_2.$

Compare this result with the heat capacity $of$ aclassical gas.