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Thermal Physics
Problem sheet 8
Each question carries 5 marks
9th November 2021 Due 16th November 2021
22. Show from calculating
$$ \frac{\Delta S}{\Delta E}\,=\,\frac{1}{T}\,=\,k\beta$$
for the case of Fermi-Dirac distribution,
$$ \Omega_{FD}\,=\,\Pi_{i=1}^\infty\left({}^{g_i}C_{N_i}\right) $$
[ notation as used in the class ]. Also for the case of equilibrium we have
$$ N_i\,=\,\frac{g_i}{e^{\alpha+\beta\epsilon_i}\,+\,1} $$
Hint: Consider changes only in two levels say with energies $\epsilon _1$ and $\epsilon_2$. Then argue that the result so obtained is independent of the choice of levles]
23. Repeat the same ( as problem 22) for the case of Bose-Einstein distribution.
24. Find the number of microstates of a particle in a box of volume $V$ and having energy between $U$ and $U\,+\,\Delta U$ if the relation between energy $E$ and the momentum $p$ is given by $ E\,=\,Cp^n$, where $C$ and $n$ are constants.
