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The entropy of water at $100^o$C is $310 \,{\rm{calories/(Kg-Kelvin)}}$ and that of steam at the same temperature is $1760\, {\rm{calories/(Kg-Kelvin)}}$.

(a) What is the latent heat of vaporization at this temperature? Express your answer in calories/Kg.

(b) The enthalpy $H\,=\,U\,+\,PV$ of steam under these conditions is $64\times10^4 {\rm{calories/Kg}}$. Calculate the enthalpy of water in calories/Kg under the same conditions.

(c) Calculate the Gibbs free energy $G\,=\,U\,-\,TS\,+\,PV$
in calories/Kg.

$$pV\,=\,A(T)\,+\,B(T)p\,+\,C(T)p^2 $$
Find $C_p(T,p)$ in terms of $C_p(T,p_0)$ and $p\,,\,p_0$ ( initial and the final pressures) and $A(T)\,,\,B(T)\,,\,C(T)$ and their derivatives with respect to temperature $T$. [ Write an appropriate expression for
$$\frac{\partial C_p}{\partial p} $$ and use it to obtain $C_p$]

Consider a closed cylinder whose walls are adiabatic. The cylinder is divided into three equal parts $A_1$, $A_2$ and $A_3$ by means of partitions $S_1$ and $S_2$, which can move along the length of the cylinder without friction. The partition $S_1$ is adiabatic and $S_2$ is conducting. Initially, each of the three parts contain one mole of Helium gas, which can be treated as an ideal gas, is at pressure $P_0$, temperature $T_0$ and volume $V_0$. Assume the specific heat at constant volume $C_v\,=\,\frac{3R}{2}$ and the specific heat at constant pressure $C_p\,=\,\frac{5R}{2}$. Now, heat is supplied to the to the left most partition $A_1$ till the temperature in part $A_3$ becomes $T_3\,=\,\frac{9T_0}{4}$ Find the final volume, pressure and temperature in terms of $V_0$, $P_0$ and $T_0$. Assume the entire process is quasistatic.
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3. ( Continuation of problem 2)

(a) What is work done by the gas in $A_1$ ?

(b) What is the heat supplied to the gas in $A_1$?

{Consider an open system of non-interacting particles obeying Maxwell-Boltzmann statistics. The system is at temperature $T$ and chemical potential $\mu$. Let $\sigma^2_N$ denote the variance of the number of particles in the system. Show that \begin{eqnarray*} \sigma^2_N &=& k_BT\left( \frac{\partial\langle N\rangle}{\partial\mu}\right)_{T,V}, \end{eqnarray*} where $\langle N\rangle$ is the mean number of particles. The angular brackets denote averaging is carried out over a grand canonical ensemble

KPN

Calculate $Z_V$ for a classical ideal gas and use it to calculate $(\Delta V)^2 \,=\,\overline{V^2}\,-\,\overline{V}^2 $.

Consider a 2 dimensional phase space ( $q,p$) with a rectangular region defined by four corners as shown.
If the region ABCD is the phase space region at time time t = 0 , find the region $A'B'C'D'$ at time t given the Hamiltonian is
$$ H\,=\,\frac{p^2}{2m}\,-\, m a q $$
and explicitly verify that the area is constant. Take the coordinates of A,B,C and D as $(q_A,p_A)\,,\,(q_B,p_A)\,,\,(q_B,p_C)$ and $(q_A,p_C)$ respectively

Consider a paramagnetic system, with variables magnetization $M$, the magnetic field $B$ and absolute temperature $T$. ( We assume it's dependence on pressure as negligible). The equation of state is ( which will be obtained from statistical mechanics later in the course) is
$$ M\,=\,C\frac{B}{T}, $$
where $C$ is a constant ( referred to as the Curie constant, who had experimentally obtained this relation.
The system's internal energy is ( for a one-dimensional system)
$$ U\,=\,-MB.$$
The work done on the system by external surrounding is $-MdB$

(a) Write the expression for $DQ$ in terms of $dM$ and $dB$

(b) Write the equation for entropy change $dS$ in terms of $dM$ and $dB$

(c) Obtain the entropy $S$

Using the identity $\Gamma(N\,+\,1)\,=\,N!=\,\int_0^\infty x^Ne^{-x} dx $
and write the integral in the form
$$ \int_0^\infty e^{Ng(x)}dx $$
Find the maximum of $g(x)$ ( which occurs say at $x_0$ ). Assume the integral
is dominated by the contribution from the neighbourhood of $x_0$ for large N. Expanding $g(x)$ up to second order in $(x-x_0)$ derive the Stirling's approximation ( for $n\rightarrow\,\infty$)
$$ \rm{ln}N!\,=\,N\rm{ln}N\,-\,N\,+\,\frac{1}{2}\rm{ln}(2\pi N)$$

( More exact formula , just for information, is
$$ N!\,=\,\left(\sqrt{2\pi}\right)e^{-N}N^{N+1/2}\left[1\,+\,\frac{1}{12N}\,+\,\frac{1}{288N^2}\,+\,\ldots\right]$$

Show that constant volume thermometers using an ideal gas as well as a van der Waals gas both yield the same
temperature scale when uniform scales are adopted. Equation of state for ideal gas is \(P V = nRT\) and for van der
Wall gas is
\begin{equation*}
\Big(p + \frac{an^2}{V^2}\Big )(V − nb) = nRT
\end{equation*}