Browse & Filter

An ideal gas is enclosed in a cylindrical volume with a movable piston. It can be taken from a state 1 to state 2 by several processes as shown in \(P-V \) diagram below. It is given that if the walls are adiabatic the system obeys the equation \(P^3 V^5=\text{const}\) The energy of the state in state 1 is given to be 100J. The pressures and volume at for the two states are given as follows. \[P_1=10^5 \,\text{Pa}, V_1=10^{-3}\, \text{m}^3; P_2 =\Big(\frac{10^5}{32}\Big) \,\text{Pa}, V_2= b\times 10^{-3}\, \text{m}^3.\]  {\textwidth}{ Several paths taking the system from state 1 to state 2 are shown in figure below.

• AB followed by BC
• AD followed by DC
• Linear path connecting A and C
• Some other path AQC

Your are required to compute the internal energy of the system in state 2 by computing work done by selecting a path from one of the above options. If you choose option (d), some other path, specify the path. Compute the work done along the path chosen and find the internal energy of system in state 2.

Consider a system of $N$ atoms. Assuming that they can exist in two states only. The ground state having energy zero and an excited state having energy $\epsilon$.

• Find the number of micro states with total energy $U$.
• Write an expression for entropy and using Stirling approximation for the factorial $$ \ln (N!) \approx N \ln N - N$$ find the temperature of the system and hence show that $$ U = \frac{N\epsilon}{1+ e^{\epsilon/kT}}$$
• What is fraction of atoms are in the excited state at very large temperature $(T >> kT)$?

Two heat reservoirs, each of fixed volume $V$ are at temperatures $T_1$ and $T_2$ ( $T_1\,>\,T_2$) initially. An engine operates between the two till both of them reach the same temperature $T_3$. The specific heat $c_V$ of the reservoirs can be assumed to constant through out. The engine is back to its inital state.

• Show that $$ T_3^2\,\geq\,T_1T_2 $$
• Calculate the maximum work that can be extracted from the two reservoirs. ( Answer in terms of $T_1$ and $T_2$)

The tension $\tau$ in an elastic rubber band is given by $$ \tau\,=\,aT\left(\frac{L}{L_0(T)}-\,\left(\frac{L_0(T)}{L}\right)^2\right),$$ where $a$ is a constant, $L_0(T)$is the unstretched length at zero tension, and is a function of temperature only.

• Write the first law using the work done when it is elongated and gets a supply of heat. ( Be careful of signs!)
• Use the first law to write $dF$, where $F$ is the free energy of the rubber band.
• Solve for the free energy $F$ and show that $$ F(T,L)\,-\,F(T,L_0(T))\,=\,aT\left(\frac{L^2}{2L_0(T)}\,+\,\frac{L_0(T)^2}{L^2}\,-\,\frac{3L_0(T)}{2}\right)$$ and the entropy $S$ $$ S(T,L)\,-\,S(T,L_0(T))\,=\,a\left(\frac{3L_0}{2}\,-\,\frac{L_0^2}{L}\,-\,\frac{L^2}{2L_0}\right)\,-\,aT\left(\frac{3}{2}\,-\,\frac{2L_0}{L}\,+\,\frac{L^2}{2L_0^2}\right)\frac{dL_0(T)}{dT} $$
• Find the heat $Q$ transferred to the elastic band when it is stretched from $L_0$ to $L$ isothermally.
• Show that $$ \left(\frac{\partial T}{\partial L}\right)_S \,=\,\frac{aTL_0^2}{c_LL^2}\left(-1\,+\,\left(\frac{L}{L_0}\right)^3\,+\,\frac{Ta}{L_0}\frac{dL_0}{dT}\left(2\,+\,\left(\frac{L}{L_0}\right)^3\right)\right)$$ where $$c_L\,=\,\left(\frac{DQ}{\partial T}\right)_L.$$

The tension $\tau$ in an elastic rubber band is given by
$$ \tau\,=\,aT\left(\frac{L}{L_0(T)}-\,\left(\frac{L_0(T)}{L}\right)^2\right),$$
where $a$ is a constant, $L_0(T)$is the unstretched length at zero tension, and is a function of temperature only.

(a) Write the first law using the work done when it is elongated and gets a supply of heat. ( Be careful of signs!)

(b) Use the first law to write $dF$, where $F$ is the free energy of the rubber band.

(c) Solve for the free energy $F$ and show that
$$ F(T,L)\,-\,F(T,L_0(T))\,=\,aT\left(\frac{L^2}{2L_0(T)}\,+\,\frac{L_0(T)^2}{L^2}\,-\,\frac{3L_0(T)}{2}\right)$$
and the entropy $S$
$$ S(T,L)\,-\,S(T,L_0(T))\,=\,a\left(\frac{3L_0}{2}\,-\,\frac{L_0^2}{L}\,-\,\frac{L^2}{2L_0}\right)\,-\,aT\left(\frac{3}{2}\,-\,\frac{2L_0}{L}\,+\,\frac{L^2}{2L_0^2}\right)\frac{dL_0(T)}{dT} $$

(d)Find the heat $Q$ transferred to the elastic band when it is stretched from $L_0$ to $L$ isothermally.

(e) Show that
$$ \left(\frac{\partial T}{\partial L}\right)_S \,=\,\frac{aTL_0^2}{c_LL^2}\left(-1\,+\,\left(\frac{L}{L_0}\right)^3\,+\,\frac{Ta}{L_0}\frac{dL_0}{dT}\left(2\,+\,\left(\frac{L}{L_0}\right)^3\right)\right)$$

where
$$c_L\,=\,\left(\frac{DQ}{\partial T}\right)_L.$$

Consider the following expressions for entropy. Which ones can possibly be a fundamental equation and which ones violate one or more of postulates II,III and IV?

• $S\,=\,K_1\left(NVU\right)^{1/3}$
• $ S\,=\,K_2\left(\frac{NU}{V}\right)^{2/3} $
• $ S\,=\,K_3\left(\frac{V^3}{NU}\right)$
• $ S\,=\,N{\rm{\ln}}\left(\frac{UV}{N^2K_4}\right)$

$K_i$'s are positive constants so that dimensions match. $S,U,N,V $ are the entropy, internal energy, number of particles and volume respectively.

Estimate how long it would take a Nitrogen molecule of air in a room to travel 1 m?
Assume that the cross section of a molecule as $10^{-20}m$ and the temperature is $300^o$ K.

The fundamental equation for a system is given by
\begin{equation*}
u = \Lambda \frac{s^{3/2}}{v^{1/2}}
\end{equation*}
where \(\Lambda\) is a constant.
Prove the following equations
\begin{eqnarray}
T &=& \frac{5}{2} \frac{\Lambda S^{3/2}}{NV^{1/2}}\\
P V^{2/3} &=& N \frac{N \Lambda 2^{1/2}}{5*{3/2}} T^{5/3}\\
\mu &=& - \Big(\frac{2}{5}\Big)^{5/2} \frac{2}{\Lambda ^{2/3}} \Big(\frac{V}{N}\Big)^{1/3} T^{5/3}.
\end{eqnarray}

A box is divided into two equal parts by a non-conducting partition having a hole of diameter $D$. Initially, some Helium gas is preset on both sides at temperature $T_1$ K and $T_2$K, by heating the walls of the box.

(a) What is the steady state when $D\,>>\,L_1$ and $D\,>>\,L_2$ where $L_1$ and $L_2$ are mean free paths at temperature $T_1$ and $T_2$ respectively.

(b) What is the steady state when $D\,<<\,L_1$ and $D\,<<\,L_2$ where $L_1$ and $L_2$ are mean free paths at temperature $T_1$ and $T_2$ respectively.