Browse & Filter

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.

Let
$$\frac{\partial (x,y)}{\partial (a,b)}\,\equiv\,\left|\begin{array}{ll}
\frac{\partial x}{\partial a}&\frac{\partial y}{\partial a}\\
\frac{\partial x}{\partial b}&\frac{\partial y}{\partial b}\\
\end{array}\right|$$

Then show that
$$ \frac{\partial (x,y)}{\partial (a,b)}\frac{\partial (a,b)}{\partial (c,d)}\,=\,\frac{\partial (x,y)}{\partial (c,d)} $$

Remarks : 1. This can be generalised to higher dimensions.

2. This can be found in books - and is very useful in changing variables in multiple integrals.

For a black body of temperature \(T\), Planck's law can be used to show that the
wavelength for which the emissivity \(e(\lambda, T)\)
is maximum is given by
\[ \beta h c = 4.965 \lambda_\text{max}\]
Solar radiation has a maximum intensity near
\(\lambda = 5\times 10^{-5}\)cm. Assume
that the the sun's surface is in thermal equilibrium. Solve the above
equation numerically and determine its temperature.