[QUE/TH-08001] TH-PROBLEM

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