Riemann surface
matlab

Problem 2.1 Show that the Helmholtz free energy $F(T, V)$ of a system whose volume is $V$ in contact with a heat bath at temperature $T$ is a minimum at equilibrium. Analogously, show that the Gibbs free energy $G(T, P)$ of a system in contact with a bath at temperature $T$ and pressure $P$ is a minimum at equilibrium.

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Problem 2.4 The principal specific heats $C_P$ and $C_1$ of any substance can be expressed in terms of its temperature $T$. volume $V$, adiabatic compressibility $\kappa_S \equiv-V^{-1}(\partial V / \partial P)_S$, isothermal compressibility $\kappa_T \equiv-V^{-1}(\partial V / \partial P)_T$ and thermal expansivity $\alpha \equiv V^{-1}(\partial V / \partial T)_p$. Obtain these expressions as follows.

(a) Regarding the entropy $S$ as a function of $T$ and $V$. show that
$$T \mathrm{~d} S=C_V \mathrm{~d} T+\frac{\alpha T}{\kappa_T} \mathrm{~d} V .$$
(b) Regarding $S$ as a function of $T$ and $P$, show that
$$T \mathrm{~d} S=C_P \mathrm{~d} T-\alpha T V \mathrm{~d} P .$$
(c) Using these results, prove that
$$C_P-C_1=\frac{T V \alpha^2}{\kappa_T} .$$
(d) Express $C_P$ and $C_V$ in terms of $T, V, \alpha, \kappa_T$ and $\kappa s$.

Assuming that the radius of curvature is $R$, the subtending angle of the strip is $\theta$, and the change of thickness is negligible, we have
$$\begin{gathered} l_2=\left(R+\frac{x}{4}\right) \theta, \quad l_1=\left(R-\frac{x}{4}\right) \theta \ l_2-l_1=\frac{x}{2} \theta=\frac{x}{2} \frac{l_1+l_2}{2 R}=\frac{x l_0}{4 R}\left[2+\left(\alpha_1+\alpha_2\right) \Delta T\right] . \end{gathered}$$
From (1) and (2) we obtain
$$l_2-l_1=l_0 \Delta T\left(\alpha_2-\alpha_1\right)$$
(3) and (4) then give
$$R=\frac{x}{4} \frac{\left[2+\left(\alpha_1+\alpha_2\right) \Delta T\right]}{\left(\alpha_2-\alpha_1\right) \Delta T}$$

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