In this problem you will investigate the behavior of a van der Waals fluid
near the critical point. It is easiest to work in terms of reduced variables
throughout.
(a) Expand the van der Waals equation in a Taylor series in
\(\left(V-V_{c}\right)\), keeping terms through order \(\left(V-V_{c}\right)^{3}
.\) Argue that, for \(T\) sufficiently close to \(T_{c}\) the term quadratic in
\(\left(V-V_{c}\right)\) becomes negligible compared to the others and may be
dropped.
(b) The resulting expression for \(P(V)\) is antisymmetric about the point
\(V=V_{c}\) Use this fact to find an approximate formula for the vapor pressure
as a function of temperature. (You may find it helpful to plot the isotherm.)
Evaluate the slope of the phase boundary, \(d P / d T\), at the critical point.
(c) Still working in the same limit, find an expression for the difference in
volume between the gas and liquid phases at the vapor pressure. You should
find \(\left(V_{g}-V_{l}\right) \propto\left(T_{c}-T\right)^{\beta},\) where
\(\beta\) is known as a critical exponent. Experiments show that \(\beta\) has a
universal value of about \(1 / 3,\) but the van der Waals model predicts a
larger value.
(d) Use the previous result to calculate the predicted latent heat of the
transformation as a function of temperature, and sketch this function.
(e) The shape of the \(T=T_{c}\) isotherm defines another critical exponent,
called \(\delta\) :
\(\left(P-P_{c}\right) \propto\left(V-V_{c}\right)^{\delta} .\) Calculate
\(\delta\) in the van der Waals model. (Experimental values of \(\delta\) are
typically around 4 or \(5 .\) )
(f) A third critical exponent describes the temperature dependence of the
isothermal compressibility,
$$\kappa \equiv-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T},$$
This quantity diverges at the critical point, in proportion to a power of
\(\left(T-T_{c}\right)\) that in principle could differ depending on whether one
approaches the critical point from above or below. Therefore the critical
exponents \(\gamma\) and \(\gamma^{\prime}\) are defined by the relations $$\kappa
\propto\left\\{\begin{array}{ll}
\left(T-T_{c}\right)^{-\gamma} & \text { as } T \rightarrow T_{c} \text { from
above } \\
\left(T_{c}-T\right)^{-\gamma^{\prime}} & \text { as } T \rightarrow T_{c}
\text { from below }
\end{array}\right.$$
Calculate \(\kappa\) on both sides of the critical point in the van der Waals
model, and show that \(\gamma=\gamma^{\prime}\) in this model.
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