PHZ3400-09 Problem Set 2

Problem 1

Consider the famous Van der Waals equation describing the liquid-gas transition:

$(P+{\frac {aN^{2}}{V^{2}}})(V-Nb)=NkT$.

a) Show that introducing the average volume per particle $v=V/N$, this equation can be expressed as a cubic polynomial in $v$.

b) By looking for extrema ($dP/dv=0$) of the $P(v)$ isotherms, show that the pressure is monotonic function of the volume, above a certain critical temperature $T_{c}$. Show that the temperature, volume, and pressure of the critical point are given by:

$T_{c}={\frac {8}{27}}{\frac {a}{b}}\!$, $V_{c}=3Nb\!$, and $P_{c}={\frac {1}{27}}{\frac {a}{b^{2}}}\!$.

c) Show that the Van der Waals equation can be written in universal form

$({\frac {P}{P_{c}}}+3({\frac {V_{c}}{V}})^{2})(3{\frac {V}{V_{c}}}-1)=8{\frac {T}{T_{c}}}\!$,

and that

${\frac {P_{c}V_{c}}{T_{c}}}={\frac {3}{8}}$.

d) The isothermal compressibility is defined as:

$\mathrm {K} =-\left({\frac {dV}{dP}}\right)_{T}$.

Examine the system along the critical isotherm $T=T_{c}$, and show that the compressibility diverges as the critical point is approached. The phenomenon of critical opalescence is a direct consequence of very large compressibility close to the critical point - leading to huge density fluctuations. This behavior is a precursor of liquid-gas phase separation below the critical temperature.