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.

Problem 2

The same phenomenon can be studied using a recently-developed JAVA applet that can be accessed at:

http://badmetals.magnet.fsu.edu/sim_clps.html

To visualize phase separation, carry out the following steps:

a) Start the simulation by first selecting the lattice size to be 60 and click "Go".

b) Adjust the temperature to be T=3. You will then see the particles performing a vigorous thermal motion.

c) Turn off the long-range Coulomb repulsion by clicking on the "Coulomb Repulsion off" tab at the bottom right. At T=3 this will not affect the system. Then select the number of particles at N=1800, V=1, and R_o=1.

d) Now gradually reduce the temperature to T=2. Do you see any change? Cool further... To observe the configuration of particles, you can click "Pause" to freeze the motion, and "Resume" to start it again. Do this at several temperatures, and print the screen for each T.

e) Below a certain (critical) temperature, the particles will start to display phase separation, by forming droplets. As you watch the motion, the droplets merge together and grow. This phenomenon is called spinodal decomposition. By observing these motions, try to estimate the critical temperature Tc for your system.

d) Now set T=0. What happens?

e) Finally, click "Coulomb repulsion on". How do the particles rearrange? Print the screen again and describe the pattern.

f) Now change the interaction range to R_o = 3. What happens now? Print the screen and describe your observation.