Van der waal gas in isothermal free expansion

[mewmew]

Homework Statement

A gas obeys the Van Der Waal equation of state. The gas undergoes a free expansion from volume Vi to Vf at a constant temperature T. Find the change in entropy of the gas.

Homework Equations

du=Tds-Pdv+mdn

The Attempt at a Solution

I can solve the problem assuming du= 0 but this shouldn't be right. I can't seem to find out how to find out how much heat enters to keep the gas at a constant temperature. Everything shows that for a free expansion q=0, but here this is not the case.

 

[OlderDan]

Homework Statement

A gas obeys the Van Der Waal equation of state. The gas undergoes a free expansion from volume Vi to Vf at a constant temperature T. Find the change in entropy of the gas.

Homework Equations

du=Tds-Pdv+mdn

The Attempt at a Solution

I can solve the problem assuming du= 0 but this shouldn't be right. I can't seem to find out how to find out how much heat enters to keep the gas at a constant temperature. Everything shows that for a free expansion q=0, but here this is not the case.

Take a look at this

http://www.physics.rutgers.edu/ugrad/351/Lecture 15.ppt

This does not do your problem exactly, but it does give a derivation of the vdW energy as an additional term -N²a/V added to the ideal gas energy. It uses that result for a calculation of the change in temperature in a free expansion of a vdW gas from an initial volume to a final volume where V is so large that the addition term is neglected. If seems to me this could be easily modified to keep the additional term for any final volume. So you should be able to express a differential change in temperature in terms of a differential change in volume and relate that to the amount of heat needed to prevent that change in temperature and turn it into an isothermal process.

There is an equation for S in terms of T and V that may be just what you are looking for. If indeed S is a state variable, then the equation for S should be valid. Here is another site that seems to reach the same copnclusion. The difference in entropies at T2,V2 and T1,V1 is given as depending only on the initial and final states.

http://theory.phy.umist.ac.uk/~judith/stat_therm/node51.html