4. Consider one mole of a van der Waals gas, which obeys the equation (p + a/V^2

Question

4. Consider one mole of a van der Waals gas, which obeys the equation (p + a/V^2)*(V b) = RT, where a and b are constants. Assume V >> b, and that b is of the order of magnitude as a/RT.

(a) By working to second order in the small quantities, show that V ~RT/p + b a/RT + 2abp/(R^2*T^2) a^2*p/(R^3*T^3).

(b) Using the results in (a), find an approximate expression, to first order in small quantities, for T as a function of p on the “inversion curve” for the Joule-Thomson process. (The inversion curve is defined to be the locus of points in the p T plane which have µ (T /p)H = 0.) You should find T ~ 2a/Rb 2bp/R.

(c)drew the inversion curve for a van der Waals gas. Argue that, in the region where the approximation here is likely to be valid, the inversion curve found in (b) has roughly the same shape as the complete inversion curve.

Explanation / Answer

Van der waals equation; for calculating the pressure of a nonideal gas: (P + a/V^2 ) (V - nb) = nRT

pv=nRT, We Fix the voulume in a small quantity so V=nRT/P,

APPLY THE VANDEWAALS EQUN. WITH THE CONSTANTS

WE MAY GET,

V ~RT/p + b a/RT + 2abp/(R^2*T^2) a^2*p/(R^3*T^3).

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