Question 6.16: In the previous Example Problem, it was assumed that CP=CV. ...
In the previous Example Problem, it was assumed that C_{P}=C_{V}.
a. Use equation 3.38 C_{P,m}=C_{V,m}+TV_{m}\frac{\beta^{2}}{\kappa}, and the experimentally determined value C_{P,m}=27.98 J mol^{-1}K^{-1} to obtain a value for C_{V,m} for Hg\left( l \right) at 300. K.
b. Did the assumption that C_{P,m}\approx C_{V,m} in Example Problem 6.15 introduce an appreciable error in your calculation of \Delta U and \Delta H?
C_{P}=C_{V}+TV\frac{\beta^{2}}{\kappa} or C_{P,m}=C_{V,m} + TV_{m}\frac{\beta^{2}}{\kappa} (3.38)
a. C_{P,m}-C_{V,m}=\frac{TV_{m}\beta^{2}}{\kappa}
=\frac{300. k ×\frac{0.20059 Kg mol^{-1}}{13534 Kgm^{3}}×\left( 1.81×10^{-4} K^{-1} \right)^{2}}{3.91 × 10^{-6} bar^{-1}× \left( 1 bar/10^{5} Pa \right)}
=3.73 J K^{-1} mol^{-1}
C_{V,m} =C_{P,m}-\frac{TV_{m}\beta^{2}}{\kappa}
=27.98 J K^{-1} mol^{-1} -3.73 J K^{-1} mol^{-1} =24.25 J K^{-1} mol^{-1}
b. \Delta U is in error by \sim 15\% and \Delta U is unaffected by assuming that C_{P,m}\approx C_{V,m}. It is a reasonable approximation to set C_{P} =C_{V} for a liquid or solid.