In this problem, we will estimate fy for a fluid that obeys the van der Waals...

 

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In this problem, we will estimate fy for a fluid that obeys the van der Waals EOS using the following equation we derived in class: RT In Plow (-_ vidP. Plow Because the van der Waals EOS is explicit in P (but not v), performing the vidP integral is a little more complicated than for the virial EOS we considered in class. However, following a series of mathematical manipulations, here we will derive that: (vi- b)P] b 2a =- In + RT Vi b RT vi where vi is the molar volume at the system T and P, and can be readily obtained from the vdW EOS. (a) To perform the integral on the r.h.s. of the above boxed equation, we will first change the variable of integration from P to vi by using the EOS. To do so, obtain an expression for dP (at constant T). (b) Our next task is to identify the new limits for the integral. Leave the upper limit to be the system volume, vi, but recognize that the volume in the lower limit can be approximated by RT/Plow. Explain why. (c) Integrate with respect to vi. (d) Recognize that as Plow 0, RT/Plow becomes large, so that RT/Plow >> b is a good approximation. (e) Terms containing In Plow should cancel from both sides of the equation. Now set Plow 0 to obtain the expression given above. (f) Finally, to get a sense for the deviation from ideal gas behavior at the critical point, use your answer above to estimate (f/P) at the critical temperature and pressure. In this problem, we will estimate fy for a fluid that obeys the van der Waals EOS using the following equation we derived in class: RT In Plow (-_ vidP. Plow Because the van der Waals EOS is explicit in P (but not v), performing the vidP integral is a little more complicated than for the virial EOS we considered in class. However, following a series of mathematical manipulations, here we will derive that: (vi- b)P] b 2a =- In + RT Vi b RT vi where vi is the molar volume at the system T and P, and can be readily obtained from the vdW EOS. (a) To perform the integral on the r.h.s. of the above boxed equation, we will first change the variable of integration from P to vi by using the EOS. To do so, obtain an expression for dP (at constant T). (b) Our next task is to identify the new limits for the integral. Leave the upper limit to be the system volume, vi, but recognize that the volume in the lower limit can be approximated by RT/Plow. Explain why. (c) Integrate with respect to vi. (d) Recognize that as Plow 0, RT/Plow becomes large, so that RT/Plow >> b is a good approximation. (e) Terms containing In Plow should cancel from both sides of the equation. Now set Plow 0 to obtain the expression given above. (f) Finally, to get a sense for the deviation from ideal gas behavior at the critical point, use your answer above to estimate (f/P) at the critical temperature and pressure.