A compound has the follOwing physical properties:

Question

Critical Temperature = 1000 K
Critical Pressure = 20 atm
Boiling temperature at atmospheric pressure = 300 K
Molar Volume of saturated liquid at atmospheric pressure = 0.1 L/mol
Coefficient of Thermal Expansion for LIQUID is constant at 0.001 K-1
Isothermal Compressibility of LIQUID is constant at 5x10-4atm-1
Ideal Gas Heat Capacity is Cp*= 4R
Heat capacity in the liquid phase is constant at Cv=0.3 L atm/mol K,
In the gas/vapor phase, can be described by the equation of state:
PV = RT + (BP3)
Where B = 0.05 L mol-1atm-2
A) A stream of vapor enters a turbine at T=800 K and P=10 atm, and it leaves at T=500 K and P=2 atm. Find ?H and ?S for the vapor in this process.
B) Derive general equations (i.e., valid for any fluid, not just this compound) that express ((??H)/(??V))_Tand ((??H)/?T)_?Ventirely in terms of measureable properties. Your expressions can include P, V, T, CP, CV, and/or their derivatives, as well as the gas constant R.
C) The compound described above, initially in the liquid phase at T=300 K and P=1 atm, is placed in a rigid container and cooled to T=275 K. What is the change in molar enthalpy of the compound for this process?

Answer 1

Apply definition of change in enthalpy of a gas including residual properties:



Evaluate ?H_1^Rand ?H_2^R, integrating residual enthalpy expression with respect to P:





Where ?V is derived from the equation of state:



Evaluate ((??V)/?T)_P:



Substitute ((??V)/?T)_P and ?V into ?H^R expression all in terms of R, P, B, and T:



Evaluate ?H_2^ig-?H_1^ig:



Substitute into change in enthalpy of a gas equation:



Apply definition of change in entropy of a gas including residual properties:



Evaluate residual entropy of gas, integrating with respect to dP:





Substitute into total change in entropy of a gas equation:





To derive expression for ((??H)/(??V))_Tin terms of measurable properties, start by applying the expansion rule:



Where ((??H)/(??S))_P=T and ((??H)/?P)_?S=?V from the fundamental property relationship for change in enthalpy:



Where ((??S)/(??V))_T=(?P/?T)_?Vfrom Maxwell’s relationships:



One could also obtain a valid answer in terms of isothermal compressibility and coefficient of





Substitute into ((??H)/(??V))_Texpression:



To derive expression for ((??H)/?T)_?Vin terms of measurable properties, start by applying the expansion rule:



Where ((??H)/(??S))_P=T and ((??H)/?P)_?S=?V from the fundamental property relationship for change in enthalpy:





Again one can also express a valid answer in terms of isothermal compressibility and coefficient of thermal expansion, and this will be useful in part C. (?P/?T)_?V=?_V/K_T as derived previously:



The process involves the changing of enthalpy at constant molar volume (rigid container) with changing temperature. Therefore, ((??H)/?T)_?V must be evaluated: