P8.1. Develop an expression for the Gibbs energy departure function based on the Redlich-Kwong (1958) equation of state:
(ANS. (G – Gig)/RT = –ln(1 – bρ) – aln(1 + bρ)/(bRT3/2) + Z – 1 – lnZ)
P8.2. For certain fluids, the equation of state is given by Z = 1 – bρ/Tr.
Develop an expression for the enthalpy departure function for fluids of this type. (ANS. –2bρ/Tr)
P8.3. In our discussion of departure functions we derived Eqn. 8.14 for the internal energy departure for any equation of state.
a. Derive the analogous expression for 
.
b. Derive an expression for in terms of a, b, ρ, and T for the equation of state:
P8.4. Even in the days of van der Waals, the second virial coefficient for square-well fluids (λ = 1.5) was known to be B/b = 4 + 9.5 [exp(NAε/RT) – 1]. Noting that ex ~ 1 + x + x2/2 + …, this observation suggests the following equation of state:
Derive an expression for the Helmholtz energy departure function for this equation of state. (ANS. –4ln(1 – bρ) – 9.5NAεbρ/kT)
P8.5. Making use of the Peng-Robinson equation, calculate ΔH, ΔS, ΔU, and ΔV for 1 gmol of 1,3-butadiene when it is compressed from 25 bar and 400 K to 125 bar and 550 K. (ANS. ΔH = 12,570 J/mol; ΔS = 17.998 J/mol-K; ΔU = 11,690 J/mol; ΔV = –640.8 cm3/mol)
P8.6. Ethane at 425 K and 100 bar initially is contained in a 1 m3 cylinder. An adiabatic, reversible turbine is connected to the outlet of the tank and exhausted to atmosphere at 1 bar absolute.
a. Estimate the temperature of the first gas to flow out of the turbine. (ANS. 185 K)
b. Estimate the rate of work per mole at the beginning of this operation. (ANS. 8880 J/mol)
P8.7. Ethylene at 350°C and 50 bar is passed through an adiabatic expander to obtain work and exits at 2 bar. If the expander has an efficiency of 80%, how much work is obtained per mole of ethylene, and what is the final temperature of the ethylene? How does the final temperature compare with what would be expected from a reversible expander? (ANS. 11 kJ/mole, 450 K versus 404 K)
P8.8. A Rankine cycle is to operate on methanol. The boiler operates at 200°C (Psat = 4.087 MPa), and a superheater further heats the vapor. The turbine outlet is saturated vapor at 0.1027 MPa, and the condenser outlet is saturated liquid at 65°C (Psat = 0.1027 MPa). What is the maximum possible value for the cycle thermal efficiency (ηθ = –W/QH)? (ANS. 26%)
P8.9. An ordinary vapor-compression cycle is to be designed for superconductor application using N2 as refrigerant. The expansion is to 1 bar. A heat sink is available at 105 K. A 5 K approach should be sufficient. Roughly 100 Btu/hr must be removed. Compute the coefficient of performance (COP) and compare to the Carnot COP. Also, estimate the compressor’s power requirement (hp) assuming it is adiabatic and reversible. (ANS. 1.33, 0.3)
P8.10. Suppose ethane was compressed adiabatically in a 70% efficient continuous compressor. The downstream pressure is specified to be 1500 psia at a temperature not to exceed 350°F. What is the highest that the upstream temperature could be if the upstream pressure is 200 psia? (Hint: Neglect the upstream departure function.) (ANS. 269 K)
P8.11. As part of a liquefaction process, ammonia is throttled to 80% quality at 1 bar. If the upstream pressure is 100 bar, what must be the upstream temperature?
P8.12. An alternative to the pressure equation route from the molecular scale to the macroscopic scale is through the energy equation (Eqn. 7.51). The treatment is similar to the analysis for the pressure equation, but the expression for the radial distribution function must now be integrated over the range of the potential function.
a. Suppose that u(r) is given by the square-well potential (R = 1.5) and g(r) = 10 – 5(r/σ) for r > σ. Evaluate the internal energy departure function where ρσ3 = 1 and ε/kT = 1. (ANS. –5.7π)
b. Suppose that the radial distribution function at intermediate densities can be reasonably represented by: g ~ (1 + 2(σ/r)2) at all temperatures. Derive an expression for the attractive contribution to the compressibility factor for fluids that can be accurately represented by the Sutherland potential. (ANS. 3πρNAσ3NAε/RT)
8.12. Homework Problems
8.1. What forms does the derivative (∂CV/∂V)T have for a van der Waals gas and a Redlich-Kwong gas? (The Redlich-Kwong equation is given in Problem P8.1.) Comment on the results.
8.2. Estimate CP, CV, and the difference CP – CV in (J/mol-K) for liquid n-butane from the following data.2
8.3. Estimate CP, CV, and the difference CP – CV in (J/mol-K) for saturated n-butane liquid at 298 K n-butane as predicted by the Peng-Robinson equation of state. Repeat for saturated vapor.
8.4. Derive the integrals necessary for departure functions for U, G, and A for an equation of state written in terms of Z = f(T,P) using the integrals provided for H and S in Section 8.6.
a. Derive the enthalpy and entropy departure functions for a van der Waals fluid.
b. Derive the formula for the Gibbs energy departure.
8.6. The Soave-Redlich-Kwong equation is presented in problem 7.15. Derive expressions for the enthalpy and entropy departure functions in terms of this equation of state.
8.7. In Example 8.2 we wrote the equation of state in terms of Z = f (T,ρ). The equation of state is also easy to rearrange in the form Z = f (T,P). Rearrange the equation in this form, and apply the formulas from Section 8.6 to resolve the problem using departures at fixed T and P.
8.8. The ESD equation is presented in problem 7.19. Derive expressions for the enthalpy and entropy departure functions in terms of this equation of state.
8.9. A gas has a constant-pressure ideal-gas heat capacity of 15R. The gas follows the equation of state,
over the range of interest, where a = –1000 cm3/mole.
a. Show that the enthalpy departure is of the following form:
b. Evaluate the enthalpy change for the gas as it undergoes the state change:
T1 = 300 K, P1 = 0.1 MPa, T2 = 400 K, P2 = 2 MPa
8.10. Derive the integrated formula for the Helmholtz energy departure for the virial equation (Eqn. 7.7), where B is dependent on temperature only. Express your answer in terms of B and its temperature derivative.
8.11. Recent research suggests the following equation of state, known as the PC-SAFT model.
a. Derive an expression for Z.
b. Derive the departure function for (U-Uig).
Note: ηP = bρ; m = constant proportional to molecular weight; ai, bi are constants.
8.12. Recent research in thermodynamic perturbation theory suggests the following equation of state.
a. Derive the departure function for (A – Aig)T,V.
b. Derive the departure function for (U – Uig).
Hint: substitute u = 0.7 + Texp(10ηP); ηP = bρ.
8.13. A gas is to be compressed in a steady-state flow reversible isothermal compressor. The inlet is to be 300 K and 1 MPa and the gas is compressed to 20 MPa. Assume that the gas can be modeled with equation of state
where a = 385.2 cm3-K/mol and b = 15.23 cm3/mol. Calculate the required work per mole of gas.
8.14. A 1 m3 isolated chamber with rigid walls is divided into two compartments of equal volume. The partition permits transfer of heat. One side contains a nonideal gas at 5 MPa and 300 K and the other side contains a perfect vacuum. The partition is ruptured, and after sufficient time for the system to reach equilibrium, the temperature and pressure are uniform throughout the system. The objective of the problem statements below is to find the final T and P.
The gas follows the equation of state
where b = 20 cm3/mole; a = 40,000 cm3K/mole; and CP = 41.84 + 0.084T(K) J/mol-K.
a. Set up and simplify the energy balance and entropy balance for this problem.
b. Derive formulas for the departure functions required to solve the problem.
b. Determine the final P and T.
8.15. P-V-T behavior of a simple fluid is found to obey the equation of state given in problem 8.14.
a. Derive a formula for the enthalpy departure for the fluid.
b. Determine the enthalpy departure at 20 bar and 300 K.
c. What value does the entropy departure have at 20 bar and 300 K?
8.16. Using the Peng-Robinson equation, estimate the change in entropy (J/mole-K) for raising butane from a saturated liquid at 271 K and 1 bar to a vapor at 352 K and 10 bar. What fraction of this total change is given by the departure function at 271 K? What fraction of this change is given by the departure function at 352 K?
8.17. Suppose we would like to establish limits for the rule T2 = T1(P2/P1)R/CP by asserting that the estimated T2 should be within 5% of the one calculated using the departure functions. For ω = 0 and Tr = [1, 10] at state 1, determine the values of Pr where this assertion holds valid by using the Peng-Robinson equation as the benchmark.
8.18. A piston contains 2 moles of propane vapor at 425 K and 8.5 MPa. The piston is taken through several state changes along a pathway where the total work done by the gas is 2 kJ. The final state of the gas is 444 K and 3.4 MPa. What is the change, ΔH, for the gas predicted by the Peng-Robinson equation and how much heat is transferred? Note: A reference state is optional; if one is desired, use vapor at 400 K and 0.1 MPa.
8.19. N.B. Vargaftik3 (1975) lists the experimental values in the following table for the enthalpy departure of isobutane at 175°C. Compute theoretical values and their percent deviations from experiment by the following
a. The generalized charts
b. The Peng-Robinson equation
8.20. n-pentane is to be heated from liquid at 298 K and 0.01013 MPa to vapor at 360 K and 0.3 MPa. Compute the change in enthalpy using the Peng-Robinson equation of state. If a reference state is desired, use vapor at 310 K, 0.103 MPa, and provide the enthalpy departure at the reference state.
8.21. For each of the fluid state changes below, perform the following calculations using the Peng-Robinson equation: (a) Prepare a table and summarize the molar volume, enthalpy, and entropy for the initial and final states; (b) calculate ΔH and ΔS for the process; and (c) compare with ΔH and ΔS for the fluid modeled as an ideal gas. Specify your reference states.
a. Propane vapor at 1 bar and 60°C is compressed to a state of 125 bar and 250°C.
b. Methane vapor at –40°C and 0.1013 MPa is compressed to a state of 10°C and 7 MPa.
8.22. 1 m3 of CO2 initially at 150°C and 50 bar is to be isothermally compressed in a frictionless piston/cylinder device to a final pressure of 300 bar. Calculate the volume of the compressed gas, ΔU, the work done to compress the gas, and the heat flow on compression assuming
a. CO2 is an ideal gas.
b. CO2 obeys the Peng-Robinson equation of state.
8.23. Solve problem 8.22 for an adiabatic compression.
8.24. Consider problem 3.11 using benzene as the fluid rather than air and eliminating the ideal gas assumption. Use the Peng-Robinson equation. For the same initial state,
a. The final tank temperature will not be 499.6 K. What will the temperature be?
b. What is the number of moles left in the tank at the end of the process?
b. Write and simplify the energy balance for the process. Determine the final temperature of the piston/cylinder gas.
8.25. Solve problem 8.24 using n-pentane.
8.26. A tank is divided into two equal halves by an internal diaphragm. One half contains argon at a pressure of 700 bar and a temperature of 298 K, and the other chamber is completely evacuated. Suddenly, the diaphragm bursts. Compute the final temperature and pressure of the gas in the tank after sufficient time has passed for equilibrium to be attained. Assume that there is no heat transfer between the tank and the gas, and that argon:
a. Is an ideal gas
b. Obeys the Peng-Robinson equation
8.27. The diaphragm of the preceding problem develops a small leak instead of bursting. If there is no heat transfer between the gas and tank, what is the temperature and pressure of the gas in each tank after the flow stops? Assume that argon obeys the Peng-Robinson equation.
8.28. A practical application closely related to the above problem is the use of a compressed fluid in a small can to reinflate a flat tire. Let’s refer to this product as “Fix-a-flat.” Suppose we wanted to design a fix-a-flat system based on propane. Let the can be 500 cm3 and the tire be 40,000 cm3. Assume the tire remains isothermal and at low enough pressure for the ideal gas approximation to be applicable. The can contains 250 g of saturated liquid propane at 298 K and 10 bar. If the pressure in the can drops to 0.85 MPa, what is the pressure in the tire and the amount of propane remaining in the can? Assuming that 20 psig is enough to drive the car for a while, is the pressure in the tire sufficient? Could another tire be filled with the same can?
8.29. Ethylene at 30 bar and 100°C passes through a throttling valve and heat exchanger and emerges at 20 bar and 150°C. Assuming that ethylene obeys the Peng-Robinson equation, compute the flow of heat into the heat exchanger per mole of ethylene.
8.30. In the final stage of a multistage, adiabatic compression, methane is to be compressed from –75°C and 2 MPa to 6 MPa. If the compressor is 76% efficient, how much work must be furnished per mole of methane, and what is the exit temperature? How does the exit temperature compare with that which would result from a reversible compressor? Use the Peng-Robinson equation.
a. Ethane at 280 K and 1 bar is continuously compressed to 310 K and 75 bar. Compute the change in enthalpy per mole of ethane using the Peng-Robinson equation.
b. Ethane is expanded through an adiabatic, reversible expander from 75 bar and 310 K to 1 bar. Estimate the temperature of the stream exiting the expander and the work per mole of ethane using the Peng-Robinson equation. (Hint: Is the exiting ethane vapor, liquid, or a little of each? The saturation temperature for ethane at 1 bar is 184.3 K.)
8.32. Our space program requires a portable engine to generate electricity for a space station. It is proposed to use sodium (Tc = 2300 K; Pc = 195 bar; ω = 0; CP/R = 2.5) as the working fluid in a customized form of a “Rankine” cycle. The high-temperature stream is not superheated before running through the turbine. Instead, the saturated vapor (T = 1444 K, Psat = 0.828 MPa) is run directly through the (100% efficient, adiabatic) turbine. The rest of the Rankine cycle is the usual. That is, the outlet stream from the turbine passes through a condenser where it is cooled to saturated liquid at 1155 K (this is the normal boiling temperature of sodium), which is pumped (neglect the pump work) back into the boiler.
a. Estimate the quality coming out of the turbine.
b. Compute the work output per unit of heat input to the cycle, and compare it to the value for a Carnot cycle operating between the same TH and TC.
8.33. Find the minimum shaft work (in kW) necessary to liquefy n-butane in a steady-state flow process at 0.1 MPa pressure. The saturation temperature at 0.1 MPa is 271.7 K. Butane is to enter at 12 mol/min and 0.1 MPa and 290 K and to leave at 0.1 MPa and 265 K. The surroundings are at 298 K and 0.1 MPa.
8.34. The enthalpy of normal liquids changes nearly linearly with temperature. Therefore, in a single-pass countercurrent heat exchanger for two normal liquids, the temperature profiles of both fluids are nearly linear. However, the enthalpy of a high-pressure gas can be nonlinearly related to temperature because the constant pressure heat capacity becomes very large in the vicinity of the critical point. For example, consider a countercurrent heat exchanger to cool a CO2 stream entering at 8.6 MPa and 115°C. The outlet is to be 8.6 MPa and 22°C. The cooling is to be performed using a countercurrent stream of water that enters at 10°C. Use a basis of l mol/min of CO2.
a. Plot the CO2 temperature (°C) on the ordinate versus H on the abscissa, using H = 0 for the outlet state as the reference state.
b. Since dHwater/dx = dHCO2/dx along a differential length, dx, of countercurrent of heat exchanger, the corresponding plot of T versus H for water (using the inlet state as the reference state) will show the water temperature profile for the stream that contacts the CO2. The water profile must remain below the CO2 profile for the water stream to be cooler than the CO2. If the water profile touches the CO2 profile, the location is known as a pinch point and the heat exchanger would need to be infinitely big. What is the maximum water outlet temperature that can be feasibly obtained for an infinitely sized heat exchanger?
c. Approximately what water outlet temperature should be used to ensure a minimum approach temperature for the two streams of approximately 10°C?
8.35. An alternative to the pressure equation route from the molecular scale to the macroscopic scale is through the energy equation (Eqn. 7.50). The treatment is similar to the analysis for the pressure equation, but the expression for the radial distribution function must now be integrated over the range of the potential function. Suppose that the radial distribution function can be reasonably represented by:
g = 0 for r < σ
g ~ 1 + ρNAσ6ε/(r3kT) for r > σ
at all temperatures and densities. Use Eqn. 7.50 to derive an expression for the internal energy departure function of fluids that can be accurately represented by the following:
a. The square-well potential with λsw = 1.5
b. The Sutherland potential
Evaluate each of the above expressions at ρNAσ3 = 0.6 and ε/kT = 1.
8.36. Starting with the pressure equation as shown in Chapter 6, evaluate the internal energy departure function at ρNAσ3 = 0.6 and ε/kT = 1 by performing the appropriate derivatives and integrations of the equation of state obtained by applying
g = 0 for r < σ
g ~ 1 + ρNAσ6ε/(r3kT) for r > σ
at all temperatures and densities:
a. The square-well potential with λsw = 1.5
b. The Sutherland potential
c. Compare these results to those obtained in problem 8.35 and explain why the numbers are not identical.
8.37. Molecular simulation can be used to explore the accuracy and significance of individual contributions to an equation of state. Use the DMD module at Etomica.org to explore Xe’s energy departure.
a. The simulation results below have been tabulated at ηP = 0.167, λ = 1.7. Plot U/NAε versus βε for these data along with those at ηP = 0.375 from homework problem 7.25.
b. Prepare a plot of Xe’s simulated U – Uig versus 1000/T using your best ε and σ at ηP = 0.375 and showing the isochoric data for Xe from Webbook.nist.gov at 22.14 mol/L on the same axes.
c. The data for U – Uig exhibit a linear trend with βε. The data for Z also exhibit a linear trend with βε. What trends do these two data sets indicate for (A – Aig)TV/RT? Are they consistent? Explain.
d. Use the trapezoidal rule and the energy equation (Eqn. 7.49) to estimate A – Aig and plot as a dashed line. How accurate are your estimates (AAD%) and how could you improve them?
SW results at ηP = 0.167, λ = 1.7.
8.38. Suppose two molecules had similar potential functions, but they were mirror images of one another as shown in the figure below. Which one (A or B) would have the larger internal energy departure? You may assume that the radial distribution function is the same for both potential models.
a. Reason qualitatively but refer to appropriate equations to explain your answer.
b. Compute the values of (U – Uig)/RT at a packing fraction of 0.4 and a temperature of 50 K. Assume values of the radial distribution function as follows:
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