Safari Books Online is a digital library providing ondemand subscription access to thousands of learning resources.
8.1  What forms does the derivative (∂C_{V}/∂V)_{T} have for a van der Waals gas and a RedlichKwong gas? (The RedlichKwong equation is given in Problem P8.1.) Comment on the results.  
8.2  Estimate C_{P}, C_{V}, and the difference C_{P} – C_{V} in (J/molK) for liquid nbutane from the following data.^{[2]}
 
8.3  Estimate C_{P}, C_{V}, and the difference C_{P} – C_{V} in (J/molK) for saturated nbutane liquid at 298 K nbutane as predicted by the PengRobinson 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.  
8.5 
 
8.6  The SoaveRedlichKwong 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 constantpressure idealgas heat capacity of 15R. The gas follows the equation of state,
over the range of interest, where a = –1000 cm^{3}/mole.
 
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 PCSAFT model.
 
8.12  Recent research in thermodynamic perturbation theory suggests the following equation of state.
 
8.13  A gas is to be compressed in a steadystate 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 cm^{3}K/mol and b = 15.23 cm^{3}/mol. Calculate the required work per mole of gas.  
8.14  A 1 m^{3} 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 cm^{3}/mole; a = 40,000 cm^{3}K/mole; and C_{P} = 41.84 + 0.084T(K) J/molK.
 
8.15  PVT behavior of a simple fluid is found to obey the equation of state given in problem 8.14.
 
8.16  Using the PengRobinson equation, estimate the change in entropy (J/moleK) 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 T_{2} = T_{1}(P_{2}/P_{1})^{R/C}P by asserting that the estimated T_{2} should be within 5% of the one calculated using the departure functions. For ω = 0 and T_{r} = [1, 10] at state 1, determine the values of P_{r} where this assertion holds valid by using the PengRobinson 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 PengRobinson 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. Vargaftik^{3} (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
 
8.20  npentane 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 PengRobinson 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 PengRobinson 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.
 
8.22  1 m^{[3]} of CO_{2} 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
 
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 PengRobinson equation. For the same initial state,
 
8.25  Solve problem 8.24 using npentane.  
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:
 
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 PengRobinson 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 “Fixaflat.” Suppose we wanted to design a fixaflat system based on propane. Let the can be 500 cm^{3} and the tire be 40,000 cm^{3}. 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 PengRobinson 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 PengRobinson equation.  
8.31 
 
8.32  Our space program requires a portable engine to generate electricity for a space station. It is proposed to use sodium (T_{c} = 2300 K; P_{c} = 195 bar; ω = 0; C_{P}/R = 2.5) as the working fluid in a customized form of a “Rankine” cycle. The hightemperature stream is not superheated before running through the turbine. Instead, the saturated vapor (T = 1444 K, P^{sat} = 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.
 
8.33  Find the minimum shaft work (in kW) necessary to liquefy nbutane in a steadystate 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 singlepass countercurrent heat exchanger for two normal liquids, the temperature profiles of both fluids are nearly linear. However, the enthalpy of a highpressure 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 CO_{2} 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 CO_{2}.
 
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 + ρN_{A}σ^{6}ε/(r^{3}kT) 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:
Evaluate each of the above expressions at ρN_{A}σ^{3} = 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 ρN_{A}σ^{3} = 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 + ρN_{A}σ^{6}ε/(r^{3}kT) for r > σ at all temperatures and densities:
 
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.
SW results at η_{P} = 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.
