Chapter 6: Thermodynamic Properties of Pure Fluids (Solution)
Problem 6.1: Show that
(a)\;\;C_P=T\frac{(\partial V/\partial T)_P}{(\partial T/\partial P)_S} (b)\;\;dH=C_PdT+\left[V-\frac{(C_P-C_V)\kappa}\beta\right]dP (c)\;\;T{\left(\frac{\partial P}{\partial T}\right)}_V-P={\left(\frac{\partial U}{\partial V}\right)}_T (d)\;\;{\left(\frac{\partial U}{\partial P}\right)}_T=-\left[T\;{\left(\frac{\partial V}{\partial T}\right)}_P+P\;{\left(\frac{\partial V}{\partial P}\right)}_T\right]Problem 6.2: Maxwell’s equation can be used to evaluate the latent heat of vaporization of a pure substance. Choose the appropriate equation and explain the method to estimate the latent heat of vaporisation as a function of temperature using the vapor pressure data.
Problem 6.3: Show that for a gas obeying the van der Waals equation (∂CV/∂V)T = 0.
Problem 6.4: A gas is found to obey the equation of state P(V – b) = RT. Show that its CP doesn’t change with changes in pressure at a constant temperature.
Problem 6.5: Define Joule–Thomson coefficient and explain how it could be used for determining the heat capacity of gases.
Problem 6.6: Show that
\mu=\frac{RT^2}{PC_P}{\left(\frac{\partial Z}{\partial T}\right)}_Pwhere Z is the compressibility factor. ‘The Joule–Thomson coefficient is negative at pressures greater than about 9PC, where PC is the critical pressure.’ Justify.
Problem 6.7: Show that for a gas obeying the equation of state PV(1 – bP) = RT,
(a)\;\;\triangle G=RT\ln\frac{P_2(1-bP_1)}{(P_1(1-bP_2)} (b)\;\;\triangle A=RT\ln\frac{P_2(1-bP_1)}{(P_1(1-bP_2)}-\frac{RT}{\left(1-bP_2\right)}+\frac{RT}{\left(1-bP_1\right)}Problem 6.8: With the help of Maxwell equations prove that the specific heats of ideal gases are functions of temperature only.
Problem 6.9: Using the method of Jacobians derive a relationship for the ratio of heat capacities CP and CV. Show that the slope of P-V diagram for a reversible adiabatic process is g times that for a reversible isothermal process.
Problem 6.10: Using the method of Jacobians shows that
(a)\;\;{\left(\frac{\partial H}{\partial S}\right)}_V=T\left(1+\frac{V\beta}{C_V\kappa}\right) (b)\;\;{\left(\frac{\partial G}{\partial V}\right)}_T=-\frac1\kappa (c)\;\;{\left(\frac{\partial H}{\partial S}\right)}_T=T-\frac1\beta (d)\;\;{\left(\frac{\partial P}{\partial T}\right)}_V=\frac\beta\kappa (e)\;\;-{\left(\frac{\partial P}{\partial V}\right)}_T={\left(\frac{\partial P}{\partial T}\right)}_V{\left(\frac{\partial T}{\partial V}\right)}_P (f)\;\;\kappa_S=\kappa-\frac{VT\beta^2}{C_P}Where \kappa_S=-(1/V)(\partial V/\partial P)_S , the adiabatic compressibility.
(g)\;\;{\left(\frac{\partial U}{\partial V}\right)}_T=T{\left(\frac{\partial P}{\partial T}\right)}_V-P (h)\;\;\frac{T{\left(\partial V/\partial P\right)}_P}{{\left(\partial T/\partial P\right)}_S} (i)\;\;{\left(\frac{\partial P}{\partial V}\right)}_H=\frac{C_P}{\beta^2V^2T-C_VV\kappa-V^2\beta}Problem 6.11: A pure gas flowing at a low rate through a well-insulated horizontal pipe at high pressure is throttled to a slightly lower pressure. The gas obeys the equation of state P(V – b) = RT, where b is a positive constant. Does the gas temperature rise or fall by throttling?
Problem 6.12: The following relationship between fugacity and pressure has been proposed: f = P + aP2, where a is a function of temperature only. Find an equation of state for the gas conforming to this relation. Is the equation of state realistic? Explain.
Problem 6.13: Derive an expression for the fugacity coefficient of a gas obeying the following equation of state
P=\frac{RT}{V-b}\frac a{T^{0.5}(V+b)V}where a and b are empirical constants.
Problem 6.14: Using the generalized compressibility factor method, explain how you would make a generalized condition of enthalpy departure for a range of pressures.
Problem 6.15: Describe the T-S diagram of a pure fluid. Given the equation of state or the generalized compressibility charts, explain how you would construct the T-S diagram. Give the relevant thermodynamic relations and the reference states used. What additional data are necessary?
Problem 6.16: The melting point of benzene is found to increase from 278.5 K to 278.78 K, when the external pressure is increased by 100 bar. The heat of fusion of benzene is 128 kJ/kg. What is the change in volume per kg accompanying the fusion of benzene?
Problem 6.17: Carbon tetrachloride boils at 349.75 K at 1 bar. Its latent heat of vaporization is 194.8 kJ/kg. What would be the boiling point of carbon tetrachloride at 2 bar?
Problem 6.18: The variation of the vapor pressure of benzene with temperature is given in the table below.
| T (K) | 280.6 | 288.4 | 299.1 | 315.2 | 333.6 | 353.1 |
| P (×102 bar) | 5.33 | 8.00 | 13.30 | 26.70 | 53.33 | 100 |
Estimate the latent heat of vaporization of benzene and its vapor pressure at 393 K. Is it possible to approximate the vapor pressure as an exponential function of temperature? Does the Clapeyron equation suggest such an approximation?
Problem 6.19: If the pressure inside a pressure cooker is 200 kPa, what is the boiling point of water inside it? The normal boiling point of water is 373 K and the latent heat of vaporization of water is 2257 kJ/kg at 373 K.
Problem 6.20: The vapor pressure and molar volume of water as a function of temperature is given below:
| T (K) | 490 | 500 | 510 |
| PS (kPa) | 2181 | 2637 | 3163 |
| VG (m3/kg) | 91.50×10-3 | 75.85×10-3 | 63.23×10-3 |
| VL (m3/kg) | 1.18×10-3 | 1.20×10-3 | 1.22×10-3 |
Calculate the latent heat of vaporisation of water at 500 K using (a) Clapeyron equation; (b) Clausius–Clapeyron equation.
Problem 6.21: (a) Prove the following, where m is the Joule–Thomson coefficient.
{\left(\frac{\partial C_P}{\partial P}\right)}_T=-\mu{\left(\frac{\partial C_P}{\partial P}\right)}_P-C_P{\left(\frac{\partial\mu}{\partial T}\right)}_P(b) If m = – 0.1975 + 138/T – 319 P/T2 K/bar, and CP = 6.557 + 6.1477×10–2T – 2.148×10–7T2 kJ/kmol K, evaluate the derivative (∂CP/∂P)T at 1 bar and 333 K when T is in K and P is in bar.
Problem 6.22: The volume coefficient of expansion of water at 373 K is 7.8×10–4 K–1. Calculate the change in entropy when the pressure is increased from 1 bar to 100 bar. At 373 K, the density of water is 958 kg/m3.
Problem 6.23: Calculate the change in enthalpy, entropy, and internal energy when 1 mol liquid water at 273 K and 1 bar is converted into steam at 473 K and 3 bar. List the assumptions used. Data: At 1 bar the specific heat of steam is CP = 37.002 – 8.00×10–3T + 9.24×10–6T2, where CP is in kJ/kmol and T is in K. Enthalpy of vaporization at 373 K = 40.6 kJ/kmol.
Problem 6.24: Calculate the change in internal energy, enthalpy, entropy, and free energy when one kmol hydrogen gas at 300 K and 1 bar is heated and compressed to 500 K and 100 bar. The entropy of hydrogen in the initial state is 131.5 kJ/kmol K. Enthalpy at 273 K may be taken to be zero. Assume CP = 27.3 + 4.2×10–3T at 1 bar where CP is in kJ/kmol K and T is in K. Hydrogen may be treated as an ideal gas.
Problem 6.25: Calculate the enthalpy and entropy of isobutane vapor at 360 K and 15.6 bar from the following data: Enthalpy and entropy for saturated liquid at 290 K are both zero. The average specific heat of isobutane liquid between 290 K and 295 K is 2.34 kJ/kg K. The heat of vaporisation at 295 K is 335 kJ/kg. The vapor pressure of isobutane is 3.1 bar at 295 K. The specific heat of the gas at 3.1 bar varies with temperature as given below:
| T (K) | 295 | 310 | 328 | 345 | 360 |
| CP (kJ/kg K) | 1.806 | 1.806 | 1.777 | 1.806 | 1.856 |
The specific volumes of gaseous isobutane in m3/kg are as follows:
| P (bar) | 310 K | 328 K | 345 K | 360 K | 378 K |
| 2.7 | 0.151 | 0.160 | 0.169 | 0.179 | 0.188 |
| 4.1 | 0.096 | 0.103 | 0.110 | 0.116 | 0.122 |
| 6.8 | —- | 0.057 | 0.062 | 0.066 | 0.070 |
| 10.2 | —- | —- | 0.037 | 0.041 | 0.044 |
| 13.6 | —- | —- | —- | 0.028 | 0.031 |
| 15.6 | —- | —- | —- | 0.023 | 0.026 |
Problem 6.26: Calculate CP – CV for CO2 at 1 bar and 273 K given that the van der Waals constants for CO2 are a = 0.365 J m3/mol2 and b = 42.8×10–6 m3/mol.
Problem 6.27: Calculate the CP of CO2 gas at 100 bar and 373 K given that CP at 1 bar and 373 K is 40.6 J/mol K. The van der Waals constants are a = 0.365 J m3/mol2 and b = 42.8×10–6 m3/mol. [Hint:
\triangle C_P=-\int_1^{100}T(\partial^2V/\partial T^2)_P\operatorname dPAssume various values of P and find (∂2V/∂T2)P in each case. Plot –T(∂2V/∂T2)P against P and find the area under the curve between P = 1 and P = 100 bar. The area thus determined equals ∆CP].
Problem 6.28: Wet steam at 20 bar is throttled to a pressure of 1.5 bar and a temperature of 411 K. What was the initial quality of steam? Take the data from the steam tables.
Problem 6.29: Calculate (∂U/∂P)T, (∂H/∂P)T and m of a substance at 298 K and 1 bar, if the following data are given: CP = 138 kJ/kmol K, V = 0.09 m3/kmol, (∂V/∂T)P = 9.0×10–8 m3/kmol K and (∂V/∂P)T = – 9.0×10–9 m3/kmol bar.
Problem 6.30: Show that for any gas whose volume varies linearly with temperature at a given pressure, the Joule–Thomson coefficient is zero.
Problem 6.31: At 200 K, the compressibility factor of oxygen varies with pressure as given below. Evaluate the fugacity of oxygen at this temperature and 100 bar.
| P (bar) | 1.00 | 4.00 | 7.00 | 10.00 | 40.00 | 70.00 | 100.00 |
| Z | 0.99701 | 0.98796 | 0.97880 | 0.96956 | 0.8734 | 0.7764 | 0.6871 |
Problem 6.32: (a) Calculate the fugacity of CO at 50 bar and 400 bar, if the following data are applicable at 273 K.
| P (bar) | 25 | 50 | 100 | 200 | 400 | 800 | 1000 |
| Z | 0.9890 | 0.9792 | 0.9741 | 1.0196 | 1.2482 | 1.8057 | 2.0819 |
(b) Using the values of Z at 50 and 400 bar, calculate the van der Waals constants for CO. From these determine the fugacities at these pressures and compare the results with the previous ones.
Problem 6.33: Calculate the fugacity of methane gas at 322 K and 55 bar, given that the critical constants are 190.7 K and 46.4 bar.
Problem 6.34: Calculate the fugacity of nitrogen at 800 bar from the following data at 273 K.
| P (bar) | 50 | 100 | 200 | 400 | 800 | 1000 |
| PV/RT | 0.9846 | 0.9846 | 1.0365 | 1.2557 | 1.7959 | 2.0641 |
Problem 6.35: Find the per cent increase in the fugacity of gaseous oxygen per degree rise in temperature in the neighbourhood of 298 K and 200 bar, if under these conditions, the Joule–Thomson heat is 1457 J/mol. (Joule–Thomson heat = H0 – H.)
Problem 6.36: Estimate the fugacity of gaseous propane at 12 bar and 310 K using the following data.
| P (bar) | 1.7 | 3.4 | 6.8 | 10.2 | 11.7 | 13.6 | 34 |
| V (m3/kg) | 0.3313 | 0.1609 | 0.0754 | 0.0468 | 0.0382 | 0.021 | -0.00207 |
Also, compute the fugacity of propane at 310 K and 70 bar given that the vapour pressure of propane at 310 K is 13 bar.
Problem 6.37: The experimental pressure-volume data for benzene at 675 K from very low pressures up to about 75 bar may be approximated by the equation V = 0.0561(1/P – 0.0046), where V is in m3/mol and pressure P is in bar. What is the fugacity of benzene at 1 bar and 675 K?
Problem 6.38: In the previous problem what is the fugacity of benzene at 75 bar and 675 K assuming that the fugacity of benzene at 1 bar is unity?
Problem 6.39: (a) The compressibility of a gas may be represented by PV/RT = A + BP + CP2 + DP3, where A, B, C, and D are functions of temperature and P is measured in bar. Derive an expression for fugacity as a function of pressure at a given temperature. (b) Evaluate the fugacity of nitrogen at 273 K and 300 bar, given that at 273 K the constants in the equation of state given in part (a) are A = 1.00, B = –5.314×10–4, C = 4.276×10–6 and D = –3.292×10–9.
Problem 6.40: (a) Find an expression for the fugacity coefficient of a gas that obeys the equation of state
\frac{PV}{RT}=1+\frac BV+\frac C{V^2}(b) Use the result in part (a) to estimate the fugacity of argon at 1.00 bar and 273 K if the constants B and C are respectively –21.13×10–6 m3/mol and 1054×10–12 m6(mol)–2.
Problem 6.41: Derive an expression for the fugacity coefficient of a gas obeying the equation of state z = a + bP + cP2, where P is in bar. Determine fugacity of oxygen at 293 K and 100 bar, given that a = 1.0; b = –0.753×10–3, and c = 0.15×10–5.