5.9 The melting point of iodine is 113oC. The vapor pressures of the solid and liquid states are given by lnpsat = A - B/T + DlnT, where:
Bs = 8240, Bl = 7381, Ds = -2.51, Dl = 5.18, As = 34.16, and Al = 47.83
(a) Calculate the triple point temperature and pressure of iodine. Why does this result differ from the reported melting temperature?
(b) Consider iodine vapor initially at 0.04 atm and 150oC. For the following processes, determine which condensed phase first appears and at what p and T. Sketch the p-T projection for iodine and identify the paths of the two processes. Assume that the vapor obeys the ideal gas law.
(c) The initial vapor is isothermally compressed.
(d) The initial vapor is cooled at constant volume.
5.10 At 800 K and 1 atm pressure, the Gibbs free energy of a liquid is 1 kJ/mole greater than that of the solid phase. The heat of fusion of the substance is 25 kJ/mole and the heat capacities of the liquid and solid are equal.
(a) What is the melting point of the substance at 1 atm pressure?
(b) If the molar volume of the liquid is 1 cm3/mole smaller than that of the solid, at what pressure does the substance melt at a temperature of 800 K?
5.11 It is an axiom of thermodynamics that the change in a property during a process from state 1 to state 2 is independent of the path followed. This is to be demonstrated for the following process: 1 mole of a superheated vapor at temperature T1 is cooled at constant pressure to the saturation temperature T2 and further cooled to condense nL moles of liquid. The enthalpy difference h2 - h1 is to be calculated for this path and for an alternate (hypothetical) path. The second path is condensing nL moles of liquid from the vapor at a constant temperature T1 then cooling the two-phase mixture to T2.
(a) Draw a diagram showing these two paths like the one in Problem 5.8
(b) Derive the equations for h2 - h1 for the two paths and show that they are identical. Assume that the heat capacities of the vapor and the liquid are temperature independent and that the heat of vaporization varies with T according to Eq (5.7).
5.12 In the earth's crust, CaCO3 occurs in two crystalline modifications, calcite(C) and aragonite(A). At very high pressures (as occurs deep below the surface), the effect of total pressure on the Gibbs free energies of these two mineral forms must be considered. The free energy change corresponding to the reaction A ^ C is given by:
gC - gA = -50 + 9x10-3T + 0.07(p - 1) J/mole where T is in Kelvins and p is in atm.
(a) Identify each of the numerical terms in the above equation in terms of thermodynamic quantities characterizing the transformation from A to C. For the last term, use the equation on the bottom of p. 12 of the reader, with AsA-C independent p.
(b) Which phase is stable at the earth's surface?
(c) The earth's temperature increases linearly with depth with a gradient of 1 K/m. A lithostatic pressure gradient is also created by a rock density of 5 g/cm3. At what depth does the form stable at the surface transform to the other crystal form? Neglect the effect of temperature on AhA-C.
5.13 The line containing point A in the carbon p-T diagram in the slides (or Fig. 5.7 of the reader) shows that the pressure required for the graphite-to-diamond transition increases with temperature. At 14.5 kbar and 300 K, compare the slope dpg-d/dT obtained from the plot and that obtained from the Clapyron equation. Assume that
Avg-d = -2x10-6 m3/mole is independent of temperature and pressure. Account for the effect of pressure on Ahg-d. Neglect the effect of pressure on the entropy of transition. At 1 atm,
5.14 The constant-pressure cooling of an air-water vapor mixture from 308 K, 70% relative humidity, to 278 K, two-phase, was worked out in class. Calculate the system volume at the beginning and end of each step. Assume that the system contains one mole of air and that the total pressure is 1 atm.
5.15 At constant temperature, the vapor pressure of a pure substance (liquid), psat, is slightly dependent on the total pressure p. The latter can be increased by adding an inert gas to the vessel containing the liquid and vapor at equilibrium. Use the method of Sect.
5.3 to determine the effect of p on psat. Recognize that because the gas is a mixture of the inert species and the vapor, dgg must be replaced by d^g, where is the chemical potential of the vapor in the gas phase.
(a) If the vapor pressure with no inert gas present is p°sat, what is the vapor pressure if inert gas is added to raise the total pressure to p?
(c) What is the percentage change in the vapor pressure of water from the steam-tables value at 25oC (gas phase is pure vapor) when liquid water is in equilibrium with water vapor in air at a total pressure of 1 atm?
5.16 A sealed, rigid container of volume V is partially filled with a mass m of liquid water. The remainder of the volume is filled with air. Initially, the system temperature is To, its pressure is po and the air has zero relative humidity. A quantity Q of heat is added to the contents of the vessel.
(a) Derive the equations from which the final temperature T, pressure p, and steam quality x can be derived. The vapor pressure of water is given by Eq (8.14). Assume that the specific heats of air and liquid water are known. The internal energy change upon vaporization can be approximated by Ahvap.
(b) Solve for T, p, and x using the following input values:
V = 1 liter; m = 200 g; T = 25oC; p, = 1 atm; Q = 50.8 J; Estimate the heat capacities How is the input heat partitioned among sensible heat to the moist air and the heat for vaporization of water?
(c) Solve the problem using the steam tables instead of the equations developed in part (a).Use the properties given in (b).
5.17 A condenser operates as a steady-flow device, receiving an input vapor with an inert gas(air) molar flow rate n t at a temperature To and a relative humidity The condenser extracts heat from the gas at a rate Q and the exit gas is at temperature Te. The unit operates at a constant total pressure p.
(a) Derive the equations for:
i) The unit's heat flux and the exit temperature.
ii) The fraction of the water vapor in the inlet gas condensed in the unit.
The saturation pressure curve is given by Eq (8.14). Use A and B for the constants in the derivation.
(b) Solve the equations of part (a) for the heat flux Q for the following conditions:
n : = 0.2 moles/s; ^ = 85%; To = 105oC; p = 2 atm; Te = 75oC. The heat capacities are to be taken from Fig. 2.3.
5.18 An air-water vapor mixture is contained in a closed volume initially at 1 atm total pressure and 7oC. The relative humidity of the mixture is 50%. The vessel is heated at constant volume to 25.7oC. Calculate the following:
(a) The dew point of the air in the vessel before heating (state 1)
(b) The relative humidity and dew point of the mixture after heating (state 2)
(c) The heat absorbed per mole of gas.
The specific heats at constant pressure for air and water vapor are 3.5R and 4R, respectively. The gas components can be considered to be ideal.
5.19 Saturated air initially at 1 atm total pressure and 40oC is heated. What is the relative humidity at 50oC if the heating is conducted at:
(a) constant volume
(b) constant pressure hint: do not use the steam tables for this problem. Instead, assume that air and water vapor behave ideally
5.20 At the end of the day, the temperature is 30oC and the relative humidity is 30%. By 3 AM, the temperature has dropped to 20oC. What is the relative humidity at this time?
5.21 The triple point of CO2 is -57oC, 5 atm. At -78oC, the vapor pressure of the solid is 1 atm. The enthalpy of melting is 8300 J/mole. Determine:
(a) the enthalpy of vaporization, Ahvap
(b) the vapor pressure of the liquid at -23oC
5.22 An insulated, constant-pressure vessel contains moist air at 1 atm, 21oC and 50% relative humidity. 0.003 moles of liquid water at 15oC are injected into the vessel. Find the final temperature and relative humidity.
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