To assess the importance of the second term, the following properties of water are employed: AhM = 6008 J/mole-K at Tm = 273 K and ACp = Cpl - Cps = 37 J/mole-K. For T = 263 K (-10oC), Eq (3.27b) gives AgM = 220 J/mole while Eq (3.27a) yields AgM = 213 J/mole. The ~ 3% effect of the second term is small enough to warrant use of the simpler form given by Eq (3.27b) for applications requiring evaluation of AgM at temperatures other than the melting point.
In the right-hand graph of Fig. 3.7, the solid experiences a phase transition prior to melting. The solid-solid transition is a consequence of free-energy minimization just as is the melting process. Solid phase SI has a lower entropy than SII, and so is stable at low temperature. These two curves cross at the SI-SII phase-transition temperature, Ttr. The SII phase remains stable up to the melting temperature, where its free energy curve crosses the liquid curve.
3.1 A diatomic gas in a rigid container of 2 m3 volume initially at 293 K and 200 kPa is heated until it reaches 500 K. Although the p-v-t behavior of the gas is ideal, its heat capacity is the ideal-gas value plus a term aT, where a = 0.03 J/mole-K2. The temperature of the thermal reservoir that supplies heat to the system is 600 K.
(a) How much heat is absorbed during the process?
(b) What is the entropy change of the system? Why is it positive?
(c) What is the total entropy change (system + surrounding) during the process?
(d) Is the process reversible? If not, identify the source of the irreversibility.
3.2 One mole of a monatomic ideal gas at 273 K and 0.1 MPa is subjected to the following cycle, each step of which is conducted reversibly:
(a) isothermal increase in volume by a factor of 10
(b) adiabatic pressurization by a factor of 100
(c) return to the initial state along a straight-line path on the p-v diagram. Calculate the work and heat in each step and verify that the sum of the work values is equal to the sum of the heat additions.
3.3 Consider a frictionless, massless piston of area A in a cylinder containing a fixed quantity of air. The top of the piston is connected to a rigid fitting above the assembly by a spring with a force constant k (see figure). The system is moved reversibly from state 1 to state 2 by adding heat.
(a) What is the variation of p with V during this process? This relation may be expressed in terms of p1 and V1. Hint: a force balance on the piston is needed. Neglect psurr. The spring is initially compressed.
(b) How much work is done by the air inside the cylinder in moving from state 1 to state 2?
(c) How much energy is stored in the spring during the process? Is the stored energy equal to the work done by the system (Hint; you need to find the cylinder volume without gas when the spring is at its equilibrium length then derive the equation relating the system volume to the spring displacement).
(d) What is the final temperature? How much heat has been added to the gas?
3.4 Air in an insulated cylinder is reversibly compressed from a volume of 0.17 m3 to a volume of 0.034 m3 the initial temperature and pressure are 10oC and 0.2 MPa, respectively. For air, Cp and Cv can be taken to be 7R/2 and 5R/2, respectively.
(a) Calculate the work needed to accomplish this process.
(b) Calculate the final temperature of the air
(c) Show that the result of part (a) satisfies the First Law.
3.5 Two adiabatic vessels each containing a diatomic ideal gas are connected by a valve. The initial state is shown in the diagram, with pio > p20. The valve is opened and the gas in vessel 1 expands reversibly (as in the blowdown of a gas in a cylinder). Vessel 2 contains a frictionless, adiabatic piston which is pushed to the right as the gas in it is compressed reversibly. The final state is in mechanical equilibrium (pif = p2f = pi) but not in thermal equilibrium (Tif ^ Ti ^ T^).
Vessel 2
Vessel 1
T0 Pio Vi i"Ho moles
T0 Pio Vi i"Ho moles
T0 P20
n20 moles
Initial State
Final State (equilibrium)
Final State (equilibrium)
(a) Write all of the relevant equations which determine the parameters of the final state.
(b) Solve the equations with the initial conditions: Vi = 0.1 m3; V2 = 0.2 m3; To = 298
3.6 A monatomic ideal gas at temperature T1 and pressure p1 in an insulated container of volume V is connected by a valve to an insulated cylinder-piston that initially contains no gas. The valve is opened and the gas in the container flows into the cylinder. The massless piston rises reversibly against external pressure psurr until the volume of gas in the cylinder is Vc. At this point the valve is closed. The process is adiabatic and the gas that remains in the cylinder at the end can be assumed to have expanded reversibly. What fraction of the gas remains in the container at the end of the expansion and what are the temperatures of this gas and of that in the cylinder? Hint: use the results of the cylinder blowdown example.
3.7 Helium gas in a 1 m3 container is initially at 20oC. What quantity of heat transferred from the surroundings is required to raise the gas temperature by 1oC in the following three cases, in which the container is:
(a) rigid and initially at 1atm pressure
(b) rigid and initially at 3 atm pressure
(c) fitted with a frictionless piston and the pressure is maintained at 1 atm.
3.8 Two insulated tanks containing water are connected by a valve. The conditions in the tanks are:
Tank A: Va = 0.6 m3; pa = 200 kPa; Ta = 500oC Tank B: Vb = 0.3 m3; pb = 500 kPa; xb = 0.9 (quality)
The valve is opened and the two tanks achieve a uniform state. Assume that the process is adiabatic.
(a) Is the process reversible or irreversible? Give reasons
(b) What two thermodynamic properties of the system are conserved (i.e., remain unchanged) in this process?
(c) Is the final state a superheated vapor or a two-phase mixture.?
(d) What is the final pressure?
(e) What is the entropy change?
3.9 2000 kJ of heat are removed from a container at constant pressure. The container initially was filled with 1 kg of steam at 0.6 MPa and 400 oC. What is the final temperature and quality of the steam? To avoid a trial-and-error solution, use the p-h diagram for steam (below) and the steam tables.
3.10 For an isothermal process, what is the work required to compress a solid from a low pressure pi where the specific volume is vi to a higher pressure p2 where the specific volume is V2? The coefficient of compressibility of the solid is p and the equation of state is given in differential form by Eq (2.16a).
3.11 A 1.5 liter rigid vessel contains 30 g of water and no air or other gas. The initial state is liquid-vapor equilibrium at 25oC. Heat is added to the vessel until the water is just in the saturated-vapor state. What is the final temperature and how much heat must be added to reach the final state?
3.12 How much heat is required to melt 1 mole of a metal starting at 25oC? The process is conducted at 1 atm pressure, and the relevant properties of the metal are:
Cv = 24 J/mole-K; Cp = Cv + 0.01T; AhM = 10 kJ/mole; Tm = 1000 K.
3.13 A reversible process involving 1 mole of a diatomic ideal gas consists of two steps. In step 1, starting from an initial state at 200oC and 0.3 MPa, heat is added at constant pressure until the temperature reaches 400oC. In step 2, the gas is expanded adiabatically until the temperature returns to 200oC.
(a) How much heat is added in Step 1?
(b) How much work is done in both steps combined?
3.14 One mole of an ideal gas is contained in a cylinder fitted with a massless, frictionless piston to which a spring is attached. The gas is initially at 0.1 Mpa pressure and 298 K.
(a) Initially the spring is at its equilibrium length and so exerts no force on the piston. What is the height Ho of the gas in the cylinder?
(b) The cylinder is heated until the height of the piston increases from Ho to 0.25 m. What are the final pressure and temperature of the gas?
3.15 The heat capacity of a metal is given by: C = 22.6 + 6.6 x 10-3T J/mole-K where T is the temperature in K. Two 1-mole pieces of the metal, one at TA0 = 873 K and the other at TB0 = 573K, are placed in an adiabatic container maintained at constant pressure.
(a) What process occurs within the container? Is it spontaneous and why?
(b) Derive the equation for TB as a function of TA during the process. Sketch this function.At equilibrium, what is the common final temperature of the two pieces?
(c) Derive the equation for the entropy change of the contents of the container as a function of TA. Sketch this function. At what temperature is the entropy change a maximum? What is the entropy increase of the system at equilibrium?
(d) What is the entropy change of the surroundings during this process? How does the entropy of the universe increase as a result of this process?
3.16 A rigid container with a volume of 0.2 m3 is divided into two equal volumes by a partition. Both sides contain nitrogen, volume A at 2 Mpa and 200oC and volume B at 200 kpa and 100oC. The partition is ruptured and the gas comes to equilibrium at 70oC, which is the temperature of the surroundings. Determine the work done and the entropy changes of the system and the surroundings during the process. Is the process reversible? Assume N2 behaves ideally.
3.17 The following problem is analogous to the irreversible expansion of an ideal gas shown in Fig.1.17. The object is to demonstrate that attainment of equilibrium resulting from fluid expansion in an isolated system always occurs with an increase in entropy, no matter what substance is involved.
Consider 1 kg of liquid water initially at 120oC and 5 MPa pressure held in the small section of rigid adiabatic container separated by a membrane from a section 100 times larger. The larger section is initially a vacuum. The membrane is ruptured and the water uniformly fills the entire container. (a) What is the final state of the water?
(c) Show that the entropy of the system has increased in the process of changing from the initial state to the equilibrium state.
3.18 An ideal diatomic gas is contained in two tanks connected by a valve. Tank A is 600 liters in volume, insulated, and initially has gas at 1.4 MPa and 300oC. Tank B is not insulated, is 300 liters in volume, and initially contains gas at 200 kpa and 200oC.
The valve is opened and gas flows from tank A to tank B until the temperature in tank A reaches 250oC, at which time the valve is closed. During this process, heat transfer from tank B to the surroundings(at 25oC) maintains the gas in this tank at 200oC at all times. Assuming that the gas remaining in tank A has undergone a reversible adiabatic expansion, determine the following:
(a) The final pressure in tank A
(b) The number of moles of gas transferred from tank A to tank B and the final pressure in tank B.
(c) The entropy change of the gas and that of the gas plus the surroundings.
3.19 This problem is an irreversible version of Problem 3.3. An ideal gas is contained in a cylinder by a piston of area A. A spring is connected at one end to the piston and at the other to a rigid fitting in the cylinder housing. The force constant of the spring is denoted by k. The spring and cylinder are encased in a rigid, adiabatic container, thus forming an isolated system. The heat capacities of the spring, the piston, and the cylinder walls are negligible compared to that of the gas. Initially, the gas conditions are po, To, and vo and the spring is at its equilibrium length.The piston is held in place by blocks, which are removed to allow the gas to expand against the spring. In the final state, the gas has expanded to a volume v and its final pressure and temperature are P and T, respectively.
(a) Show that the final values of the ratios v/vo and T/To are functions of the specific heat ratio of the gas y = CP/Cv and a dimensionless parameter a = A2P0/kv0.
(b) For y = 5/3 and a = 1, what is the final state of the gas?
(c) Has the gas expanded isentropically in this process?
3.20 An ideal monatomic gas in a closed cylinder is separated into two parts by a frictionless piston that is held in place by stops.
3.20 An ideal monatomic gas in a closed cylinder is separated into two parts by a frictionless piston that is held in place by stops.
The initial conditions of the gas in the two sections are:
pio = 2 atm, V10 = 1 cm3, T10 = 400 K p2o = 1 atm, V20 = 3 cm3, T20 = 3 00 K.
The cylinder walls and the piston are perfect insulators, so that no heat is transferred from one section of the gas to the other, or between the surroundings and the gas. By removing stops holding the piston in place, the system is allowed to achieve mechanical equilibrium at a final pressure pf. Determine the state of the gas in the two parts of the cylinder after pressure equilibrium has been achieved.
Hint: derive equations for: the total volume, the number of moles in each compartment, the 1st law, and the entropy increase . The principle of maximum entropy in an isolated system needs to be used. A trial-and-error solution is required.
3.21 Pressure equilibration in the process described in Problem 3.20 can be conducted reversibly (but still adiabatically ) by allowing the piston to do work on the surroundings via a shaft.
(a) What is the final common pressure of the gas in the two chambers?
(b) How much net (external) work is done by the gas?
3.22 The heat capacity of a solid is CPS = A + BT. Its liquid phase heat capacity is a constant CPL. The heat of fusion is AhM and the melting temperature is TM. What is the change in entropy per mole when heated from T1 as a solid to T2 in the liquid region?
3.23 An ideal monatomic gas undergoes the cycle shown below.
(a) What are the "iso" constraints on each process?
(b) What pressure p3 assures that the 3 ^ 1 step returns the system to its initial state?
(c) What is the net work of the cycle?
(a) complete the following table:
State v x104. m3/mole |
p. MPa |
T. K s/R |
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