Exercise 2.18
a. Using Eq. (2.4-27) find the final pressure for the adiabatic expansion of Example 2.17. Use the ideal gas law to find the initial pressure.
b. Use the ideal gas law and the final temperature from Example 2.17 to find the final pressure.
Exercise 2.20
Since dw >−PdV for an irreversible compression process, show that the final temperature for an irreversible adiabatic compression of an ideal gas must be higher than the final temperature of the reversible adiabatic compression with the same initial state and the same final volume.
P R O B L E M S
Section 2.4: Calculation of Amounts of Heat and Energy Changes
2.23 a. Calculate the Joule coefficient for carbon dioxide at 298.15 K, using the truncated virial equation of state.
ApproximatedB2/dT by the quotient of finite differences of the values for 0◦C and 50◦C. Use the value ofCP,min Table A.8 to obtain a value ofCV,m. State any assumptions or approximations.
b. Find the final temperature if 1.000 mol of carbon dioxide at 298.15 K and 33 atm is expanded into an evacuated vessel with a volume of 20.00 L.
2.24 A sample of 3.00 mol of argon is heated from 25.00◦C to 100.00◦C, beginning at a pressure of 1.000 atm
(101,325 Pa).
a. Findq,w, and∆Uif the heating is done at constant volume.
b. Findq,w, and∆Uif the heating is done at constant pressure.
2.25 Find the final pressure if 2.000 mol of nitrogen is expanded adiabatically and reversibly from a volume of 20.00 L to a volume of 40.00 L, beginning at a pressure of 2.500 atm. Assume nitrogen to be ideal with
CV,m5R/2.
2.26 A sample of 1.000 mol of neon gas is expanded from a volume of 5.000 L and a temperature of 400.0 K to a volume of 8.000 L.
a. Find the final temperature if the expansion is adiabatic and reversible. Assume that the gas is ideal and that CV3nR/2constant.
b. Find∆U,q, andwfor the expansion of part a.
c. Find∆U,q,w, and the final temperature if the expansion is adiabatic but at a constant external pressure of 1.000 atm, starting from the same state
as in part a and ending at the same volume as in part a.
d. Find∆U,q, andwif the expansion is reversible and isothermal, starting at the same state as in part a and ending at the same volume as in part a. State any assumptions and approximations.
2.27 Find the final temperature and the final volume if 2.000 mol of nitrogen is expanded adiabatically and reversibly from STP to a pressure of 0.600 atm. Assume nitrogen to be ideal withCV,m5R/2.
2.28 1.000 mol of carbon dioxide is expanded adiabatically and reversibly from 298.15 K and a molar volume of 5.000 L mol−1to a volume of 20.00 L mol−1.
a. Find the final temperature, assuming the gas to be ideal withCV,m5R/2constant.
b. Find the final temperature, assuming the gas to be described by the van der Waals equation with CV,m5R/2constant.
2.29 A sample of 20.00 g of acetylene, C2H2, is expanded reversibly and adiabatically from a temperature of 500 K and a volume of 25.00 L to a volume of 50.00 L. Use the value ofCV,mobtained from the value in Table A.8 for 500 K with the assumption that acetylene is an ideal gas.
a. Find the percent difference between this value ofCV,m
that you obtain and 5R/2.
b. Find the final temperature.
c. Find the values of∆U,q, andwfor the process.
2.30 a. A sample of 2.000 mol of argon gas is adiabatically and reversibly expanded from a temperature of 453.15 K and a volume of 15.0 L to a final
temperature of 400.0 K. Find the final volume, ∆U,
w, and qfor the process. Assume argon to be ideal and assume thatCV,m 3
2R.
b. Consider an irreversible adiabatic expansion with the same initial state and the same final volume, carried out withP(transferred)1.000 atm. Find the final temperature,∆U,w, andqfor this process.
2.31 a. Find the final temperature,∆U,q, andwfor the reversible adiabatic expansion of O2gas from 373.15 K and a molar volume of 10.00 L to a molar volume of 20.00 L. Assume the gas to be ideal with
CV,m 5R/2.
b. Repeat the calculation of part a for argon instead of oxygen. Assume thatCV,m3R/2.
c. Explain in physical terms why your answers for parts a and b are as they are.
2.32 a. 2.000 mol of O2gas is compressed isothermally and reversibly from a pressure of 1.000 atm and a
temperature of 100.0◦C to a pressure of 3.000 atm. Find
∆U,q,w, and∆Hfor this process. State any
assumptions or approximations. Assume that the gas is ideal.
b. The same sample is compressed adiabatically and reversibly from a pressure of 1.000 atm and a
temperature of 100.0◦C to a pressure of 3.000 atm. Find
∆U,q, andwfor this process. State any assumptions or approximations. Assume thatCV,m5RT /2 and that the gas is ideal.
2.33 For each of the following monatomic gases, calculate the percent difference between the tabulated value ofCP,min Table A.8 in the appendix and 5R/2 for each temperature in the table.
a. Ar b. H c. He d. O e. C
2.34 For each of the following diatomic gases, calculate the percent difference between the tabulated value ofCP,min Table A.8 in the appendix and 7R/2 for each temperature in the table.
a. CO b. O2
c. NO d. HCl
2.35 A sample of 1.000 mol of water vapor originally at 500.0 K and a volume of 10.0 L is expanded reversibly and adiabatically to a volume of 20.0 L. Assume that the water vapor obeys the van der Waals equation of state and that its heat capacity at constant volume is described by
Eq. (2.4-25) withα22.2 J K−1mol−1and β10.3×10−3J K−2mol−1.
a. Find the final temperature.
b. Find the value ofwand∆U.
c. Compare your values with those obtained if water vapor is assumed to be an ideal gas and its heat capacity at constant pressure is constant and equal to its value at 500 K.
2.36 a. A sample of 2.000 mol of H2gas is reversibly and isothermally expanded from a volume of 20.00 L to a volume of 50.00 L at a temperature of 300 K. Findq,w, and∆Ufor this process.
b. The same sample of H2gas is reversibly and adiabatically (without any transfer of heat) expanded from a volume of 20.00 L and a temperature of 300 K to a final volume of 50.00 L. Find the final temperature.
Findq,w, and∆Ufor this process.
2.37 A sample of 2.000 mol of N2gas is expanded from an initial pressure of 1.000 atm and an initial temperature of 450.0 K to a pressure of 0.400 atm.
a. Find the final temperature if the expansion is adiabatic and reversible. Assume thatCV5nR/2, so that γ7/51.400.
b. Find∆U,q, andwfor the expansion of part a.
c. Find∆U,q,w, and the final temperature if the expansion is adiabatic but at a constant external pressure of 0.400 atm, starting from the same state as in part a and ending at the same volume as in
part a.
d. Find∆U,q, andwif the expansion is reversible and isothermal, ending at the same pressure as in part a.
2.5 Enthalpy
Many liquid systems in the laboratory are contained in vessels that are open to the atmosphere and are thus maintained at a nearly constant pressure.4For convenient analysis of constant-pressure processes we define a new variable, denoted byH and called the enthalpy:
HU+PV (definition of the enthalpy) (2.5-1) The enthalpy is a state function becauseU,P, andVare state functions.
Consider a simple system with a pressure that remains equal to a constant external pressure. We will refer to these conditions simply asconstant-pressure conditionsand assume thatP(transferred)P Pext. For a process under such conditions,
dw −PextdV −PdV (constant pressure) (2.5-2) This expression fordwis the same as that for reversible processes, Eq. (2.1-11). We do not assert that all processes that occur at constant pressure are reversible processes, but only that the reversible expression fordwapplies. If the volume changes fromV1
toV2at constant pressure, w
c
dw −
c
PextdV −
c
PdV −P(V2−V1)
(simple system, constant pressure) (2.5-3) The heat transferred to the system is given by
dqdU−dwdU+PdV (simple system, constant pressure) (2.5-4) From Eq. (2.5-1)
dH dU+PdV +VdP (2.5-5)
At constant pressure theVdPterm vanishes, so that
dqdH (simple system, constant pressure) (2.5-6) For a finite process,
q∆H (simple system, constant pressure) (2.5-7) Although qis generally path-dependent, it is path-independent for constant-pressure processes, for whichq∆H. Enthalpy changes of constant-pressure processes are sometimes called “heats” of the processes.
4The extreme observed sea-level barometric pressures are 1083.8 mbar (1.069 atm) and 877 mbar (0.866 atm):Guinness Book of World Records, Guinness, 1998, p. 95.