102
2 Work, Heat, and Energy: The First Law of Thermodynamics
where P is the pressure and V is the volume of the system. At constant pressure, ∆H
is equal to q for a simple system. Various calculations of ∆H values were carried out,
including ∆H values for temperature changes, phase changes, and chemical reactions.
A D D I T I O N A L P R O B L E M S
2.66 a. Calculate q, w, ∆U, and ∆H if 100.0 g of water is
vaporized at 100.0 ◦ C and 1.000 atm. Assume that the
water vapor is described by the van der Waals equation
of state and that the density of the liquid is
1.00 g cm −1 .
b. Repeat the calculation assuming gas ideality and
neglecting the volume of the liquid compared with that
of the vapor. How usable do you think this
approximation is?
2.67 It is shown in the theory of hydrodynamics 8 that the speed
of sound in a fluid, v s , is given by
v
2
s
V m C P
Mκ T C V
V m γ
Mκ T
where κ T is the isothermal compressibility, V m is the molar
volume, and M is the molar mass.
a. Find the speed of sound in air at 298.15 K and
1.000 atm, assuming a mean molar mass of
0.029 kg mol −1 and C V,m 5R/2.
b. Find the speed of sound in helium at 298.15 K and
1.000 atm.
c. The speed of sound in ethane at 10 ◦ C is equal to
308 m s −1 . Find the ratio C P /C V for ethane at this
temperature.
2.68 A sample of 1.000 mol of N 2 gas is expanded adiabatically
from a volume of 10.00 L and a temperature of 400.0 K to
a volume of 20.00 L. Assume that N 2 is ideal, with
C V,m 5R/2.
a. Find the final temperature if the expansion is carried
out reversibly.
b. Find the final temperature if the expansion is carried
out with a constant external pressure of 1.000 atm.
c. Find the final temperature if the gas expands into a
vacuum.
d. Find ∆U and w for each of the processes of parts a, b,
and c.
8 H. Lamb, Hydrodynamics, 6th ed., Cambridge University Press,
New York, 1932.
e. For the reversible expansion of part a, show that if the
integral
w rev −
c
PdV
is carried out using Eq. (2.4-27) the same value of w is
obtained as from
w ∆U − q ∆U
2.69 a. The pressure on a sample of 1.000 mol of liquid water
is increased from 1.00 bar to 100.00 bar at a constant
temperature of 20.00 ◦ C. Find ∆H, ∆U, q, and w. State
any assumptions.
b. The pressure on the same sample of water is increased
adiabatically from 1.00 bar to 100.00 bar, beginning at
20.00 ◦ C. Find ∆H, ∆U, q, and w. State any
assumptions.
2.70 Assume that 1.000 mol of octane, C 8 H 18 , is completely
combusted with the stoichiometric amount of oxygen.
Assume that the reaction goes to completion.
a. Assume that the reaction takes place at a constant
temperature of 298.15 K under standard conditions,
with liquid water as one of the products. Find the
values of ∆H, ∆U, q, and w.
b. Assume that part of the energy of reaction is used to raise
the products of the reaction by 100.0 m near the surface
of the earth. Find the values of ∆H, ∆U, q, and w.
c. Assume that the reaction is carried out adiabatically and
at constant pressure starting at 298.15 K. Find the final
temperature of the products. If the final temperature is
above 100 ◦ C, the water will be in the form of water
vapor.
2.71 Label the following statements as true or false. If a
statement is true only under certain conditions, label it as
false.
a. If a process is carried out at constant pressure, dq
behaves like an exact differential.
b. Although dw is an inexact differential, dw rev is exact.
c. dw −PdV is correct for all simple systems.
2 Work, Heat, and Energy: The First Law of Thermodynamics
where P is the pressure and V is the volume of the system. At constant pressure, ∆H
is equal to q for a simple system. Various calculations of ∆H values were carried out,
including ∆H values for temperature changes, phase changes, and chemical reactions.
A D D I T I O N A L P R O B L E M S
2.66 a. Calculate q, w, ∆U, and ∆H if 100.0 g of water is
vaporized at 100.0 ◦ C and 1.000 atm. Assume that the
water vapor is described by the van der Waals equation
of state and that the density of the liquid is
1.00 g cm −1 .
b. Repeat the calculation assuming gas ideality and
neglecting the volume of the liquid compared with that
of the vapor. How usable do you think this
approximation is?
2.67 It is shown in the theory of hydrodynamics 8 that the speed
of sound in a fluid, v s , is given by
v
2
s
V m C P
Mκ T C V
V m γ
Mκ T
where κ T is the isothermal compressibility, V m is the molar
volume, and M is the molar mass.
a. Find the speed of sound in air at 298.15 K and
1.000 atm, assuming a mean molar mass of
0.029 kg mol −1 and C V,m 5R/2.
b. Find the speed of sound in helium at 298.15 K and
1.000 atm.
c. The speed of sound in ethane at 10 ◦ C is equal to
308 m s −1 . Find the ratio C P /C V for ethane at this
temperature.
2.68 A sample of 1.000 mol of N 2 gas is expanded adiabatically
from a volume of 10.00 L and a temperature of 400.0 K to
a volume of 20.00 L. Assume that N 2 is ideal, with
C V,m 5R/2.
a. Find the final temperature if the expansion is carried
out reversibly.
b. Find the final temperature if the expansion is carried
out with a constant external pressure of 1.000 atm.
c. Find the final temperature if the gas expands into a
vacuum.
d. Find ∆U and w for each of the processes of parts a, b,
and c.
8 H. Lamb, Hydrodynamics, 6th ed., Cambridge University Press,
New York, 1932.
e. For the reversible expansion of part a, show that if the
integral
w rev −
c
PdV
is carried out using Eq. (2.4-27) the same value of w is
obtained as from
w ∆U − q ∆U
2.69 a. The pressure on a sample of 1.000 mol of liquid water
is increased from 1.00 bar to 100.00 bar at a constant
temperature of 20.00 ◦ C. Find ∆H, ∆U, q, and w. State
any assumptions.
b. The pressure on the same sample of water is increased
adiabatically from 1.00 bar to 100.00 bar, beginning at
20.00 ◦ C. Find ∆H, ∆U, q, and w. State any
assumptions.
2.70 Assume that 1.000 mol of octane, C 8 H 18 , is completely
combusted with the stoichiometric amount of oxygen.
Assume that the reaction goes to completion.
a. Assume that the reaction takes place at a constant
temperature of 298.15 K under standard conditions,
with liquid water as one of the products. Find the
values of ∆H, ∆U, q, and w.
b. Assume that part of the energy of reaction is used to raise
the products of the reaction by 100.0 m near the surface
of the earth. Find the values of ∆H, ∆U, q, and w.
c. Assume that the reaction is carried out adiabatically and
at constant pressure starting at 298.15 K. Find the final
temperature of the products. If the final temperature is
above 100 ◦ C, the water will be in the form of water
vapor.
2.71 Label the following statements as true or false. If a
statement is true only under certain conditions, label it as
false.
a. If a process is carried out at constant pressure, dq
behaves like an exact differential.
b. Although dw is an inexact differential, dw rev is exact.
c. dw −PdV is correct for all simple systems.