26
1 The Behavior of Gases and Liquids
1.28 Evaluate each of the partial derivatives in Problem 1.26 for
carbon dioxide at 298.15 K and 10.000 bar.
1.29 a. Derive an expression for the isothermal compressibility of a gas obeying the van der Waals equation of
state. Hint: Use the reciprocal identity, Eq. (B-8).
b. Evaluate the isothermal compressibility of carbon
dioxide gas at a temperature of 298.15 K and a molar
volume of 0.01000 m 3 mol −1 . Compare with the value
obtained from the ideal gas law.
1.30 Write the expressions giving the compression factor Z
as a function of temperature and molar volume for the
van der Waals, Dieterici, and Redlich–Kwong equations
of state.
1.31 a. For the van der Waals equation of state at temperatures
below the Boyle temperature, find an expression for a
value of the pressure other than P 0 for which
PV m RT .
b. Find the value of this pressure for nitrogen gas at
298.15 K.
1.32 a. By differentiation, find an expression for the isothermal
compressibility of a gas obeying the Dieterici equation
of state.
b. Find the value of the isothermal compressibility of
nitrogen gas at 298.15 K and V m 24.4 L. Compare
with that of an ideal gas.
1.33 a. By differentiation, find an expression for the coefficient
of thermal expansion of a gas obeying the van der
Waals equation of state.
b. Find the value of the coefficient of thermal expansion
of nitrogen gas at 298.15 K and V m 24.4 L mol −1 .
1.34 By differentiation, find an expression for the coefficient of
thermal expansion of a gas obeying the Dieterici equation
of state.
1.35 Manipulate the Dieterici equation of state into the virial
form. Use the identity
e
−x
1 − x +
x 2
2!
−
x 3
3!
+ · · · + (−1)
n x n
n!
+ · · ·
where n! n(n − 1)(n − 2)(n − 3) . . . (3)(2)(1). Write
expressions for the second, third, and fourth virial
coefficients.
1.36 Write an expression for the isothermal compressibility of a
nonideal gas obeying the Redlich–Kwong equation of state.
1.37 The experimental value of the compression factor
Z PV m /RT for hydrogen gas at T 273.15 K and
V m 0.1497 L mol −1 is 1.1336. Find the values of Z
predicted by the van der Waals, Dieterici, and
Redlich–Kwong equations of state for these conditions.
Calculate the percent error for each.
1.38 The parameters for the van der Waals equation of state for
a mixture of gases can be approximated by use of the
mixing rules:
a a 1 x
2
1 + a 12 x 1 x 2 + a 2 x
2
2
b b 1 x
2
1 + b 12 x 1 x 2 + b 2 x
2
2
where x 1 and x 2 are the mole fractions of the two
substances and where a 1 , b 1 , a 2 , and b 2 are the van der
Waals parameters of the two substances. The quantities a 12
and b 12 are defined by
a 12 (a 1 a 2 )
1/2
and
b 12
b
1/3
1 + b
1/3
2
3
3
a. Using these mixing rules and the van der Waals
equation of state, find the pressure of a mixture of
0.79 mol of N 2 and 0.21 mol of O 2 at 298.15 K and at a
mean molar volume (defined as V /n total ) of
0.00350 m 3 mol −1 . Compare your answer with the
pressure of an ideal gas under the same conditions.
b. Using the van der Waals equation of state, find the
pressure of pure N 2 at 298.15 K and at a molar volume
of 0.00350 m 3 mol −1 .
c. Using the van der Waals equation of state, find the
pressure of pure O 2 at 298.15 K and at a molar volume
of 0.00350 m 3 mol −1 .
1.39 Find the value of the isothermal compressibility of carbon
dioxide gas at 298.15 K and a molar volume of
24.4 L mol
−1 ,
a. According to the ideal gas law.
b. According to the truncated virial equation of state
PV m
RT
1 +
B 2
V m
For carbon dioxide at 298.15 K,
B 2 −12.5 × 10 −5 m 3 mol −1 .
1.40 Considering P to be a function of T , V , and n, obtain the
expression for dP for a gas obeying the van der Waals
equation of state.
1 The Behavior of Gases and Liquids
1.28 Evaluate each of the partial derivatives in Problem 1.26 for
carbon dioxide at 298.15 K and 10.000 bar.
1.29 a. Derive an expression for the isothermal compressibility of a gas obeying the van der Waals equation of
state. Hint: Use the reciprocal identity, Eq. (B-8).
b. Evaluate the isothermal compressibility of carbon
dioxide gas at a temperature of 298.15 K and a molar
volume of 0.01000 m 3 mol −1 . Compare with the value
obtained from the ideal gas law.
1.30 Write the expressions giving the compression factor Z
as a function of temperature and molar volume for the
van der Waals, Dieterici, and Redlich–Kwong equations
of state.
1.31 a. For the van der Waals equation of state at temperatures
below the Boyle temperature, find an expression for a
value of the pressure other than P 0 for which
PV m RT .
b. Find the value of this pressure for nitrogen gas at
298.15 K.
1.32 a. By differentiation, find an expression for the isothermal
compressibility of a gas obeying the Dieterici equation
of state.
b. Find the value of the isothermal compressibility of
nitrogen gas at 298.15 K and V m 24.4 L. Compare
with that of an ideal gas.
1.33 a. By differentiation, find an expression for the coefficient
of thermal expansion of a gas obeying the van der
Waals equation of state.
b. Find the value of the coefficient of thermal expansion
of nitrogen gas at 298.15 K and V m 24.4 L mol −1 .
1.34 By differentiation, find an expression for the coefficient of
thermal expansion of a gas obeying the Dieterici equation
of state.
1.35 Manipulate the Dieterici equation of state into the virial
form. Use the identity
e
−x
1 − x +
x 2
2!
−
x 3
3!
+ · · · + (−1)
n x n
n!
+ · · ·
where n! n(n − 1)(n − 2)(n − 3) . . . (3)(2)(1). Write
expressions for the second, third, and fourth virial
coefficients.
1.36 Write an expression for the isothermal compressibility of a
nonideal gas obeying the Redlich–Kwong equation of state.
1.37 The experimental value of the compression factor
Z PV m /RT for hydrogen gas at T 273.15 K and
V m 0.1497 L mol −1 is 1.1336. Find the values of Z
predicted by the van der Waals, Dieterici, and
Redlich–Kwong equations of state for these conditions.
Calculate the percent error for each.
1.38 The parameters for the van der Waals equation of state for
a mixture of gases can be approximated by use of the
mixing rules:
a a 1 x
2
1 + a 12 x 1 x 2 + a 2 x
2
2
b b 1 x
2
1 + b 12 x 1 x 2 + b 2 x
2
2
where x 1 and x 2 are the mole fractions of the two
substances and where a 1 , b 1 , a 2 , and b 2 are the van der
Waals parameters of the two substances. The quantities a 12
and b 12 are defined by
a 12 (a 1 a 2 )
1/2
and
b 12
b
1/3
1 + b
1/3
2
3
3
a. Using these mixing rules and the van der Waals
equation of state, find the pressure of a mixture of
0.79 mol of N 2 and 0.21 mol of O 2 at 298.15 K and at a
mean molar volume (defined as V /n total ) of
0.00350 m 3 mol −1 . Compare your answer with the
pressure of an ideal gas under the same conditions.
b. Using the van der Waals equation of state, find the
pressure of pure N 2 at 298.15 K and at a molar volume
of 0.00350 m 3 mol −1 .
c. Using the van der Waals equation of state, find the
pressure of pure O 2 at 298.15 K and at a molar volume
of 0.00350 m 3 mol −1 .
1.39 Find the value of the isothermal compressibility of carbon
dioxide gas at 298.15 K and a molar volume of
24.4 L mol
−1 ,
a. According to the ideal gas law.
b. According to the truncated virial equation of state
PV m
RT
1 +
B 2
V m
For carbon dioxide at 298.15 K,
B 2 −12.5 × 10 −5 m 3 mol −1 .
1.40 Considering P to be a function of T , V , and n, obtain the
expression for dP for a gas obeying the van der Waals
equation of state.