Physical Chemistry Third Edition

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 compressi-

bility 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 factorZ

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 thanP0 for which

PVmRT.

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 andVm 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 andVm 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

xn

n!

+ ···

wheren!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

ZPVm/RTfor hydrogen gas atT 273 .15 K and

Vm 0 .1497 L mol−^1 is 1.1336. Find the values ofZ

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:

aa 1 x^21 +a 12 x 1 x 2 +a 2 x^22

bb 1 x^21 +b 12 x 1 x 2 +b 2 x^22

wherex 1 andx 2 are the mole fractions of the two

substances and wherea 1 ,b 1 ,a 2 , andb 2 are the van der

Waals parameters of the two substances. The quantitiesa 12

andb 12 are defined by

a 12 (a 1 a 2 )^1 /^2

and

b 12 

(

b^11 /^3 +b^12 /^3

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 asV/ntotal)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

PVm

RT

 1 +

B 2

Vm

For carbon dioxide at 298.15 K,

B 2 − 12. 5 × 10 −^5 m^3 mol−^1.

1.40 ConsideringPto be a function ofT,V, andn, obtain the

expression fordPfor a gas obeying the van der Waals

equation of state.