2.9 Problems for This Chapter
121
P =
RT
V − β
−
α
V (V + β) + β(V − β)
,
(2.8.36b)
have proven to be quite successful for simple molecular fluids, especially in the
liquid/vapour region. These two equations have been employed extensively by
chemical engineers for molar densities ρ ranging from well below 1 mol L −1 to
as much as 15 mol L −1 over a fairly extensive range of temperatures. Both models
represent a considerable improvement over the Van der Waals equation of state for
molar densities exceeding about 7 mol L −1 .
2.9 Problems for This Chapter
1. Obtain an expression for the work W done by an ideal classical gas during an
isothermal expansion from initial volume V i to a final volume V f , and show that
the Helmholtz energy A decreases by the same amount.
2. The form for the entropy of an ideal classical gas given in Eq. (2.3.13b) ensures
that the entropy is an extensive thermodynamic quantity. Verify the extensivity
of S by doubling both the number of ideal gas particles N and the volume V
of the container, while holding the pressure and temperature (which are the
relevant intensive thermodynamic variables) constant. Determine whether or
not the Helmholtz energy A is an extensive quantity.
3. Consider ideal gases A and B, both at temperature T , sharing a container of total
volume V , but initially separated into volumes V A and V B by an impermeable
partition that is then removed to allow the two gases to mix. Show that the
pressures P A and P B of the two gases in the final mixture satisfy the Dalton law
of partial pressures. How are the partial pressures of the mixture components
related to their initial pressures?
4. Consider a container of volume V , initially separated by an impenetrable wall
of negligible volume into two compartments having unequal volumes V A and
V B and containing ideal gases A and B, respectively. If the number of molecules
are N A and N B (in general, N A = N B ), and both gases are at temperature
T , obtain an expression for the entropy difference, ((S) i → f, between the
initial state (with the partition) and the final state (without the partition) once
equilibrium has been re-established. Show that if N A = N B ≡ N and V A =
V B ≡ V , the entropy difference becomes 2Nk B ln 2.
5. Show that if in Problem 4 the ideal gases A and B have both temperature T
and pressure P in common, then the entropy difference can be expressed in the
form ((S) i→f = −nR(x A ln x A +x B ln x B ), with n ≡ n A +n B the total number
of moles in the mixture, and x α ≡ n α /(n A + n B ), α = A, B, the mole fraction
of species α in the binary gas mixture.
6. Consider a container of volume V and at temperature T , initially separated
by an impenetrable wall of negligible volume into two compartments of
volumes V 1 and V 2 , each compartment containing an ideal gas A. If the
121
P =
RT
V − β
−
α
V (V + β) + β(V − β)
,
(2.8.36b)
have proven to be quite successful for simple molecular fluids, especially in the
liquid/vapour region. These two equations have been employed extensively by
chemical engineers for molar densities ρ ranging from well below 1 mol L −1 to
as much as 15 mol L −1 over a fairly extensive range of temperatures. Both models
represent a considerable improvement over the Van der Waals equation of state for
molar densities exceeding about 7 mol L −1 .
2.9 Problems for This Chapter
1. Obtain an expression for the work W done by an ideal classical gas during an
isothermal expansion from initial volume V i to a final volume V f , and show that
the Helmholtz energy A decreases by the same amount.
2. The form for the entropy of an ideal classical gas given in Eq. (2.3.13b) ensures
that the entropy is an extensive thermodynamic quantity. Verify the extensivity
of S by doubling both the number of ideal gas particles N and the volume V
of the container, while holding the pressure and temperature (which are the
relevant intensive thermodynamic variables) constant. Determine whether or
not the Helmholtz energy A is an extensive quantity.
3. Consider ideal gases A and B, both at temperature T , sharing a container of total
volume V , but initially separated into volumes V A and V B by an impermeable
partition that is then removed to allow the two gases to mix. Show that the
pressures P A and P B of the two gases in the final mixture satisfy the Dalton law
of partial pressures. How are the partial pressures of the mixture components
related to their initial pressures?
4. Consider a container of volume V , initially separated by an impenetrable wall
of negligible volume into two compartments having unequal volumes V A and
V B and containing ideal gases A and B, respectively. If the number of molecules
are N A and N B (in general, N A = N B ), and both gases are at temperature
T , obtain an expression for the entropy difference, ((S) i → f, between the
initial state (with the partition) and the final state (without the partition) once
equilibrium has been re-established. Show that if N A = N B ≡ N and V A =
V B ≡ V , the entropy difference becomes 2Nk B ln 2.
5. Show that if in Problem 4 the ideal gases A and B have both temperature T
and pressure P in common, then the entropy difference can be expressed in the
form ((S) i→f = −nR(x A ln x A +x B ln x B ), with n ≡ n A +n B the total number
of moles in the mixture, and x α ≡ n α /(n A + n B ), α = A, B, the mole fraction
of species α in the binary gas mixture.
6. Consider a container of volume V and at temperature T , initially separated
by an impenetrable wall of negligible volume into two compartments of
volumes V 1 and V 2 , each compartment containing an ideal gas A. If the