1 (38)

Prévia do material em texto

30 1 THE PROPERTIES OF GASES
Equation 1.2 is rewritten to give an expresion for the pressure
p = RT
Vm
(1 + B
Vm
+ C
V 2m
+ . . .)
�is expression for p is substituted into the right-hand side of eqn 1.3 to give
pVm = RT (1 + B′p + C′p2 + . . .)
= RT
⎛
⎝
1 + B
′RT
Vm
(1 + B
Vm
+ C
V 2m
+ . . .) + C
′R2T2
V 2m
(1 + B
Vm
+ C
V 2m
+ . . .)
2
+ . . .
⎞
⎠
(1.4)
On the right there is only one term in 1/Vm
B′R2T2
Vm
�is is identi�ed with the term in 1/Vm on the right-hand side of eqn 1.2
B′R2T2
Vm
= BRT
Vm
hence B = B′RT
�ere are two terms in 1/V 2m on the right-hand side of eqn 1.4
B′BR2T2
V 2m
+ C
′R3T3
V 2m
�ese are identi�ed with the term in 1/V 2m on the right-hand side of eqn 1.2
B′BR2T2
V 2m
+ C
′R3T3
V 2m
= CRT
V 2m
hence B′BRT + C′R2T2 = C
With the substitution B = B′RT , the relationship is C = R2T2[(B′)2 + C′] .
P1C.16 �e given equation of state is multiplied through by V 2m to give
pV 2m = RTVm + (a + bT)
�e partial di�erential with respect to T of both sides of the equation is then
computed, being careful to remember thatVm is a function ofT so, for example,
the term RTVm must be di�erentiated using the product rule. Calculating the
derivative is essentially an application of the chain rule.
2pVm (∂Vm
∂T
)
p
= RT (∂Vm
∂T
)
p
+ RVm + b
�is equation is then rearranged by collecting the terms in ∂Vm/∂T together
(∂Vm
∂T
)
p
(2pVm − RT) = RVm + b hence (∂Vm
∂T
)
p
= RVm + b
2pVm − RT