Page 91 - Modeling of Chemical Kinetics and Reactor Design

Thermodynamics of Chemical Reactions 61
=
 ∂ U  ∂ U ik  ∂ U 
dU = dS + dV + ∑   dn (2-5)
 ∂ S   ∂ V   ∂ n  i
,
,
,,
Vn i S n i i=1 i SV n j , i j ≠
If the number of moles are held constant, then from Equation 2-3
 ∂U =  ∂U =−
 ∂S  T and  ∂V  p (2-6)
,
,
Vn i S n i
The chemical potential µ is defined by
i

 ∂U 
µ ≡
 ∂n 
i   (2-7)
i
,,
SV n j
Equation 2-5 thus becomes
dU = TdS − pdV + ∑ µ i dn i (2-8)

i=1
The Gibbs function is expressed as:

G = U + pV – TS (2-9)

Differentiating Equation 2-9 gives

dG = dU + pdV + Vdp – TdS – SdT (2-10)

Substituting Equation 2-3 into Equation 2-10 yields

dG = TdS – pdV + pdV + Vdp – TdS – SdT (2-11)

dG = Vdp – SdT (2-12)

Since Equation 2-12 holds for a system of constant composition, it
is expressed as

G = G(T, P, n , n . . . n (2-13)
1 i k)
Partial differentiation of Equation 2-13 gives