MIT 3.00 Fall 2002
164
c W.C Carter
°
Lecture 24
Implications of Equilibrium and Gibbs-Duhem
Last Time
Drawing Curves Correctly
Stability, Global Stability, Metastability, Instability
Equilibrium States With More Than One Variable
For a system of fixed composition, δU (S, V ) can be expanded25
δU =
∂U
∂U
dS +
dV
∂S · ∂V
¸
1 ∂ 2U
∂ 2U
∂ 2U
2
2
+
(dS) + 2
dSdV +
(dV ) + . . .
2 ∂S 2
∂S∂V
∂V 2
For a local equilibrium
∂U
= T◦
∂S
so that
µ
(dS, dV )
∂2U
∂S 2
∂2U
∂S∂V
and
∂2U
∂S∂V
∂2U
∂V 2
∂U
= −P◦
∂V
¶µ
dS
dV
(24-1)
(24-2)
¶
>0
(24-3)
The matrix is called the Hessian of the system and for the inequality to be true it must be
“positive definite” for a two-by-two matrix.
25
Assuming that U (S, V ) has continuous derivatives near the point (S, V ) that it is being expanded around.
MIT 3.00 Fall 2002
165
c W.C Carter
°
Necessary conditions for a local minimum are:
∂ 2U
>0
∂S 2
and
∂ 2U ∂ 2U
−
∂S 2 ∂V 2
µ
∂ 2U
∂S∂V
(24-4)
¶2
>0
(24-5)
evaluated at the extrema.
Therefore:
µ ¶
∂ 2U
∂T
T
=
=
>0
(24-6)
2
∂S
∂S V
CV
CV > O for stability (If you add heat to a system, then its entropy must rise)
The second part (Eq. 24-5) that must also positive can be written in terms of the Jacobian
¡ ¢ ¡ ∂U ¢
∂( ∂U
, ∂V S )
∂(T, −P )
∂S V
=
>0
(24-7)
∂(S, V )
∂(S, V )
µ
¶
∂P
T
<0
∂V T CV
µ
¶
∂P
<0
∂V T
(24-8)
MIT 3.00 Fall 2002
166
c W.C Carter
°
for a stable equilibrium.
More Mathematical Thermodynamics: Homogeneous Functions
Consider U (S, V, Ni ), if I scale all the extensive variables by multiplying each of the extensive variables with the same “scale factor” λ then
U (λS, λV, λNi ) = λU (S, V, Ni )
(24-9)
Functions that have the property of Equation 24-9, like U , are called “homogeneous degree
one” (HD1) function of their variables.
Notice that G is not a completely homogeneous function:
G(λT, λP, λNi ) 6= λG(T, P, Ni )
(24-10)
i.e., increasing the pressure is not like changing an extensive variable.
However,
G(T, P, λNi ) = λG(T, P, Ni )
(24-11)
G is HD1 only in the Ni .
Notice that (here lies a common mistake!)
G(T, P, λXi ) 6= λG(T, P, Xi )
G is a different function than G.
Consider carefully, what can be deduced from Equation 24-11.
Taking the derivative with respect to λ
(24-12)
MIT 3.00 Fall 2002
167
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°
C
X
i=1
∂G ∂(λNi )
= G(T, P, Ni )
∂(λNi ) ∂λ
(24-13)
We get the following very important equation:
C
X
µi Ni = G(T, P, Ni )
(24-14)
i=1
This corresponds to what has been discussed about the relation of the Gibbs free energy.
It corresponds to the internal degrees of freedom.
The Gibbs-Duhem Relation
Consider
G=
C
X
µ i Ni
i=1
and compare it to our previous expression for dG:
(24-15)
MIT 3.00 Fall 2002
168
c W.C Carter
°
It follows that (This is another important equation):
0 = −SdT + V dP −
C
X
Ni dµi
(24-16)
i=1
This is the Gibbs-Duhem Equation. It will be used again and again.
Notice that Equation 24-16 has the following form:
~
0 = Y~ · dX
(24-17)
At equilibrium, a small virtual change in the system is normal to the size of the system.