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5.60 Thermodynamics & Kinetics
Spring 2008
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5.60 Spring 2008
Lecture #13
page 1
The Gibbs Free Energy
• With the free energies
Helmholtz free energy
Gibbs free energy
A = U - TS
G = H - TS
we’ve introduced all our state functions. For closed systems,
U (S ,V
From
and
)
⇒
dU =TdS − pdV H (S , p )
⇒
dH =TdS +Vdp A (T ,V )
⇒
dA = −SdT − pdV G (T , p )
⇒
dG = −SdT +Vdp Fundamental equations
∂A ⎞
⎛ ∂A ⎞ dV
⎟ dT + ⎜
⎟
⎝ ∂T ⎠V
⎝ ∂V ⎠T
⎛ ∂G ⎞
⎛ ∂G ⎞
dG = ⎜
⎟ dp
⎟ dT + ⎜
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
dA = ⎛⎜
⎛ ∂A ⎞ = −S
⎜
⎟
⎝ ∂T ⎠V
⎛ ∂A ⎞ = − p
⎜
⎟
⎝ ∂V ⎠T
⎛ ∂G ⎞
⎜
⎟ = −S
⎝ ∂T ⎠ p
⎛ ∂G ⎞
⎜
⎟ =V
⎝ ∂p ⎠T
The Maxwell relations:
∂ 2A
∂2A
=
∂V ∂T
∂T ∂V
and
∂ 2G
∂2G
=
∂p ∂T
∂T ∂p
now allow us to find how S depends on V and p.
⇒
⎛ ∂S ⎞ = ⎛ ∂p ⎞
⎜
⎟
⎜
⎟
⎝ ∂V ⎠T ⎝ ∂T ⎠V
⎛ ∂S ⎞
⎛ ∂V ⎞
⎜
⎟ = −⎜
⎟
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
These can be obtained from an equation of state.
5.60 Spring 2008
Lecture #13
page 2
We can now also relate T and H to p-V-T data.
⎛ ∂p ⎞
⎜
⎟ −p
⎝ ∂T ⎠V
⎛ ∂U ⎞
⎜
⎟ =T
⎝ ∂V ⎠T
⎛ ∂S ⎞
⎜
⎟ − p =T
⎝ ∂V ⎠T
⎛ ∂H ⎞
⎜
⎟ =T
∂
p
⎝
⎠T
⎛ ∂S ⎞
⎛ ∂V ⎞
⎜
⎟ +V = V −T ⎜
⎟
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
•
For an ideal gas
→ U and H from equations of state!
pV = nRT
nR p
⎛ ∂p ⎞
=
⎜
⎟ =
T
⎝ ∂T ⎠V V
⇒
⎛ ∂U ⎞
⎜
⎟ =0
⎝ ∂V ⎠T
nR V
⎛ ∂V ⎞
=
⎜
⎟ =
p T
⎝ ∂T ⎠ p
⇒
⎛ ∂H ⎞
⎜
⎟ =0
⎝ ∂p ⎠T
This proves that for an ideal gas U(T) and H(T), functions of T only.
We had assumed this was true from Joule and Joule-Thomson
expansion experiments. Now we know it is rigorously true.
•
For a van der Waals gas
⎛ p + a ⎞ V − b = RT
⎜
⎟
V 2⎠
⎝
(
p=
RT
a
⎛ ∂U ⎞
−p = 2 ≠0
⎜
⎟ =
V
⎝ ∂V ⎠T V − b
)
RT
a
− 2
V −b V
⇒
U( T, V )
5.60 Spring 2008
•
Lecture #13
The special role of G(T,p):
⎛ ∂G ⎞
⎟ ,
⎝ ∂T ⎠ p
V =⎜
⎛ ∂G ⎞
⎟
⎝ ∂T ⎠ p
H = G +TS
⇒
H = G −T ⎜
U = H − pV
⇒
U = G −T ⎜
A =U −TS
⇒
A =G − p⎜
∂S ⎞
⎟
⎝ ∂T ⎠ p
If you know G(T,p),
you know everything!
⎛ ∂G ⎞
⎟
⎝ ∂p ⎠T
S = −⎜
C p =T ⎛⎜
page 3
⇒
⎛ ∂G ⎞
⎛ ∂G ⎞
⎟
⎟ − p⎜
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
⎛ ∂G ⎞
⎟
⎝ ∂p ⎠T
⎛ ∂2G ⎞
2 ⎟
⎝ ∂T ⎠ p
C p = −T ⎜
Can get all the thermodynamic functions from G(T,p)!
•
G ( T, p )
for liquids, solids, and gases (ideal)
From
V =⎜
⎛ ∂G ⎞
⎟
⎝ ∂p ⎠T
p2
⇒
G (T , p2 ) = G (T , p1 ) + ∫ Vdp
•
Liquids and solids
p1
⇒
V is small
G (T , p2 ) = G (T , p1 ) +V ( p2 − p1 ) ≈ G (T , p1 )
⇒
G (T
) 5.60 Spring 2008
•
Lecture #13
page 4
Ideal gases
G (T , p2 ) = G (T , p1 ) + ∫
p2
p1
p
RT
dp = G (T , p1 ) + RT ln 2
p
p1
Take p1 = p o = 1 bar
G (T , p ) = G o (T ) + RT ln
From
p
p0
⎛ ∂G ⎞
S = −⎜
⎟
⎝ ∂T ⎠ p
or G (T , p ) = G o (T ) + RT ln p
(p in bar)
⇒
S (T , p ) = S o (T ) − R ln p