Bridgeman Table

David Keffer, Department of Chemical Engineering, University of Tennessee
Bridgman Tables
Convenient for Single-component Pressure Explicit Equations of State
and Constant Volume Heat Capacity
I. Volume
1.
⎛ ∂p
⎞
( ∂p)
V = −(
∂V)
p = ⎜ ⎟
⎝ ∂T
⎠ V
2. ( ∂T)
V = −(
∂V)
T = 1
3.
C
( ∂S)
( V) V
V = − ∂ S =
T
4. ( ∂U)
V = −(
∂V)
U = CV
5.
⎛ ∂p
⎞
( ∂H)
V = −(
∂V)
H = CV
+ V⎜
⎟
⎝ ∂T
⎠ V
6. ( ∂A)
V = −(
∂V)
A = −S
7.
⎛ ∂p
⎞
( ∂G)
V = −(
∂V)
G = −S
+ V⎜
⎟
⎝ ∂T
⎠ V
1

David Keffer, Department of Chemical Engineering, University of Tennessee
II. Temperature
1. ( ∂V)
T = −(
∂T)
V = −1
2.
3.
4.
5.
⎛ ∂p
⎞
( ∂p)
T = −(
∂T)
p = −⎜
⎟
⎝ ∂V
⎠ T
⎛ ∂p
⎞
( ∂S)
T = −(
∂T)
S = −⎜
⎟
⎝ ∂T
⎠ V
⎛ ∂p
⎞
( ∂U)
T = −(
∂T)
U = p − T⎜
⎟
⎝ ∂T
⎠ V
⎛ ∂p
⎞ ⎛ ∂p
⎞
( ∂H)
T = −(
∂T)
H = −T⎜
⎟ − V⎜
⎟
⎝ ∂T
⎠V
⎝ ∂V
⎠ T
6. ( ∂A)
T = −(
∂T)
A = p
7.
⎛ ∂p
⎞
( ∂G)
T = −(
∂T)
G = −V⎜
⎟
⎝ ∂V
⎠ T
2

David Keffer, Department of Chemical Engineering, University of Tennessee
III. Pressure
1.
2.
3.
4.
5.
6.
7.
⎛ ∂p
⎞
( ∂V)
p = −(
∂p)
V = −⎜
⎟
⎝ ∂T
⎠ V
⎛ ∂p
⎞
( ∂T)
p = −(
∂p)
T = ⎜ ⎟
⎝ ∂V
⎠ T
2
CV
⎛ ∂p
⎞ ⎛ ∂P
⎞
( ∂S)
p = −(
∂p)
S = ⎜ ⎟ − ⎜ ⎟
T ⎝ ∂V
⎠T
⎝ ∂T
⎠ V
2
⎛ ∂p
⎞ ⎛ ∂P
⎞ ⎛ ∂p
⎞
( ∂U)
p = −(
∂p)
U = CV⎜
⎟ − T⎜
⎟ + p⎜
⎟
⎝ ∂V
⎠T
⎝ ∂T
⎠V
⎝ ∂T
⎠ V
2
⎛ ∂p
⎞ ⎛ ∂P
⎞
( ∂H)
p = −(
∂p)
H = CV⎜
⎟ − T⎜
⎟
⎝ ∂V
⎠T
⎝ ∂T
⎠ V
⎛ ∂p
⎞ ⎛ ∂p
⎞
( ∂A)
p = −(
∂p)
A = p⎜
⎟ − S⎜
⎟
⎝ ∂T
⎠V
⎝ ∂V
⎠ T
⎛ ∂p
⎞
( ∂G)
p = −(
∂p)
G = −S⎜
⎟
⎝ ∂V
⎠ T
3

David Keffer, Department of Chemical Engineering, University of Tennessee
IV. Entropy
1.
2.
3.
4.
2
CV
⎛ ∂p
⎞ ⎛ ∂P
⎞
( ∂p)
S = −(
∂S)
p = − ⎜ ⎟ + ⎜ ⎟
T ⎝ ∂V
⎠T
⎝ ∂T
⎠ V
⎛ ∂p
⎞
( ∂T)
S = −(
∂S)
T = ⎜ ⎟
⎝ ∂T
⎠ V
C
( ∂V)
( S) V
S = − ∂ V = −
T
C
( ∂U)
( S) p V
S = − ∂ U =
T
5.
( ∂H)
S
= −(
∂S)
H
= −
V
T
⎡
2
⎛ ∂p
⎞ ⎛ ∂P
⎞
⎤
⎢CV⎜
⎟ − T⎜
⎟ ⎥
⎢
⎣
⎝ ∂V
⎠T
⎝ ∂T
⎠V
⎥
⎦
6.
7.
CV
⎛ ∂p
⎞
( ∂A)
S = −(
∂S)
A = p − S⎜
⎟
T ⎝ ∂T
⎠ V
2
V ⎡
⎛ ∂p
⎞ ⎛ ∂P
⎞
⎤
⎛ ∂p
⎞
( ∂G)
S = −(
∂S)
G = − ⎢CV
⎜ ⎟ − T⎜
⎟ ⎥ − S⎜
⎟
T ⎢ V T T V ⎥ ⎝ ∂T
⎣
⎝ ∂ ⎠ ⎝ ∂ ⎠
⎦ ⎠ V
4

David Keffer, Department of Chemical Engineering, University of Tennessee
V. Internal Energy
1.
2.
2
⎛ ∂p
⎞ ⎛ ∂P
⎞ ⎛ ∂p
⎞
( ∂p)
U = −(
∂U)
p = −CV⎜
⎟ + T⎜
⎟ − p⎜
⎟
⎝ ∂V
⎠T
⎝ ∂T
⎠V
⎝ ∂T
⎠ V
⎛ ∂p
⎞
( ∂T)
U = −(
∂U)
T = −p
+ T⎜
⎟
⎝ ∂T
⎠ V
3. ( ∂V)
U = −(
∂U)
V = −CV
4.
5.
C
( ∂S)
( U) p V
U = − ∂ S = −
T
⎡
2
⎛ ∂p
⎞ ⎤ ⎛ ∂p
⎞ ⎛ ∂P
⎞
( ∂H)
U = −(
∂U)
H = −p⎢CV
+ V⎜
⎟ ⎥ − VCV⎜
⎟ + VT⎜
⎟
⎣ ⎝ ∂T
⎠V
⎦ ⎝ ∂V
⎠T
⎝ ∂T
⎠ V
⎛ ∂p
⎞
( U A v − ST⎜
⎟
⎝ ∂T
⎠ V
6. ∂A)
= −(
∂U)
= p[ C + S]
2
⎛ ∂p
⎞ ⎛ ∂p
⎞ ⎛ ∂P
⎞
( U G
⎜ ⎟ − VCV⎜
⎟ + VT⎜
⎟
⎝ ∂T
⎠V
⎝ ∂V
⎠T
⎝ ∂T
⎠ V
7. ∂G)
= −(
∂U)
= pS − [ pV + ST]
5

David Keffer, Department of Chemical Engineering, University of Tennessee
VI. Enthalpy
1.
2.
3.
2
⎛ ∂p
⎞ ⎛ ∂P
⎞
( ∂p)
H = −(
∂H)
p = −CV⎜
⎟ + T⎜
⎟
⎝ ∂V
⎠T
⎝ ∂T
⎠ V
⎛ ∂p
⎞ ⎛ ∂p
⎞
( ∂T)
H = −(
∂H)
T = T⎜
⎟ + V⎜
⎟
⎝ ∂T
⎠V
⎝ ∂V
⎠ T
⎛ ∂p
⎞
( ∂V)
H = −(
∂H)
V = −CV
− V⎜
⎟
⎝ ∂T
⎠ V
4.
( ∂S)
H
= −(
∂H)
S
=
⎡
2
V ⎛ ∂p
⎞ ⎛ ∂P
⎞
⎤
⎢CV⎜
⎟ − T⎜
⎟ ⎥
T ⎢⎣
⎝ ∂V
⎠T
⎝ ∂T
⎠V
⎥
⎦
5.
⎡
2
⎛ ∂p
⎞ ⎤ ⎛ ∂p
⎞ ⎛ ∂P
⎞
( ∂U)
H = −(
∂H)
U = p⎢CV
+ V⎜
⎟ ⎥ + VCV⎜
⎟ − VT⎜
⎟
⎣ ⎝ ∂T
⎠V
⎦ ⎝ ∂V
⎠T
⎝ ∂T
⎠ V
⎛ ∂p
⎞ ⎛ ∂p
⎞
( H A V ⎜ ⎟ − SV⎜
⎟
⎝ ∂T
⎠V
⎝ ∂V
⎠ T
6. ∂A)
= −(
∂H)
= pC + [ pV − ST]
7.
⎡
2
⎛ ∂P
⎞
⎤
⎢ T⎜
⎟ ⎥
⎢ ⎝ ∂T
⎠V
⎥⎛
∂p
⎞ ⎛ ∂p
⎞
( ∂G)
H = −(
∂H)
G = −V
⎢
CV
− + S ⎜ ⎟ − ST⎜
⎟
⎛ ∂P
⎞ ⎥
⎝ ∂V
⎠T
⎝ ∂T
⎥
⎠
⎢ ⎜ ⎟
V
⎢ ⎝ ∂V
⎠
⎣
T ⎥
⎦
6

David Keffer, Department of Chemical Engineering, University of Tennessee
VII. Helmholtz Free Energy
1.
⎛ ∂p
⎞ ⎛ ∂p
⎞
( ∂p)
A
= −(
∂A)
p
= −p⎜
⎟ + S⎜
⎟
⎝ ∂T
⎠ ⎝ ∂V
⎠
V
T
2. ∂T)
= −(
∂A)
= −p
(
A
T
3. ∂V)
= −(
∂A)
S
(
A V
=
4.
CV
⎛ ∂p
⎞
( ∂S)
A = −(
∂A)
S = −p
+ S⎜
⎟
T ⎝ ∂T
⎠ V
⎛ ∂p
⎞
( A U V + ST⎜
⎟
⎝ ∂T
⎠ V
5. ∂U)
= −(
∂A)
= −p[ C + S]
⎛ ∂p
⎞ ⎛ ∂p
⎞
( A H V ⎜ ⎟ + SV⎜
⎟
⎝ ∂T
⎠V
⎝ ∂V
⎠ T
6. ∂H)
= −(
∂A)
= −pC
+ [ ST − pV]
7.
⎡ ⎛ ∂p
⎞ ⎤ ⎛ ∂p
⎞
( ∂G)
A = −(
∂A)
G = S⎢p
+ V⎜
⎟ ⎥ − pV⎜
⎟
⎣ ⎝ ∂V
⎠T⎦
⎝ ∂T
⎠ V
7

David Keffer, Department of Chemical Engineering, University of Tennessee
VIII. Gibbs Free Energy
1.
2.
3.
4.
⎛ ∂p
⎞
( ∂p)
G
= −(
∂G)
p
= S⎜
⎟
⎝ ∂V
⎠
⎛ ∂p
⎞
( ∂T)
G
= −(
∂G)
T
= V⎜
⎟
⎝ ∂V
⎠
⎛ ∂p
⎞
( ∂V)
G
= −(
∂G)
V
= S − V⎜
⎟
⎝ ∂T
⎠
T
T
2
V ⎡
⎛ ∂p
⎞ ⎛ ∂P
⎞
⎤
⎛ ∂p
⎞
( ∂S)
G = −(
∂G)
S = ⎢CV⎜
⎟ − T⎜
⎟ ⎥ + S⎜
⎟
T ⎢ V T T V ⎥ ⎝ ∂T
⎣
⎝ ∂ ⎠ ⎝ ∂ ⎠
⎦ ⎠ V
V
2
⎛ ∂p
⎞ ⎛ ∂p
⎞ ⎛ ∂P
⎞
( G U
⎜ ⎟ + VCV⎜
⎟ − VT⎜
⎟
⎝ ∂T
⎠V
⎝ ∂V
⎠T
⎝ ∂T
⎠ V
5. ∂U)
= −(
∂G)
= −pS
+ [ pV + ST]
6.
7.
⎡
2
⎛ ∂P
⎞
⎤
⎢ T⎜
⎟ ⎥
⎢ ⎝ ∂T
⎠V
⎥⎛
∂p
⎞ ⎛ ∂p
⎞
( ∂H)
G = −(
∂G)
H = V
⎢
CV
− + S ⎜ ⎟ + ST⎜
⎟
⎛ ∂P
⎞ ⎥
⎝ ∂V
⎠T
⎝ ∂T
⎥
⎠
⎢ ⎜ ⎟
V
⎢ ⎝ ∂V
⎠
⎣
T ⎥
⎦
⎡ ⎛ ∂p
⎞ ⎤ ⎛ ∂p
⎞
( ∂A)
G = −(
∂G)
A = −S⎢p
+ V⎜
⎟ ⎥ + pV⎜
⎟
⎣ ⎝ ∂V
⎠T⎦
⎝ ∂T
⎠ V
8