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Derivation of Thermodynamic Derivatives Using Jacobians

​
X
energy E
Y
pressure P
Z
entropy S
∂E
∂P
S
=
βPV
c
v
c
p
Rules from the book by M. Tribus are explored to deduce specified thermodynamic derivatives of the thermodynamical variables (energy
E
, enthalpy
H
, free energy of Helmholtz
A
, free energy of Gibbs
G
, entropy
S
, temperature
T
, pressure
P
, and volume
V
) as functions of measurable quantities (temperature
T
, pressure
P
, volume
V
, coefficient of thermal expansion
α
, coefficient of compressibility
β
, heat capacity at constant pressure
c
p
, and heat capacity at constant volume
c
v
). Sometimes the entropy
S
will remain outside the derivatives.
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