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B Review of Traditional Calculus
Should we wish to change the basis of our description of f from the two variables
(x, y) to two different variables (u, y), for example, then we can achieve this goal
by defining a function g(u, y) such that g = f − ux. With this definition of g, we
can write its total differential dg as
dg = df − u dx − x du = (u dx + v dy) − u dx − x du ,
or, upon simplification,
dg = −x du + v dy .
(B.2.4)
This total differential tells us that g is a function of the two independent variables
(u, y), and that its partial derivatives with respect to these variables are
−x =
∂g
∂u
y
, v =
∂g
∂y
u
.
We shall now consider application of the Legendre transform to thermodynamics.
A thermodynamic expression that corresponds to Eq. (B.2.3) is the differential
expression for the combined first and second laws, viz.,
dU = T dS − P dV ,
in which the internal energy U is represented as a function of two independent
extensive thermodynamic variables, specifically the entropy S and the volume V ,
of the (macroscopic) system. Two different Legendre transforms of U(S, V ) can be
considered.
(a) If f (x, y) is identified with U (x, y), x is identified with V , y with S, and u, v
are identified, respectively, with −P , T , then g ≡ H = U + P V is a Legendre
transform of U(S, V ) with a total differential dH = T dS + V dP , so that the
enthalpy H is a function of the independent variables S and P , one an extensive,
the other an intensive thermodynamic variable.
(b) Similarly to what we have just considered, if we instead identified x with S, y
with V , and u with T , v with −P , then g ≡ A = U − T S is also a Legendre
transform of the internal energy U , with differential dA = −SdT − P dV , so
that the Helmholtz energy A is a function of the independent variables T and
V , again one intensive and one extensive thermodynamic variable.
Notice that in example (a) we replaced the extensive thermodynamic variable V
by the intensive thermodynamic variable P , while in example (b) we replaced the
extensive thermodynamic variable S by the intensive thermodynamic variable T . It
will now likely be clear that were we to make the identifications x ≡ V , y ≡ T ,
u ≡ −S, v ≡ −P , corresponding to f ≡ A, then introduce the Legendre transform
G ≡ A + P V to obtain the total differential dG = −SdT + V dP , the resultant
Gibbs energy G would be a function of two intensive thermodynamic variables,
B Review of Traditional Calculus
Should we wish to change the basis of our description of f from the two variables
(x, y) to two different variables (u, y), for example, then we can achieve this goal
by defining a function g(u, y) such that g = f − ux. With this definition of g, we
can write its total differential dg as
dg = df − u dx − x du = (u dx + v dy) − u dx − x du ,
or, upon simplification,
dg = −x du + v dy .
(B.2.4)
This total differential tells us that g is a function of the two independent variables
(u, y), and that its partial derivatives with respect to these variables are
−x =
∂g
∂u
y
, v =
∂g
∂y
u
.
We shall now consider application of the Legendre transform to thermodynamics.
A thermodynamic expression that corresponds to Eq. (B.2.3) is the differential
expression for the combined first and second laws, viz.,
dU = T dS − P dV ,
in which the internal energy U is represented as a function of two independent
extensive thermodynamic variables, specifically the entropy S and the volume V ,
of the (macroscopic) system. Two different Legendre transforms of U(S, V ) can be
considered.
(a) If f (x, y) is identified with U (x, y), x is identified with V , y with S, and u, v
are identified, respectively, with −P , T , then g ≡ H = U + P V is a Legendre
transform of U(S, V ) with a total differential dH = T dS + V dP , so that the
enthalpy H is a function of the independent variables S and P , one an extensive,
the other an intensive thermodynamic variable.
(b) Similarly to what we have just considered, if we instead identified x with S, y
with V , and u with T , v with −P , then g ≡ A = U − T S is also a Legendre
transform of the internal energy U , with differential dA = −SdT − P dV , so
that the Helmholtz energy A is a function of the independent variables T and
V , again one intensive and one extensive thermodynamic variable.
Notice that in example (a) we replaced the extensive thermodynamic variable V
by the intensive thermodynamic variable P , while in example (b) we replaced the
extensive thermodynamic variable S by the intensive thermodynamic variable T . It
will now likely be clear that were we to make the identifications x ≡ V , y ≡ T ,
u ≡ −S, v ≡ −P , corresponding to f ≡ A, then introduce the Legendre transform
G ≡ A + P V to obtain the total differential dG = −SdT + V dP , the resultant
Gibbs energy G would be a function of two intensive thermodynamic variables,