This lesson is an introduction to relations that may be used for calculating differences in properties that are readily measured within a single, homogeneousphase, assuming a simple compressible substance (namely, the Maxwell relations). In order to develop such expressions, it is first necessary to present a general mathematical relation that will prove useful in this procedure.
The physical significance of partial derivatives as they relate to the properties of a pure substance can be explained through the use of a P–v–T diagram. The following diagram shows the surface of the superheated region of a pure substance (colored in light green).
Constant temperature, constant pressure, and constant specific volume planes all intersect at point 2 on the surface. Thus, the partial derivative (∂P/∂v)T is the slope of curve 1,2,3 at point 2. Line 4,5 represents the tangent to curve 1,2,3 at point 2. A similar interpretation can be made of the partial derivatives (∂P/∂T)v and (∂v/∂T)P.
If it is desired to evaluate the partial derivative along a constant-temperature line, the rules for ordinary derivatives can be applied. Thus, it can be written for a constant-temperature process:
(
∂ P
∂ v
)
T
=
dPT
dvT
and the integration can be performed as usual. Returning to the consideration of the relation:
dz = M dx + N dy
If x, y, and z are all point functions, the differentials are exact differentials. If this is the case, the following important relation holds:
(
∂ M
∂ y
)
x
=
(
∂ N
∂ x
)
y
the proof is:
(
∂ M
∂ y
)
x
=
∂ 2 z
∂x∂y
and:
(
∂ N
∂ x
)
y
=
∂ 2 z
∂y∂x
Because the order of differentiation makes no difference when point functions are involved, it may follow that:
