Mathematical Theorems

The equation of state provides the reationship between theromodynamic properties for ideal gas as well as for incompressible substances. However, simple equation of state is not enough to describe the thermodynamic properties for systems with real and complex materials. Hence, more mathematical equations are required to represent the realistic processes. Amongst the various thermodynamic properties, p, v, T are directly measurable whereas u, h, s are not directly measurable. This necessitates for developing thermodynamic relations apart from the equation of states.

The following three theorems are the base for developing the various thermodynamic relations.

Theorem-1 If a relation exists among the variables x, y, and z, then z may be expressed as a function of x and y .

or,

1.70

If,

1.71

and

1.72

Then,

dz = M dx + N dy

1.73

where M and N are functions of x and y. Differentiating M partially with respect to y and N with respect to x,

1.74

And,

1.75