Thermodynamic Relations

Some properties like temperature, pressure, volume, and mass can be measured directly while other properties like density and specific volume can be determined from these using some simple relations.

However, properties like internal energy, enthalpy, and entropy are not so easy to determine as they cannot be measured directly or related to easily measurable properties through some simple relations.

Thus, it is important to develop some fundamental relations between commonly encountered thermodynamic properties and express the properties that cannot be measured directly in terms of easily measurable properties.

Partial Derivatives and Associated Relations

Most basic thermodynamic relations involve differentials. Let us consider a function f that depends on a single variable x, i.e., f = f(x) as shown below:

The derivative of the function at a point defined as Slope expressed as:

Therefore, the derivative of a function f(x) with respect to x represents the rate of change of f with x.

Let us now consider a function that depends on two (or more) variables, such as z = z(x, y). This time the value of z depends on both x and y.

It is sometimes necessary to examine the dependence of z on only one of the variables by allowing one variable to change while holding the others constant and observing the change in the function.

The variation of z(x, y) with x when y is held constant is called the partial derivative of z with respect to x, and it is expressed as:

To obtain a relation for the total differential change in z(x, y) for simultaneous changes in x and y would be:

The above equation is the fundamental relation for the total differential of a dependent variable in terms of its partial derivatives with respect to the independent variables. This relation can easily be extended to include more independent variables.

An important relation for partial derivatives is used in calculus to test whether a differential dz is exact or inexact. In thermodynamics, this relation forms the basis for the development of the Maxwell relations which was discussed in a previous blog entry here where dz is replaced by dφ.

We can now develop two important relations for partial derivatives – the reciprocity and the cyclic relations.

The function z = z(x, y) can also be expressed as x = x(y, z) if y and z are taken to be the independent variables. Then the total differential of x becomes:

Eliminating dx from the above equation by combining the dz and dx equations we get:

The variables y and z are independent of each other and thus can be varied independently. For example, y can be held constant (dy = 0), and z can be varied over a range of values (dz ≠ 0). Therefore, for this equation to be valid at all times, the terms in the brackets must equal zero, regardless of the values of y and z. 

Setting the terms in each bracket of the above equation equal to zero gives two equations:

This equation is called the reciprocity relation, and it shows that the inverse of a partial derivative is equal to its reciprocal.

This equation is called the cyclic relation, and it is frequently used in thermodynamics.