3.8 Combined mathematical statement of the first and second laws of thermodynamics

The second law of thermodynamics was shown in the preceding paragraph to be expressed as

where the sign of equality refers to reversible and the sign of inequality to irreversible processes.

This equation can be written in the following form:

(3.151)

and, respectively, for a unit mass of substance

(3.152)

It will be recalled that in accordance with the equation expressing the first law of thermodynamics,

and for unit mass of substance,

or, which is the same,

and for unit mass of substance,

Substituting the values of dQ and dq from these equations respectively in Eqs. (3.151) and (3.152), we get:

(3.153)

or

(3.154)

and also

(3.155)

or

(3.156)

The above relationships are referred to as the combined mathematical statement of the first and second laws of thermodynamics.

If the only kind of work done by a system is the work of expansion, Eqs. (3.155) and (3.156) take the form

(3.157)

and

(3.158)

Below in the text, these equations will be used mainly with systems in equilibrium, for which the combined equations of the first and second laws of thermodynamics take the following form:

(3.159)

and

(3.160)