The entropy of a pure substance is determined from tables in the same manner as other properties such as v, u, and h. The value of entropy at a specified state is determined just like any other property.

• Compressed liquid: from tables
• Saturated liquid (s_f): from tables
• Saturated mixture: s = s_f + xs_fg
  where
        x = quality
        s_fg = s_g - s_f
• Saturated vapor (s_g): from tables
• Superheated vapor: from tables

If the compressed liquid data are not available, the entropy of the compressed liquid can be approximated by the saturated liquid at the same temperature.

      s_{compressed@T,P} = s_f@T

For an incompressible substances, the specific volume does not change during a process. That is,

      dv = 0

The internal energy of an incompressible substance is

      du = c(T) dT

Most solids and liquids can be approximated as incompressible substances. The entropy change of solids and liquids can be obtained by applying the above two equations to the first Tds equation.

The entropy change during a process 1-2 is

If the temperature variation (T₂ - T₁) is small over the entire process, an average specific heat (c_ave) is used and treated as a constant (constant-specific-heats assumption). Then the entropy change can be evaluated by

The property relations for ideal gases are:

      pv = RT
      du = c_vdT
      dh = c_PdT

The first relation of entropy change for ideal gases is obtained by replacing P by RT/v and du by c_vdT in the first Tds equation:

      Tds = du + Pdv
            = c_vdT + RTdv/v

      ds = c_vdT/T + Rdv/v

Integrating ds from state 1 to state 2 gives the first relation of entropy change for ideal gases.

The second relation of entropy change for ideal gases is obtained by replacing v by RT/P and dh by c_PdT in the second Tds equation:

      Tds = dh - vdP
            = c_PdT - RTdP/P

      ds = c_PdT/T - RdP/P

Integrating ds from state 1 to state 2 gives the second relation of entropy change for ideal gases.

A common approximation is made by assuming that the specific heats of ideal gases are constants if the temperature variation is small (constant-specific-heats assumption). Replacing c_v and c_P with constants c_v,av and c_P,av in the first and second relations of entropy change and integrating from state 1 to state 2 gives,

Function s⁰ is a Function of
Temperature only

Ideal-gas Properties of Air Table

Approximate Analysis and
Exact Analysis of Entropy Change
during a Cooling Process

If the temperature variation is large, the variation of specific heats with temperature should be considered and relations between c_v(T), c_P(T) and temperature are needed to integrate the first and second relations. For convenience, a function s⁰, standard-state entropy, is defined as

s⁰ is a function of temperature only and is tabulated with temperature. Absolute zero is used as the reference temperature.

Replacing T as T₁ and T₂ in the above definition gives

Then the entropy change of ideal gases during a process 1-2 is