The term "isentropic" means constant entropy. A process during which the entropy remains constant is called an isentropic process, which is characterized by

      ΔS = 0 or s₁ = s₂ for a process 1-2

If a process is both reversible and adiabatic, then it is an isentropic process.

An isentropic process is an idealization of an actual process, and serves as a limiting case for an actual process.

An Isentropic Process of Ideal Gases
on a T-s Diagram

The relations of entropy change for ideal gases are

       (1)
and
       (2)

By setting Δs to 0 in the above equations, the relations for an ideal gas which undergoes an isentropic process can be obtained. Setting equation (1) to zero gives,

Constant Specific Heat Used in
Small Temperature Interval

If the constant-specific-heats assumption is valid, the above equation can be integrated and rearranged to give

where
      k = specific heat ratio, k = c_P/c_v and R = c_P - c_v

The second relation can be obtained by setting equation (2) to zero.

Also, if the constant-specific-heats assumption is valid, the above equation becomes

The third relation can be obtained by combining the first and the second relations. That is,

The three relations of an isentropic process for ideal gases with constant specific heats in compact form are

      Tv^k-1 = constant
      TP^(1-k)/k = constant
      Pv^k = constant

Using P_r Data to Determine
Final Temperature during an
Isentropic Process

Ideal Gas Properties Table -- Air

If the constant-specific-heats assumption is not valid, the entropy change of ideal gases during a process 1-2 is

Setting the above equation to zero and rearranging, one obtains

If exp(s⁰/R) is defined as the relative pressure P_r, then the above equation becomes

      (P₂/P₁)_{s = constant} = P_r2/P_r1

Values of relative pressure are tabulated against temperature in tables.

Using v_r Data to Determine
Final Temperature during an
Isentropic Process

Ideal Gas Properties Table -- Air

In an automotive engine, the ratio v₂/ v₁ is used instead of the ratio P₂/ P₁. The ideal-gas relation gives

      v₂/ v₁ = T₂P₁/T₁P₂

Replacing P₂/P₁ by P_r2/P_r1 in the above equation gives

T/P_r is defined as relative specific volume v_r and its value is also tabulated with temperature. Thus,

      (v₂/v₁)_{s = constant} = v_r2/ v_r1

Isentropic Process of an
Incompressible Substance is
also Isothermal

The entropy change of incompressible fluids or solids during a process 1-2 with constant specific heats is

      Δs = c_aveln(T₂/T₁)

Setting Δs to zero gives

      T₂ = T₁

That is, an isentropic process of an incompressible substance is also isothermal.