Editing Isentropic process (section)

=== Derivation of the isentropic relations ===

For a closed system, the total change in energy of a system is the sum of the work done and the heat added:
: <math>dU = \delta W + \delta Q.</math>
The reversible work done on a system by changing the volume is
:<math>\delta W = -p \,dV,</math>
where <math>p</math> is the [[pressure]], and <math>V</math> is the [[Volume (thermodynamics)|volume]]. The change in [[enthalpy]] (<math>H = U + pV</math>) is given by
:<math>dH = dU + p \,dV + V \,dp.</math>

Then for a process that is both reversible and adiabatic (i.e. no heat transfer occurs), <math> \delta Q_\text{rev} = 0</math>, and so <math>dS = \delta Q_\text{rev}/T = 0 </math> All reversible adiabatic processes are isentropic.  This leads to two important observations:
: <math>dU = \delta W + \delta Q = -p \,dV + 0,</math>
:<math>dH = \delta W + \delta Q + p \,dV + V \,dp = -p \,dV + 0 + p \,dV + V \,dp = V \,dp.</math>

Next, a great deal can be computed for isentropic processes of an ideal gas.  For any transformation of an ideal gas, it is always true that
:<math>dU = n C_v \,dT</math>, and <math>dH = n C_p \,dT.</math>

Using the general results derived above for <math>dU</math> and <math>dH</math>, then
: <math>dU = n C_v \,dT = -p \,dV,</math>
: <math>dH = n C_p \,dT = V \,dp.</math>

So for an ideal gas, the [[heat capacity ratio]] can be written as
:<math>\gamma = \frac{C_p}{C_V} = -\frac{dp/p}{dV/V}.</math>

For a calorically perfect gas <math>\gamma</math> is constant. Hence on integrating the above equation, assuming a calorically perfect gas, we get
: <math> pV^\gamma = \text{constant},</math>
that is,
: <math>\frac{p_2}{p_1} = \left(\frac{V_1}{V_2}\right)^\gamma.</math>

Using the [[Equation of state#Classical ideal gas law|equation of state]] for an ideal gas, <math>p V = n R T</math>,
: <math> TV^{\gamma-1} = \text{constant}.</math>     
(Proof: <math>PV^\gamma = \text{constant} \Rightarrow PV\,V^{\gamma-1} = \text{constant} \Rightarrow nRT\,V^{\gamma-1} = \text{constant}.</math> But ''nR'' = constant itself, so <math>TV^{\gamma-1} = \text{constant}</math>.)

: <math> \frac{p^{\gamma-1}}{T^\gamma} = \text{constant} </math>

also, for constant <math>C_p = C_v + R</math> (per mole),
: <math> \frac{V}{T} = \frac{nR}{p}</math> and <math>p = \frac{nRT}{V}</math>

: <math> S_2-S_1 = nC_p \ln\left(\frac{T_2}{T_1}\right) - nR\ln\left(\frac{p_2}{p_1}\right)</math>

: <math> \frac{S_2-S_1}{n} = C_p \ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{T_2 V_1}{T_1 V_2}\right ) = C_v\ln\left(\frac{T_2}{T_1}\right)+ R \ln\left(\frac{V_2}{V_1}\right)</math>

Thus for isentropic processes with an ideal gas,
: <math> T_2 = T_1\left(\frac{V_1}{V_2}\right)^{(R/C_v)}</math> or <math> V_2 = V_1\left(\frac{T_1}{T_2}\right)^{(C_v/R)}</math>