#### Adiabatic Reversible Expansion and Compression: Screencasts

The relation between temperature and pressure for an adiabatic reversible process of an ideal gas is derived for a constant heat capacity.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

The relation between temperature and pressure for an adiabatic reversible process of an ideal gas is derived for a temperature-dependent heat capacity.

We suggest that after watching this screencast, you list the important points as a way to increase retention.

##### Important Equations:

These equations are for both closed systems and steady-state flow systems.

\[\Delta S = 0 = \int_{T_1}^{T_2} C_P\,dT – R\,ln\left( \frac{P_2}{P_1} \right) = \int_{T_1}^{T_2} C_V\,dT + R\,ln\left( \frac{V_2}{V_1} \right) \]

Adiabatic reversible compression or expansion for ideal gas with constant heat capacity

\[\Delta S = 0 = C_P\,ln\left(\frac{T_2}{T_1}\right) – R\,ln\left(\frac{P_2}{P_1}\right) = C_V\,\left(\frac{T_2}{T_1}\right) + R\,ln\left(\frac{V_2}{V_1}\right)\]

\[\frac{T_2}{T_1} = \left( \frac{P_2}{P_1}\right)^{\frac{R}{C_P}}\]

\[\frac{T_2}{T_1} = \left( \frac{V_2}{V_1}\right)^{\frac{R}{C_V}}\]

\(W_{EC} = \Delta U = C_V(T_2 – T_1)\)

\(W_S = \Delta H = C_P(T_2 – T_1)\)

where \(T_2\) = final or outlet temperature (K)

\(T_1\) = initial or feed temperature (K)

\(P_2\) = final or outlet pressure (MPa)

\(P_1\) = initial or feed pressure (MPa)

\(V_2\) = final or outlet molar volume (m^{3}/mol)

\(V_1\) = initial or inlet molar volume (m^{3}/mol)

\(R\) = ideal gas constant (J/(mol K))

\(C_P\) = constant-pressure heat capacity (J/(mol K))

\(C_V\) = constant-volume heat capacity (J/(mol K))

\(W_{EC}\) = expansion/compression work (J/mol)

\(W_S\) = shaft work (J/mol)

\(\Delta U\) = change in molar internal energy (J/mol)

\(\Delta H\) = change in molar enthalpy (J/mol)

Adiabatic reversible compression or expansion for ideal gas with temperature-dependent heat capacity

\[C_P = A + BT + CT^2 + DT^3\] where \(A\), \(B\), \(C\), and \(D\) are constants and \(T\) is absolute temperature

\[R\,ln\left(\frac{P_2}{P_1}\right) = A\,ln\left(\frac{T_2}{T_1}\right) + B(T_2 – T_1) + \frac{C}{2} (T_2^2 – T_1^2) + \frac{D}{3} (T_2^3 – T_1^3)\]

\[W_{EC} = \Delta U = \int_{T_1}^{T_2} C_V \,dT\]

\[W_S = \Delta H = \int_{T_1}^{T_2} C_P \,dT\]

\[W_{EC} = A(T_2 -T_1) +\frac{B}{2} (T_2^2 -T_1^2) + \frac{C}{3} (T_2^3 – T_1^3) + \frac{D}{4} (T_2^4 – T_1^4) – R\,(T_2 – T_1)\]

\[W_S = A(T_2 -T_1) +\frac{B}{2} (T_2^2 -T_1^2) + \frac{C}{3} (T_2^3 – T_1^3) + \frac{D}{4} (T_2^4 – T_1^4)\]