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#### 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 (m3/mol)

$$V_1$$ = initial or inlet molar volume (m3/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

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

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

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

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

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