# Ideal Gas Law and Closed System Formulae

## Ideal Gas Law

$$ PV = mRT $$

Where \( P \) is pressure in \( kPa \).

\( V \) is volume in \( m^3 \).

\( m \) is mass in \( kg \).

\( R \) is the gas constant in \( kJ/kgK \). This is \( 0.287 kJ/kgK \) for air.

\( T \) is temperature in \( K \).

## Specific Heat capacity

\( C_v \) is the specific heat capacity of a gas for a constant volume process. \( C_p \) is the specific heat capacity of a gas for a constant pressure process. For air, this is \( 0.718 kJ/kgK \) and \( 1.005 kJ/kgK\) respectively.

### Work - Process Dependent

#### Constant Volume

$$ Q = m C_v \Delta T $$

#### Constant Pressure

$$ Q = m C_p \Delta T $$

### Internal Energy

$$ \Delta U = m C_v \Delta T $$

Where \( U \) is in \( kJ \).

### Relationship Between Specific Heat Capacities

$$ R = C_p - C_v $$

$$ \gamma = \frac{C_p}{C_v} $$

## Non-Flow Energy Equation

*Assuming change in potential and kinetic energies are negligible.*

$$ Q - W = \Delta U $$

All units are in \( kJ \). \( Q \) and \( W \) are dependent on process.

## Closed System Processes

### Isometric

$$ v_1 = v_2 $$

$$ W = 0 $$

$$ Q = m C_v \Delta T $$

### Isobaric

$$ p_1 = p_2 $$

$$ W = P \Delta V $$

$$ Q = m C_p \Delta T $$

### Isothermal

$$ T_1 = T_2 $$

$$ W = Q = mRT ln( \frac{v_2}{v_1} ) = P_2 V_2 ln( \frac{v_2}{v_1} ) $$

### Isentropic

$$ s_1 = s_2 $$

$$ P_1 V_1 ^ \gamma = P_2 V_2 ^ \gamma = Constant $$

$$ W = \frac{P_2 V_2 - P_1 V_1}{1 - \gamma} $$

$$ Q = 0 $$

$$ (\frac{T_2}{T_1}) = (\frac{V_1}{V_2})^{\gamma - 1} = (\frac{P_2}{P_1})^{\frac{\gamma -1}{\gamma}} $$

### Polytropic

$$ P_1 V_1 ^n = P_2 V_2 ^n = Constant $$

$$ W = \frac{P_2 V_2 - P_1 V_1}{1 - n} $$

$$ W = \frac{mR \Delta T}{1-n} $$

$$ Q = \Delta U + W $$