# 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

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

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

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

$$Q = \Delta U + W$$