# Real Gases, Compressibility Factor, Vander Waal Equation

Ideal gas:

• The gas that obeys all gas laws at 'T' and 'P' is called real gas.
• Real gas obeys Ideal behaviour at low pressure and high temperature.

Compressibility factor (Z):
It is denoted by 'Z'
\tt Z=\frac{V_{obs}}{V_{ideal}}=\frac{V_{0}P}{nRT} (PV = nRT)
\tt Z=\frac{V_{0}P}{RT} (n = 1)
Z ∝ PV0

At low pressure:
\tt From\ \left(P+\frac{a}{V^2}\right)\left(V-b\right)=RT for 1 mole of gas
\tt \ \left(P+\frac{a}{V^2}\right)(V)=RT
\tt PV = RT-\frac{a}{V}
\tt \frac{PV}{RT}=1-\frac{a}{RTV}
\tt Z=1-\frac{a}{RTV} At low pressure Z=−ve, shows −ve deviation.

At High pressure:
P(V−b) = RT ⇒ PV − Pb = RT
PV = Pb + RT
\tt \frac{PV}{RT}=\frac{Pb}{RT}+1
\tt Z=1+\frac{Pb}{RT}

Abnormal behaviour of H2 and He
\tt \frac{PV}{RT}=1+\frac{Pb}{RT}\Rightarrow Z=\frac{1+Pb}{RT}

### View the Topic in this Video from 2:16 to 36:30

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1. The Vander Waal's equation for n moles of the gas is, \tt \left(P + \frac{n^{2}a}{V^{2}}\right) \left[V - nb\right] = nRT
\tt \left(P + \frac{n^{2}a}{V^{2}}\right) = Pressure correction for molecular attraction, \tt \left[V - nb\right] = Volume correction for finite size of molecules

2. Molar heat capacity of a substance is the quantity of heat required to raise the temperature of 1 mole of the substance by 1° C.
∴ Molar heat capacity = Specific heat capacity × Molecular weight, i.e., Cv = cv × M and Cp = cp × M