- Discovered by Andrews for CO
_{2} - Above a certain temperature, it is impossible to liquefy a gas.
- The temperature below which a gas can be compressed by applying only pressure is called critical temperature (T
_{c}). - The pressure applied to liquefy gas is called critical pressure P
_{c} - The volume occupied by one mole of the substance is called critical volume Vc
- The pressure v/s volume plot for CO2 is

- At low temperature, say 13.1˚C, the CO2 is entirely gaseous.
- As pressure is increased, the volume decreases from A to B.
- At B, deviations from Boyle’s law (P V) start. The volume decreases rapidly till C. between B & C, CO2 exists both as a liquid and gas and pressure remains constant.
- After C, CO2 is completely liquefied and on increasing pressure further only slight decrease in volume is seen.
- The pressure corresponding to the horizontal line BC is the vapour pressure of liquid.
- As the temperature is raised, the horizontal portion becomes smaller till 31.1˚C where it reduces to a point Tc. At Tc the boundary between liquid and gas disappears.
- Above 31.1˚C, no liquefaction is seen.
- Thus, it can be seen that liquefaction of gases is possible only at lower temperatures.

**Van der Waal’s Equation and the Critical Point:**

The constant of Van der Waal’s equation can be obtained from the critical point data.

The Van der Waal’s Equation can be arranged as follow:

P = RT / (V – b) – a / V^{2}

To investigate the horizontal point of inflection on a plot of P versus V, we obtain

dP/dV = -RT/ (V – b^{2}) + 2a / V^{3}

d^{2}P/ dV = 2RT/ (V – b)^{3} – 6a/V^{4}

At gas-liquid equilibrium, i.e. at critical-point,

P = Pc

V = Vc

T = Tc

And first order and second order derivative of P with respect to V are zero.

Therefore we can write;

Pc = RTc /(Vc – b) – a / Vc^{3}

0 = RTc /(Vc – b^{2}) + 2a / Vc^{3}

0 = 2RTc / (Vc – b)^{3} – 6a / Vc^{4}

These equations can be solved to get

a = 3P_{c}V^{2}_{c}

b = Vc/3

R = 8 P_{c}V_{c}/ 3T_{c}

(P_{c}V_{c}) / (RT_{c}) = 8/3 = 0.375

Thus,

Zc = 0.375, which is a compressibility factor at critical point.

Since R is a gas constant, and the values of a and b are different for different gases.

They can be calculated in terms of critical-point quantities Pc and Tc by substituting

Vc = 3 RT_{c} / (8P_{c})

a = 27R^{2}T^{2}_{c} / (64P_{c})

b = RTc / (8 Pc)