Liquefaction of gases

• Discovered by Andrews for CO₂ 
• 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 shown in the diagram.
  
  
• _{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 / V²

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

dP/dV = -RT/(V – b²) + 2a / V³

d²P/dV = 2RT/(V - b)³ – 6a/V⁴

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 / Vc³

0 = RTc /(Vc – b²) + 2a / Vc³

0 = 2RTc / (Vc - b)³ – 6a / Vc⁴

These equations can be solved to get

a = 3P_cV²_c

b = Vc/3

R = 8 P_cV_c/3T_c

(P_cV_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 = 27R²T²_c / (64P_c)

b =  RTc / (8 Pc)