Critical Constants of a Gas:
A gas can be liquefied by lowering temperature and increasing of pressure. But influence of temperature is more important. Most of gases are liquefied at ordinary pressure by suitable lowering of the temperature. But a gas can not be liquefied unless its temperature is below of a certain value depending upon the nature of the gas whatever high the pressure may be applied.
This temperature of the gas is called its Critical Temperature (T_{C}). A gas can only be liquefied when the temperature is kept below T_{C} of the gas.
Definitions of Critical Constants:
Critical Temperature (T_{C}):
Critical Temperature (T_{C}) is the maximum temperature at which a gas can be liquefied, that is the temperature above which a liquid cannot exist.
Critical Pressure (P_{C}):
Critical Pressure (P_{C}) is the maximum pressure required to cause liquefaction at the temperature (T_{C}).
Critical Volume (V_{C}):
Critical Volume (V_{C}) is the volume occupied by one mole of a gas at critical temperate (T_{C}) and Critical Pressure (P_{C}).
Andrews Isotherms:
In 1869, Thomas Andrews carried out an experiment in which P  V relations of carbon dioxide gas were measured at different temperatures.
Andrew's Graphs of P Vs V 
Following are observed from this graph:
 At high temperature, such as T₄, the isotherms look like those of an ideal gas.
 At low temperatures, the curves have altogether different appearances. Consider, for examples, a typical curve abcd. As the pressures increases, the volume of the gas decreases (curve a to b). At point b liquefaction commences and the volume decreases rapidly as the gas converted to a liquid with much higher density. This conversion takes place at constant pressure P. At the point C, liquefaction is complete and thus the cd is evidence of the fact that the liquid cannot be easily compressed. Thus, we note that ab represents the gaseous state, bc, liquid and vapour in equilibrium, and cd shows the liquid state only.
 At still higher temperatures we get a similar type of curve as discussed in the above point, except that the horizontal portion is reduced. The Pressure corresponding to this portion being higher then at lower temperatures.
 At temperatures T_{C} the horizontal portion is reduced to a mere point. At temperatures higher then T_{C}there is no indication of qualification at all.
Thus for every gas, there is a limit of temperature above which it cannot be liquefied, no matter what the pressure is.
Continuity of State:
It appears from the Amagat curve at T there is discontinuity or break during the transformation of gas to liquid. But it is not so.
The continuity of the states from the gas to liquid can be explains from the following isotherm pqrs at T
Schematic Representation of the Continuity of State 
The gas at A is heated to B at constant volume (V) along AB. Then the gas is gradually cooled at constant P along BC, the volume is reduced considerably. The gas is again cooled at constant V until the point D is reached.
No where in the process liquid would appear. At D, the system is highly compressed gas. But from the curve, this point is the representation of the liquid state. Hence there is hardly a distinction between the liquid state and the gaseous state. There are no line of separation between the two phases. This is known as the principle of continuity of state.
Critical Phenomena and Van der Walls Equation:
For one mole of a gas the Van der Waals equation,
(P + a/V²)(V  b) = RT
V³(b + RT/P)V²+(a/P)Vab/P = 0 
This equation has three roots in V for given values of a, b, P and T. It found that either all the three roots are real or one is real and the other two are imaginary.
Van der Waals Isotherms 
(a) At higher temperature and higher volume region, the isotherms looks much like the isotherms for a Ideal gas.
(b) At the temperature lower than T_{C} the isotherm exhibits a maximum and a minimum. For certain values of pressure, the equation gives three roots of volume as V₁, V₂ and V₃ at pressure P₁. The section AB and ED of the Van der walls curve at T₂ can be realized experimentally. ED represents Supersaturated or Super cooled vapour and AB represents super heated liquid. Both these states are metastable. These states are unstable in the seance that slight disturbances are sufficient to cause the system to revert spontaneously into the stable state with the two phases present in equilibrium.
(c) The section BCD of the Van der Walls isotherms cannot be realized experimentally. In this region the slope of the P  V curve is positive. Increasing or decreasing the volume of such a system would increase or decrease of pressure. The line BCD represents the meta stable state.
Expression of Critical Constants in Terms of Van der Waals Constants:
Again with increase of temperature the minimum and maximum points come close to each other and at the critical point (C) both maximum and minimum coalesce. The slope and curvature both becomes zero at this point.
Thus the mathematical condition of critical point is,
(dP/dV)_{T} = 0 and (d²P/dV²)_{T} = 0

Van der Waals equation for 1 mole gas is,
(P + a/V2)(V  b) = RT or, P = RT/(V  b)  a/V2 
Differentiating Van der Waals equation with respect to V at constant T,
We get Slope
(dP/dV)_{T} =  {RT/(V  b)²} + 2a/V³
And Curvature
(d²P/dV²)_{T} = {2RT/(V  b)³}  6a/V⁴ 
Hence at the critical point,
 {RT_{C}/(V_{C}  b)²} + 2a/V_{C}³ = 0
or, RT_{C}/(V_{C}  b)² = 2a/V_{C}³

and {2RT_{C}/(V_{C}  b)³}  6a/V_{C}⁴ = 0

or, 2RT_{C}/(V_{C} b)³ = 6a/V_{C}⁴
Thus, (V_{C}  b)/2 = V_{C}/3
V_{C} = 3b

Putting the value of V_{C} = 3b
in RT_{C}/(V_{C}  b)² = 2a/V_{C}³.
We have, RT_{C}/4b² = 2a/27b³
T_{C} = 8a/27Rb 
Again the Van der Walls equation at the critical state is
P_{C} = RT_{C}/(V_{C}  b)  a/V_{C}²
Putting the value of V_{C} and T_{C},
P_{C} = a/27b² 
The Critical constants for water are 647 K, 22.09 MPa and 0.0566 dm³mol⁻¹. Calculate the value of Van der Waals constants a and b.
a = 212.3 KPa dm⁶ mol⁻² and b = 0.0189 dm³ mol⁻¹
Compressibility Factor at the Critical State (Z_{C}):
The Critical coefficient is defined as,
RT_{C}/P_{C}V_{C}
RT_{C}/P_{C}V_{C}
Thus the value of Critical Coefficient =
{R × (8a/27Rb)}/{(a/27b²) × 3b}
= 8/3
= 2.66
{R × (8a/27Rb)}/{(a/27b²) × 3b}
= 8/3
= 2.66
Thus the value of Compressibility factor at the critical state (Zc)
= P_{C}V_{C}/RT_{C} = 3/8 = 0.375
= P_{C}V_{C}/RT_{C} = 3/8 = 0.375
Van der Waals constants in terms of critical constants:
Van der Waals constants can be determined from critical constants T_{C} and P_{C} of the gas. V_{C} in the expression is avoided due to difficulty in its determination.
We have, b = V_{C}/3
but P_{C}V_{C}/RT_{C} = 3/8
or, V_{C} = (3/8) × (RT_{C}/P_{C})
but P_{C}V_{C}/RT_{C} = 3/8
or, V_{C} = (3/8) × (RT_{C}/P_{C})
hence, b = (1/8)(RT_{C}/P_{C})
Again, a = P_{C} × 27b²
= 3 × P_{C} × (3b)²
= 3 P_{C}V_{C}²
= 3P_{C} × (3RT_{C}/8P_{C})²
= 3 × P_{C} × (3b)²
= 3 P_{C}V_{C}²
= 3P_{C} × (3RT_{C}/8P_{C})²
∴ a = (27/64)(R²T_{C}²/P_{C}) 
Calculate Van der Waals constants for Ethylene. (Tc = 280.8 K and P_{C}= 50 atm)
a = 0.057 lit mol⁻¹ and b = 4.47 lit² atm mol⁻²
1.47 × 10⁻⁸ cm