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Nov 6, 2018

Critical Constants of a Gas

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 (Tc). A gas can only be liquefied when the temperature is kept below Tc of the gas.

Definitions of Critical Constants:

Critical Temperature (Tc):
Critical Temperature (Tc) is the maximum temperature at which a gas can be liquefied, that is the temperature above which a liquid cannot exist.
Critical Pressure (Pc):
Critical Pressure (Pc) is the maximum pressure required to cause liquefaction at the temperature (Tc).
Critical Volume (Vc):
Critical Volume (Vc) is the volume occupied by one mole of a gas at critical temperate (Tc)  and Critical Pressure (Pc).

Andrews Isotherms:

In 1869, Thomas Andrews carried out an experiment in which P - V relations of carbon dioxide gas were measured at different temperatures
Andrews Isotherms
Andrew's Graphs of P Vs V
Following are observed from this graph:
(1) At high temperature, such as T₄, the isotherms look like those of an ideal gas.
(2)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.
(3) 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.
(4) At temperatures Tc the horizontal portion is reduced to a mere point. At temperatures higher then Tc 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 Tthere 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
Continuity of State
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 
This equation can be written as,V³ - (b + RT/P)V² + (a/P)V - ab/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.
Critical Phenomena and Van der Walls Equation
Van der Waals Isotherms
The main characteristics of the above isotherm are as follows:
(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 Tc 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 meta-stable. 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) = 0 and (d²P/dV²) = 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)ᴛ = - {RT/(V - b)²} + 2a/V³

And Curvature = (d²P/dV²)ᴛ = {2RT/(V - b)³} - 6a/V⁴

Hence at the critical point,

- {RTc/(Vc - b)²} + 2a/Vc³ = 0

or, RTc/(Vc - b)² = 2a/Vc³
 and {2RTc/(Vc - b)³} - 6a/Vc⁴ = 0

or, 2RTc/(Vc -b)³ = 6a/Vc

Thus, (Vc - b)/2 = Vc/3

or, Vc = 3b

Putting the value of Vc = 3b in RTc/(Vc - b)² = 2a/Vc³.
We have, RTc/4b² = 2a/27b³
or, Tc = 8a/27Rb
Again the Van der Walls equation at the critical state is
Pc = RTc/(Vc - b) - a/Vc²
Putting the value of Vc and Tc,
We have Pc = a/27b²
Problem:
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.
Answer:
a = 212.3 KPa dm⁶ mol⁻² and b = 0.0189 dm³ mol⁻¹
Full Details Click Here
Compressibility Factor at the Critical State (Zc):
The Critical coefficient is defined as RTc/PcVc
Thus the value of Critical Coefficient = {R × (8a/27Rb)}/{(a/27b²) × 3b} = 8/3 = 2.66
Thus the value of Compressibility factor at the critical state (Zc) = PcVc/RTc = 3/8 = 0.375
Van der Waals constants in terms of critical constants:
Van der Waals constants can be determind from critical constants Tc and Pc of the gas. Vc in the expression is avoided due to difficulty in its determination.
We have, b = Vc/3 but PcVc/RTc = 3/8 or, Vc = (3/8) × (RTc/Pc)
hence, b = (1/8)(RTc/Pc)
Again, a = Pc × 27b²  = 3 × Pc × (3b)² = 3 PcVc² = 3Pc × (3RTc/8Pc)²
∴ a = (27/64)(R²Tc²/Pc) 
Problem:
Calculate Van der Waals constants for Ethylene. (Tc = 280.8 K and Pc = 50 atm)
Answer:
a = 0.057 lit mol⁻¹ and b = 4.47 lit² atm mol⁻²
Full Details Click Here

Problem:
Argon has (Tc = - 122°C, Pc = 48 atm). What is the radius of the Argon atom?
Answer:
1.47 × 10⁻⁸ cm
Full Details Click Here
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