Compressibility of Gases - Concept of Corresponding States

Compressibility of gases: Concept of corresponding states
Background
The simplest definition of corresponding state in one sentence is "For most gases, when the
ratio of actual temperature to critical temperature or the ratio of actual pressure to critical
pressure is the same, they exhibit similar thermodynamic properties."
What is corresponding state theory?
Van der Walls discovered through experiments that most gases have comparable properties
close to the critical point. Following that, he put forth the "Theory of Corresponding State."
According to the theory of corresponding states, the thermodynamic properties of most gases
become similar or "correspond" to one another when the temperature is expressed as a
percentage of the critical temperature (reduced temperature, Tr). According to the theory of
corresponding states, the various thermodynamic properties, including enthalpy, specific
heat, compressibility factor, and phase behaviour, approach similar values when gases are
compared at the same reduced temperature (Tr). The theory of corresponding states makes
use of the idea of reduced variables, such as reduced temperature and reduced pressure. It is
possible to compare various substances and analyse their behaviour close to the critical point
by expressing the temperature and pressure in terms of these condensed variables.
The theory of corresponding states is primarily applicable to simple, non-polar gases where
intermolecular forces are primarily due to London dispersion forces. For complex molecules
with additional intermolecular forces like H bonds, the theory may not accurately predict their
behaviour.
Explanation
What are reduced properties
In thermodynamics, the reduced properties of a fluid are a set of state variables scaled by the
fluid's state properties at its critical point. Reduced properties provide a measure of the
“departure” of the conditions of the substance from its own critical conditions and are defined
as follows:
Pr = P/Pc, Tr = T/Tc and Vr = V/Vc P, T and V are temperature, pressure and volume. Subscript c
stands for critical point and r stands for reduced temperature and pressure. If Pr = Tr = Vr = 1,
the substance is at its critical condition. If we are beyond critical conditions, Tr > 1, Pr > 1 and
Vr > 1. By the same token, if all the conditions are subcritical, Tr < 1, Pr < 1 and Vr < 1. Critical
conditions become the scaling factor by which substances can be compared among each other
in terms of their “departure from criticality” or reduced properties.
How to find reduced specific volume
The reduced specific volume of a fluid is computed from the ideal gas law at the substance's
critical pressure and temperature:
Vr = V Pc/R Tc
This property is useful when the specific volume and either temperature or pressure are
known, in which case the missing third property can be computed directly.
How and why corresponding state work?
Why all gases have same compressibility, Z at reduced temperature, Tr
van der Walls discovery
The ratio of a gas's actual temperature to its critical temperature is known as the reduced
temperature, or Tr. The ratio of actual temperature to its critical pressure is defined as critical
pressure or Pr.
Van der Walls discovered that, when plotting the compressibility factors of various gases
against reduced temperature and reduced pressure , all the curves condense onto a single
curve known as the "compressibility chart" or "Z chart" when expressed in terms of reduced
temperature and pressure.
It can be seen above that all of the curves overlap and get closer to the value of 1 as the
reduced pressure gets closer to 1, showing the compressibility of all gases as a function of
reduced pressure. This indicates that gases behave similarly, and their compressibility tends to
be close to one at low temperatures.
Due to the dominance of intermolecular forces over kinetic energy at lower temperatures, the
compressibility of all gases equalises and becomes independent of the particular gas under
consideration.
An indicator of a gas's departure from ideal gas behaviour is its compressibility factor (Z),
where Z=1 denotes ideal gas behaviour. Most gases behave differently than they would under
ideal gas conditions at low temperatures and moderate pressures, and at these conditions,
different gases have different compressibility factors.
This confirms what van der Walls wrote in his theory of corresponding state
“The principle of corresponding states that gases may behave differently at different
temperatures and pressures, but at the same reduced temperature (Tr) and reduced pressure
(Pr), the compressibility factor of all gases tends to converge to the same value, regardless of
their individual chemical properties. Therefore, since the reduced pressure Pr is the same for all
gases, their compressibility factor Z will also be the same.
Fundamentals
All gases have the same compressibility at reduced temperature because at low temperatures,
all gases tend to behave similarly and follow similar physical laws. At lower temperatures, the
particles of gases move slower, and their kinetic energy decreases. Due to this reduction in
kinetic energy, the intermolecular forces between the molecules become more dominant,
causing the gases to condense and become more compressible. As the temperature is
reduced, the effect of these intermolecular forces becomes more pronounced, and the
compressibility of gases becomes more similar regardless of the specific gas being considered.
Therefore, all gases have similar compressibility at reduced temperatures.

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