The Law of Corresponding States

The Law of Corresponding States

In 1881, van der Waals showed that if pressure, volume and temperature of a gas are expressed in terms of its critical pressure, critical volume and critical temperature, then we can obtain an important generalization, viz., known as the law (principle) of corresponding states.

The Law of corresponding states can be deduced from the van der Walls reduced equation of state as follow:

Derivation of Reduced Equation of State:

Let
Reduced Pressure \(P_r\) or \(\pi\)=\(\frac{P}{P_c}\)
$$\Rightarrow{P=P_rP_c}$$   
Reduced Volume \(V_r\) or \(\phi\)=\(\frac{V}{V_c}\)
$$\Rightarrow{V=V_r V_c}$$
Reduced Temperature \(T_r\) or \(\theta\)=\(\frac{T}{T_c}\)
$$\Rightarrow{T=T_rT_c}$$

As we know van der Walls equation for one mole of gas is
$$(P+\frac{a}{V^2})(V-b)=RT$$

Substituting the values of P, V and T in terms of van der Walls constants in above equation, we have
$$(\pi P_c + \frac{a}{\phi^2V_c²})(\phi V_c-b)=R\theta T_c$$

Now substitute the valve of \(P_c,V_c\) and \(T_c\) in terms of a and b, i.e., substitute

\({P}_C=\frac{a}{27{b}^2}\)
\({V}_C = 3b\)
\({T}_C=\frac{8a}{27Rb}\)

$$\Rightarrow(\pi\frac{a}{27\,b^2} +\frac{a}{\phi^2\, 9b^2})(\phi (3b)-b)=R\theta \frac{8a}{27Rb}$$

$$\Rightarrow(\pi a+\frac{3a}{\phi^2})(3b\phi -b)=8\theta \,ab$$

$$\Rightarrow(\pi +\frac{3}{\phi^2})(3\phi -1)=8\theta$$
This equation is called as reduced equation of state

The Significance of Reduced equation of state :

As this equation does not involve the constant terms a, b and R. Hence it is a general equation applicable to all substances in the liquid and gaseous state, provided their molecules are spherical.

Law of corresponding states

It is deduced from the reduced equation of state.

It states that :
“When two or more substances have the same reduced temperature and the same reduced pressure, then they must have the same reduced volumes.”
This statement is known as the Principle (law) of corresponding states.

Two or more substances having the same  reduced temperature and the same reduced pressure and thus having the same reduced volume are said to be in corresponding states.

Note:
• This principle is only approximate.
• It fails, when the molecules of the gas are non-spherical or polar.

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