__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:__

__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__

__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.**