# Law of Corresponding State or Reduced Equation of State

## Law of Corresponding State or Reduced Equation of State

According to this law,two different gases behave similarly, if their reduced properties (i.e. P, V and T) are same.

The ratio of P to P

_{c}, V to V

_{c}and T to T

_{c}are called reduced pressure(P

_{r}), reduced volume(V

_{r}) and reduced temperature(T

_{r}) respectively.

so, P

_{r}= P/P

_{c}

or, P = P

_{r}P

_{c}--------(equation 1)

V

_{r}= V/V

_{c}

or, V = V

_{r}V

_{c}--------(equation 2)

similarly,

T

_{r}= T/T

_{c}

or, T = T

_{r}T

_{c}--------(equation 3)

Now replacing the value P, V and T in Vander Waal equation, we get-

(P

_{r}P

_{c}+ a/V

_{r}

^{2}V

_{c}

^{2})(V

_{r}V

_{c}− b) = RT

_{r}T

_{c}--------(equation 4)

Now putting the values of critical constants in the equation (4) we get-

(P

_{r}a/27b

^{2}+ a/V

_{r}

^{2}9b

^{2})(V

_{r}3b − b) = 8a.RT

_{r}/27Rb

or, (P

_{r}a/27b + a/V

_{r}

^{2}9b)(3V

_{r}− 1) = 8a.T

_{r}/27b

multiply the above equation by 27b/a we get-

**(P**

_{r}+ 3/V_{r}^{2}) (3V_{r}− 1) = 8T_{r}This equation is completely free from constants such as R, a, b hence this is applicable to all substances n fluid state.This is law of corresponding states.