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_rP_c     --------(equation 1)
V_r = V/V_c
or, V = V_rV_c    --------(equation 2)
similarly,
T_r = T/T_c
or, T = T_rT_c    --------(equation 3)
Now replacing the value P, V and T in Vander Waal equation, we get-
(P_rP_c + a/V_r²V_c²)(V_rV_c − b) = RT_rT_c    --------(equation 4)
Now putting the values of critical constants in the equation (4) we get-
(P_r a/27b² + a/V_r²9b²)(V_r3b − b) = 8a.RT_r/27Rb
or, (P_r a/27b + a/V_r²9b)(3V_r − 1) = 8a.T_r/27b
multiply the above equation by 27b/a we get-
  (P_r + 3/V_r²) (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.