Derivation of Critical Constants

Derivation of Critical Constants

We know that-
(P + a/V²)(V − b) = RT     (for one mole)
or, PV − bP + a/V − ab/V² = RT
multiplying the above equation by V²/P we get-
V³ − V²b + aV/P −ab/P −RTV²/p = 0
or, V³ − (b + RT/P)V² + aV/P − ab/P = 0
when, T = T_c and P = P_c    then-
V³ − (b + RT_c/P_c)V² + aV/P_c − ab/P_c = 0     --------(equation 1)
at critical points,
V = V_c
or, V − V_c = 0
or, (V − V_c)³ = 0
or, V³ − 3V²V_c + 3V_c²V − V_c³ = 0     --------(equation 2)
on equation the powers of V in equation (1) and equation (2), we get-
3V_c = RT_c/P_c + b     --------(equation 3)
3V_c² = a/P_c    --------(equation 4)
V_c³ = ab/P_c    --------(equation 5)
dividing equation (5) by equation (4) we get-
V_c/3 = b
or,   V_c = 3b  
putting the value of V_c in equation (4) we get-
27b² = a/P_c
or,   P_c = a/27b²  
Now putting the value of V_c and P_c in equation (3) we get-
9b − b = 27b²RT_c/a
or, 8a = 27bRT_c
or,   T_c = 8a/27Rb  

Critical Phenomenon and Critical Constants