Derivation of Critical Constants

Derivation of Critical Constants

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 = Tc and P = Pc    then-
V3 − (b + RTc/Pc)V2 + aV/Pc − ab/Pc = 0     --------(equation 1)
at critical points,
V = Vc
or, V − Vc = 0
or, (V − Vc)3 = 0
or, V3 − 3V2Vc + 3Vc2V − Vc3 = 0     --------(equation 2)
on equation the powers of V in equation (1) and equation (2), we get-
3Vc = RTc/Pc + b     --------(equation 3)
3Vc2 = a/Pc    --------(equation 4)
Vc3 = ab/Pc    --------(equation 5)
dividing equation (5) by equation (4) we get-
Vc/3 = b
or,   Vc = 3b  
putting the value of Vc in equation (4) we get-
27b2 = a/Pc
or,   Pc = a/27b2  
Now putting the value of Vc and Pc in equation (3) we get-
9b − b = 27b2RTc/a
or, 8a = 27bRTc
or,   Tc = 8a/27Rb  

Critical Phenomenon and Critical Constants


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