# Derivation of Critical Constants

## Derivation of Critical Constants

We know that-(P + a/V

^{2})(V − b) = RT (for one mole)

or, PV − bP + a/V − ab/V

^{2}= RT

multiplying the above equation by V

^{2}/P we get-

V

^{3}− V

^{2}b + aV/P −ab/P −RTV

^{2}/p = 0

or, V

^{3}− (b + RT/P)V

^{2}+ aV/P − ab/P = 0

when, T = T

_{c}and P = P

_{c}then-

V

^{3}− (b + RT

_{c}/P

_{c})V

^{2}+ 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, V

^{3}− 3V

^{2}V

_{c}+ 3V

_{c}

^{2}V − V

_{c}

^{3}= 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

_{c}

^{2}= a/P

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

V

_{c}

^{3}= ab/P

_{c}--------(equation 5)

dividing equation (5) by equation (4) we get-

V

_{c}/3 = b

or,

**V**

_{c}= 3bputting the value of V

_{c}in equation (4) we get-

27b

^{2}= a/P

_{c}

or,

**P**

_{c}= a/27b^{2}Now putting the value of V

_{c}and P

_{c}in equation (3) we get-

9b − b = 27b

^{2}RT

_{c}/a

or, 8a = 27bRT

_{c}

or,

**T**

_{c}= 8a/27Rb