Relationship Between Critical Constants And Vander Waal’s Constants
Vanderwaals equation for one mole of gas is,
\( \left( {P + \frac{{a}}{{V^2 }}} \right)\left( {V – n} \right) = RT \)
\(PV+\frac{a}{V}-Pb-\frac{ab}{V^2}=RT\)
Multiplying both sides by \(\frac{V^2}{P}\) and arranging the obtained equation in decreasing powers of V, we get
\(V^3-(b+\frac{RT}{P})V^2 +\frac{aV}{P} -\frac{ab}{P}=0\tag1\\\)
viz., a cubic equation in V, i.e.,For any fixed value of T & P, we have 3 values of V.
All three values may be real or only one may be real and other two imaginary.
At critical temperature \(({T}_c)\) , the three values of V merge into one another.
Thus at critical temperature and pressure, the volume of gas represents the critical volume of gas.
\(\therefore\) V= \({V}_c\)
⇒\(V-{V}_c=0\)
⇒\(({V-{V}_c})^3=0\)
⇒\({V}^3-3{V}_c{V}^2+3{{V}_c}^2 V-{{V}_c}^3 = 0    \tag{2}\\ \)
At \(T={T}_c\) and \(P={P}_c\)  ,equation 1 must be identical with equation 2
Put \(T={T}_c\) and \(P={P}_c\) in equation 1
\({V}^3-(b+\frac{R {T}_c}{{P}_c})V^2 +\frac{aV}{{P}_c} -\frac{ab}{{P}_c}=0 \tag{3}\\ \)
Since equation 2 and 3 are identical and hence the coefficients of equal powers of V in both equations must be equal, i.e.,
\(3 {V}_C = b + \frac{R{T}_C}{{P}_C} \tag{4}\\ \)
\(3{{V}_c}^2=\frac{a}{{P}_c}\tag5\\\)
⇒\(\fbox{\(a=3{P}_c{{V}_c}^2 \)}\)
Also \({V}_c^3=\frac{ab}{{P}_c} \tag6\\\)
Dividing equation 6 by equation 5 ,
\(\frac{{V}_C}{3}=\frac{a}{ab}\)
⇒ \({V}_c=\frac{3}{b}\)
⇒\(\fbox{\(b=\frac{{V}_C}{3}\)}\)
Substituting value of \(V_c\)Â in equation 5,we get
\({P}_c=\frac{a}{27{b}^2}\tag{6}\\ \)
⇒\({P}_c=\frac{a}{27{(\frac{{V}_C}{3})}^2}\)    \( \because\) \(b=\frac{{V}_C}{3}\)
⇒\({P}_c=\frac{a}{3{V_c}^2}\)
\(a=3{P}_C{{V}_C}^2\)
Substituting the values of \({V}_C\) and \({P}_C\) in equation 3, we get
\({T}_C=\frac{8a}{27Rb}\)
Substituting the value of a and b in above equation, we have
\({T}_C = \frac{8×3{P}_C{V}_C}{27R(\frac{{V}_C}{3})}\)
⇒\({T}_C=\frac{8}{3}\frac{{P}_C{V}_C}{R}\)
⇒\(\fbox{\({P}_c {V}_c\) = \(\frac{3}{8} R{T}_c\)}\)
viz., the relation between \({P}_c,{V}_c\) and \({T}_c\)
Vanderwaals constants in terms of critical constants
\(\fbox{\(a=3{P}_c{{V}_c}^2\)}\)
by putting value of \({V}_c\) from equation \({P}_c{V}_c=\frac{3}{8} R{T}_c\), we have
\(\fbox{\(a={\frac{27}{64}} {\frac{{R}^2}{{{T}_C}^2}{{P}_C}}\)}\)
\(\fbox{\(b=\frac{{V}_c}{3}\)}\)
by putting value of \({V}_c\) from equation \({P}_c{V}_c=\frac{3}{8} R{T}_c\), we have
\(\fbox{\(b=\frac{R{T}_c}{8{P}_c}\)}\)
