Question

Find the critical constant $(P_c,V_{c}and{T}_c)$ in terms of $A$ and $B$, also find compressibility factor ($Z$) for the following equation of state.
$PV=RT-\frac{A}{V}+\frac{2B}{V^2}$
where $A$ and $B$ are constant, $P$ = pressure and $V$ = molar volume.

• A. $V_c=\frac{6B}{A},T_c=\frac{A^2}{6RB},P_c=\frac{A^3}{108RB}$, compressibility factor $=\frac{P_c{V}_c}{R{T}_c}\frac{1}{3}$
• B. $V_c=\frac{3B}{A},T_c=\frac{A^2}{3RB},P_c=\frac{A^3}{108RB}$, compressibility factor $=\frac{P_c{V}_c}{R{T}_c}\frac{1}{3}$
• C. $V_c=\frac{6B}{A},T_c=\frac{A^2}{6RB},P_c=\frac{A^3}{54RB}$, compressibility factor $=\frac{P_c{V}_c}{R{T}_c}\frac{1}{2}$
• D. None of these

Answer 1

The correct option is A $V_c=\frac{6B}{A},T_c=\frac{A^2}{6RB},P_c=\frac{A^3}{108RB}$, compressibility factor $=\frac{P_c{V}_c}{R{T}_c}\frac{1}{3}$

$PV=RT-\frac{A}{V}+\frac{2B}{V^2}$
$V^3-\frac{RT{V}^2}{P}+\frac{A}{P}V-\frac{2B}{P}=0$
At critical point [$5\ast 1\ast 84$]$(V-V_c{)}^3=0$
$V^3-3V_c{V}^2+3V_c^{2}V-V_c^3=0$
so on ($8:54$)
$3V_c^2=A/P$ ....(i)
$3V_c=RT/P$ .....(ii)
$V_c^3=2B/P$ .....(iii)
On $\frac{\left(3\right)}{\left(1\right)}\Rightarrow \frac{V_c}{3}=\frac{2B}{A}$
$=V_c=6B/A$
from equation (i)
$3{\left(\frac{6B}{A}\right)}^2=A/P_c\Rightarrow P_c=\frac{A^2}{108B^2}$
from
equation (ii) $T_c=\frac{3P_c{V}_c}{R}=\frac{3}{R}\left(\frac{A^3}{108B^2}\right)\left(\frac{6B}{A}\right)$
$T_c=\frac{A^2}{6RB}\Rightarrow Z=\frac{P_c{V}_c}{R{T}_c}=\frac{\frac{A^3}{108B^2}.\frac{6B}{A}}{R.\frac{A^2}{6RB}}=\frac{1}{3}$