Question

At temperature $T$, a compound $A{B}_2\left(g\right)$ dissociates according to the reaction,
$2A{B}_2\left(g\right)\leftrightharpoons 2AB\left(g\right)+B_2\left(g\right)$ with a degree of dissociation ${}^{\prime }x$' which is small compared to unity. The expression for $K_p$ in terms of ${}^{\prime }{x}^{\prime }$ and total pressure $P$ is:

• A. $\frac{P{x}^3}{2}$
• B. $\frac{P{x}^2}{3}$
• C. $\frac{P{x}^3}{3}$
• D. $\frac{P{x}^2}{2}$

Answer 1

The correct option is A $\frac{P{x}^3}{2}$

Let initial moles of $A{B}_2$ is $1$

moles $2A{B}_2\left(g\right)\rightleftharpoons 2AB\left(g\right)+B_2\left(g\right)$ $\Delta ng=3-2=1$

at $t=0$ $1$ - - $nT$

at $Eq{b}^m$ $1-x$ $x$ $\frac{x}{2}$ $1+\frac{x}{2}$

as $K_P=\frac{\left[P_{B_2}\right][P_{AB}{]}^2}{[P_{A{B}_2}{]}^2}$

$\Rightarrow K_P=\frac{\left[n_{B_2}\right][n_{AB}{]}^2}{[n_{A{B}_2}{]}^2}{\left[\frac{RT}{V}\right]}^{\Delta ng}$

$\Rightarrow K_P=\left(\frac{1}{2}\right)\frac{\left[x\right][x{]}^2}{[1-x{]}^2}{\left[\frac{P_T}{n_T}\right]}^{\Delta ng}$ as $x<<1;1-x\simeq 1;1+\frac{x}{2}\simeq 1$

$\Rightarrow K_P=\frac{\left[x\right]\left[x^2\right]}{2}{\left[\frac{P}{1+x/2}\right]}^1$

$\Rightarrow K_P=\frac{x^3.P}{2}$