For the reaction  $2H{I}_{\left(g\right)}\rightleftharpoons H_{2\left(g\right)}+I_{2\left(g\right)}$ the degree of dissociation $\left(\alpha \right)$ of  $H{I}_{\left(g\right)}$ is related to equilibrium constant Kp by the expression

$2HI\to H_2+I_2$

$\begin{array}{l}initial100\\ atequilibrium1-\alpha \frac{\alpha }{2}\frac{\alpha }{2}\end{array}$

Partial pressum  $=\chi \times p_{total}$

$K_p=\frac{P_{H_2}\times P_{I_2}}{P_{HI}^2},Hence,K_p=\frac{{\left[\left(\frac{\alpha }{2}\right)P_T\right]}^2}{[{(1-\alpha )}^2.P_T^2]},$

$\begin{array}{l}{P}_{H_2}=\left(\frac{\alpha }{2}\right)P_T\\ P_{I_2}=\left(\frac{\alpha }{2}\right)P_T\\ P_{H_I}=(1-\alpha )P_T\end{array}$
By solving above equation  $\alpha =\frac{2\sqrt{K_p}}{1+2\sqrt{K_p}}$