At \(450^{\circ} \mathrm{C}\), the equilibrium constant for the reaction \(2 \mathrm{HI} \rightleftarrows \mathrm{H}_{2}+\mathrm{I}_{2}\) is \(K_{p}=50\). Find the dissociation grade of HI.

The balanced chemical reaction is: $$2\text{HI} \rightleftarrows \text{H}_2 + \text{I}_2$$ The equilibrium expression for this reaction is: $$K_p = \frac{[\text{H}_2][\text{I}_2]}{[\text{HI}]^2}$$

Let the initial concentration of HI be \(c\) moles/L, and the dissociation grade be \(\alpha\). At equilibrium, the concentration of HI will be \((c - 2\alpha)\). The concentrations of H\(_2\) and I\(_2\) will be \(\alpha\) moles/L each.

Substitute the equilibrium concentrations in the equilibrium expression: $$K_p = \frac{\alpha \cdot \alpha}{(c - 2\alpha)^2}$$

Given \(K_p = 50\), we can solve the equation for \(\alpha\): $$50 = \frac{\alpha^2}{(c - 2\alpha)^2}$$ Rearrange and rewrite the equation: $$\alpha^2 = 50(c - 2\alpha)^2$$ Since \(c\) is a constant, let's substitute \(x = \frac{\alpha}{c}\), which simplifies the equation to: $$x^2 = 50(1 - 2x)^2$$ Solve for \(x\): $$x^2 = 50 - 200x + 200x^2$$ $$199x^2 - 200x + 50 = 0$$ This is a quadratic equation. Apply the quadratic formula to find \(x\): $$x = \frac{-(-200)\pm \sqrt{(-200)^2 - 4(199)(50)}}{2(199)}$$ After solving, we obtain two possible values of \(x\): 0.4070 and 0.6127. It is important to note that the dissociation grade should be between 0 and 1. Here, both values are between 0 and 1, but only one will lead to physically meaningful results. The dissociation grade of HI is: $$\alpha = c \cdot x$$ If \(\alpha > 0.5c\), it means more than half of the initial concentration of HI has been dissociated. However, \(\alpha\) and \(c - 2\alpha\) must both be positive and greater than or equal to zero. If we choose x = 0.6127, it would imply that more than half of the HI has dissociated, which would lead to a negative concentration of HI \((c - 2\alpha)\). This would not make sense in a real scenario, so we choose x = 0.4070. Therefore, the dissociation grade of HI is: $$\alpha = 0.4070c$$