For the reaction N₂O₄(g) ⇌ 2NO₂(g), if Kp = 4atm at 300K, the degree of dissociation at a total pressure of 1 atm is:

For $\text{N}_2\text{O}_4 \rightleftharpoons 2\text{NO}_2$:

$$K_p = \frac{4\alpha^2 P}{1 - \alpha^2}$$

Substituting $K_p = 4$ and $P = 1$:

$$4 = \frac{4\alpha^2}{1 - \alpha^2}$$

$$1 - \alpha^2 = \alpha^2 \implies 2\alpha^2 = 1 \implies \alpha = \frac{1}{\sqrt{2}}$$

But $\frac{1}{\sqrt{2}} = \sqrt{\frac{1}{2}}$. Checking options: $\sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}$. The answer is $\alpha = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}$.