1. Write the equilibrium reaction

Graphite reacts with carbon dioxide as:

$\mathrm{CO_2(g) + C(s) \rightleftharpoons 2CO(g)}$

Since graphite is a solid, it does not appear in the expression for $K_p$.

2. Initial pressure of gases

Initially, only $\mathrm{CO_2}$ is present:

$P_{\mathrm{CO_2}}^{\text{initial}} = 0.5\ \text{atm}, \qquad P_{\mathrm{CO}}^{\text{initial}} = 0$

Let the decrease in pressure of $\mathrm{CO_2}$ due to reaction be $x$ atm.

Then at equilibrium:

$P_{\mathrm{CO_2}} = 0.5 - x$ $P_{\mathrm{CO}} = 2x$

3. Use total pressure at equilibrium

Given total pressure at equilibrium is $0.8$ atm:

$P_{\text{total}} = P_{\mathrm{CO_2}} + P_{\mathrm{CO}}$

So,

$0.8 = (0.5 - x) + 2x = 0.5 + x$

Thus,

$x = 0.3$

Therefore,

$P_{\mathrm{CO_2}} = 0.5 - 0.3 = 0.2\ \text{atm}$ $P_{\mathrm{CO}} = 2(0.3) = 0.6\ \text{atm}$

4. Write the expression for $K_p$

For

$\mathrm{CO_2(g) + C(s) \rightleftharpoons 2CO(g)}$

$K_p = \frac{(P_{\mathrm{CO}})^2}{P_{\mathrm{CO_2}}}$

Substitute the equilibrium pressures:

$K_p = \frac{(0.6)^2}{0.2} = \frac{0.36}{0.2} = 1.8\ \text{atm}$

5. Match with options

$K_p = 1.8\ \text{atm}$

So the correct option is D.

6. Comparison with stored answer

Stored correct answer: D

My derived answer: D

They agree.