For the given reaction: $\text{CO}_2(g) + \text{C}(s) \rightleftharpoons 2\text{CO}(g)$

The relationship between equilibrium constants $K_p$ and $K_c$ is given by the equation: $K_p = K_c(RT)^{\Delta n}$

where $\Delta n$ is the change in the number of moles of gaseous products and gaseous reactants. For the reaction, $\Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants}$ $\Delta n = 2 - 1 = 1$ (Carbon is in the solid state, so it is not included)

Given values: $K_p = 3.0$ $R = 0.083\text{ L bar K}^{-1} \text{mol}^{-1}$ $T = 1000\text{ K}$

Substitute the values into the equation: $3.0 = K_c (0.083 \times 1000)^1$ $3.0 = K_c (83)$ $K_c = \frac{3.0}{83}$ $K_c \approx 0.03614$ $K_c \approx 3.6 \times 10^{-2}$

Thus, the value of $K_c$ is $3.6 \times 10^{-2}$.