Step 1: For the given reaction, $2SO_2 (g) + O_2 (g) \rightleftharpoons 2SO_3 (g)$, calculate $\Delta n$.

$$\Delta n = (\text{number of moles of gaseous products}) - (\text{number of moles of gaseous reactants})$$ $$\Delta n = 2 - 3 = -1$$

Step 2: Use the relation between $K_p$ and $K_c$:

$$K_p = K_c(RT)^{\Delta n}$$

Step 3: Rearrange the formula to solve for $K_c$:

$$K_c = \frac{K_p}{(RT)^{\Delta n}} = K_p \times (RT)^{-\Delta n}$$

Since $\Delta n = -1$, $-\Delta n = 1$

$$K_c = K_p \times RT$$

Step 4: Substitute the given values ($K_p = 2.0 \times 10^{10}$ bar$^{-1}$, $T = 450$ K, $R = 0.0831$ L bar K$^{-1}$ mol$^{-1}$):

$$K_c = (2.0 \times 10^{10}) \times (0.0831) \times (450)$$ $$K_c = 2.0 \times 10^{10} \times 37.395$$ $$K_c = 7.479 \times 10^{11}$$