Kc/Kp for reaction: (g) + 1/2 (g) ⇌ (g) is — Chemical Equilibrium Chemistry Question

💡 Solution & Explanation

Step 1 - State the chemical equation and identify the physical states The given homogeneous gas-phase reaction represents the oxidation of carbon monoxide (\ce{CO}) gas to carbon dioxide (\ce{CO2}) gas: \[\ce{CO(g) + 1/2 O2(g) <=> CO2(g)}\] All the reactants and products are in the gaseous state, which means they will all contribute to both the concentration-based equilibrium constant ($K_c$) and the pressure-based equilibrium constant ($K_p$). Step 2 - Calculate the change in the number of gaseous moles ($\Delta n_g$) The change in the number of moles of gaseous species ($\Delta n_g$) between products and reactants is given by the formula: \[\Delta n_g = \sum n_{\text{g, products}} - \sum n_{\text{g, reactants}}\] Using the stoichiometric coefficients from the balanced chemical equation: * Number of moles of gaseous products = $1$ (from $1\text{ mol}$ of $\ce{CO2}$) * Number of moles of gaseous reactants = $1 + \frac{1}{2} = 1.5$ (from $1\text{ mol}$ of $\ce{CO}$ and $0.5\text{ mol}$ of $\ce{O2}$) Substituting these values into the formula: \[\Delta n_g = 1 - \left(1 + \frac{1}{2}\right) = 1 - 1.5 = -0.5 = -\frac{1}{2}\] Step 3 - Relate the equilibrium constants $K_p$ and $K_c$ The thermodynamic relationship between $K_p$ (equilibrium constant in terms of partial pressures) and $K_c$ (equilibrium constant in terms of molar concentrations) is: \[K_p = K_c(RT)^{\Delta n_g}\] Where: * $R$ is the universal gas constant. * $T$ is the absolute temperature in Kelvin. Substituting the calculated value of $\Delta n_g = -1/2$ into the relation: \[K_p = K_c(RT)^{-1/2}\] Step 4 - Calculate the ratio of $K_c$ to $K_p$ To find the required ratio $\frac{K_c}{K_p}$, we rearrange the relationship: \[\frac{K_p}{K_c} = (RT)^{-1/2}\] Taking the reciprocal on both sides to solve for $\frac{K_c}{K_p}$: \[\frac{K_c}{K_p} = \frac{1}{(RT)^{-1/2}} = (RT)^{1/2} = \sqrt{RT}\] Thus, the ratio of $K_c$ to $K_p$ is: \[\frac{K_c}{K_p} = \boxed{\sqrt{RT}}\] Step 5 - Explain each option * **(A) $1 / \sqrt{RT}$**: Incorrect. This represents the reciprocal ratio, which is $\frac{K_p}{K_c} = (RT)^{-1/2} = \frac{1}{\sqrt{RT}}$. * **(B) $\sqrt{RT}$**: Correct. As derived above, the ratio $\frac{K_c}{K_p}$ is mathematically equal to $(RT)^{1/2}$ or $\sqrt{RT}$. * **(C) $1/RT$**: Incorrect. This would be the correct ratio if the value of $\Delta n_g$ were $-1$. * **(D) $1$**: Incorrect. The ratio is equal to $1$ only when $\Delta n_g = 0$, meaning $K_p = K_c$, which is not the case for this reaction.