For the reaction $\mathrm{CO}{(g)} + \frac{1}{2} \mathrm{O}{2(g)} \rightarrow \mathrm{CO}_{2(g)}$, $K_p / K_c$ is:

A. $RT$ B. $(RT)^{-1}$ C. $(RT)^{-1/2}$ D. $(RT)^{1/2}$

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To determine the value of $\frac{K_p}{K_c}$ for the reaction:

$$ \mathrm{CO}{(g)} + \frac{1}{2} \mathrm{O}{2(g)} \rightarrow \mathrm{CO}_{2(g)} $$

we need to use the relationship between the equilibrium constants in terms of pressure ($K_p$) and concentration ($K_c$) for gaseous reactions. The relationship is given by the equation:

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

Here, $R$ is the gas constant, $T$ is the temperature in Kelvin, and $\Delta n$ is the change in the number of moles of gas, calculated as:

$$ \Delta n = \text{moles of gaseous products} - \text{moles of gaseous reactants} $$

Step-by-Step :

  1. Identify the moles of gaseous products and reactants:

    • Moles of gaseous products: 1 (for $\mathrm{CO}_{2(g)}$)
    • Moles of gaseous reactants: $1 + \frac{1}{2} = 1.5$ (since $\mathrm{CO}{(g)}$ is 1 and $\frac{1}{2} \mathrm{O}{2(g)}$ is 0.5)
  2. Calculate $\Delta n$: $$ \Delta n = 1 - 1.5 = -0.5 $$

  3. Use the relationship $ K_p = K_c (RT)^{\Delta n} $: $$ K_p = K_c (RT)^{-0.5} $$

  4. Isolate $\frac{K_p}{K_c}$: $$ \frac{K_p}{K_c} = (RT)^{-0.5} $$

Thus, the value of $\frac{K_p}{K_c}$ is $(RT)^{-0.5}$.

Therefore, the correct option is:

C. $(RT)^{-1/2}$

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