For the following reaction in gaseous phase,
` CO + (1)/(2) O_2 to CO_2 ,(K_c)/( K_p) ` is

Options

A. ` (RT)^(1//2)`
B. ` (RT)^(-1//2)`
C. ` (RT)`
D. ` (RT)^(-1)`

Explanation

To solve the problem of finding the relationship between \( K_c \) and \( K_p \) for the reaction:

\[ \text{CO} + \frac{1}{2} \text{O}_2 \rightleftharpoons \text{CO}_2 \]

we will follow these steps:

### Step 1: Write the expression for \( K_p \) and \( K_c \)

The equilibrium constant \( K_c \) is expressed in terms of concentrations, while \( K_p \) is expressed in terms of partial pressures. For the given reaction, we can write:

\[ K_c = \frac{[\text{CO}_2]}{[\text{CO}][\text{O}_2]^{1/2}} \]

\[ K_p = \frac{P_{\text{CO}_2}}{P_{\text{CO}} \cdot P_{\text{O}_2}^{1/2}} \]

### Step 2: Use the relationship between \( K_p \) and \( K_c \)

The relationship between \( K_p \) and \( K_c \) is given by the formula:

\[ K_p = K_c \cdot (RT)^{\Delta n_g} \]

where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( \Delta n_g \) is the change in the number of moles of gas.

### Step 3: Calculate \( \Delta n_g \)

To find \( \Delta n_g \), we need to determine the change in the number of moles of gas from reactants to products:

- Moles of gaseous products = 1 (from \( \text{CO}_2 \))
- Moles of gaseous reactants = 1 (from \( \text{CO} \)) + \( \frac{1}{2} \) (from \( \text{O}_2 \)) = \( \frac{3}{2} \)

Thus,

\[ \Delta n_g = \text{moles of products} - \text{moles of reactants} = 1 - \frac{3}{2} = -\frac{1}{2} \]

### Step 4: Substitute \( \Delta n_g \) into the equation

Now substituting \( \Delta n_g \) into the equation for \( K_p \):

\[ K_p = K_c \cdot (RT)^{-\frac{1}{2}} \]

### Step 5: Rearranging to find \( \frac{K_c}{K_p} \)

Rearranging the equation gives:

\[ \frac{K_p}{K_c} = (RT)^{-\frac{1}{2}} \]

Taking the reciprocal, we find:

\[ \frac{K_c}{K_p} = (RT)^{\frac{1}{2}} \]

### Final Answer

Thus, the relationship between \( K_c \) and \( K_p \) for the reaction is:

\[ \frac{K_c}{K_p} = (RT)^{\frac{1}{2}} \]