For the reaction
`CO(g)+(1)/(2) O_(2)(g) hArr CO_(2)(g),K_(p)//K_(c)` is

Options

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

Answer

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

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

we will follow these steps:

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

The equilibrium constant \( K_c \) for the reaction is given by:

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

The equilibrium constant \( K_p \) is given by:

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

### Step 2: Relate \( K_p \) and \( K_c \)

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

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

where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( \Delta n_g \) is the change in the number of moles of gas, calculated as:

\[ \Delta n_g = \text{(moles of gaseous products)} - \text{(moles of gaseous reactants)} \]

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

For the given reaction:

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

Thus, we can calculate \( \Delta n_g \):

\[ \Delta n_g = 1 - \frac{3}{2} = 1 - 1.5 = -0.5 \]

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

Now substituting \( \Delta n_g \) into the relationship between \( K_p \) and \( K_c \):

\[ K_p = K_c \cdot (RT)^{-0.5} \]

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

To find \( \frac{K_p}{K_c} \):

\[ \frac{K_p}{K_c} = (RT)^{-0.5} \]

This can be rewritten as:

\[ \frac{K_p}{K_c} = \frac{1}{\sqrt{RT}} \]

### Final Answer

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

\[ \frac{K_p}{K_c} = \frac{1}{\sqrt{RT}} \]