For the reaction `CO_((g))+1/2O_(2(g))hArrCO_(2(g)),(K_(p))/(K_(c))` is equivalent to

Options

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

Explanation

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

\[ CO_{(g)} + \frac{1}{2} O_{2(g)} \rightleftharpoons CO_{2(g)}, \]

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:

\[ K_c = \frac{[CO_2]}{[CO][O_2]^{1/2}} \]

And for \( K_p \):

\[ K_p = \frac{P_{CO_2}}{P_{CO} \cdot (P_{O_2})^{1/2}} \]

### Step 2: Determine \( \Delta n_g \)

To relate \( K_p \) and \( K_c \), we need to calculate \( \Delta n_g \), which is the change in the number of moles of gas from reactants to products.

For the reaction:

- Moles of gaseous products = 1 (from \( CO_2 \))
- Moles of gaseous reactants = 1 (from \( CO \)) + 0.5 (from \( O_2 \)) = 1.5

Thus,

\[ \Delta n_g = \text{Moles of products} - \text{Moles of reactants} = 1 - 1.5 = -0.5 \]

### Step 3: Relate \( K_p \) and \( K_c \)

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

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

Substituting \( \Delta n_g \):

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

### Step 4: Rearranging the equation

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

\[ \frac{K_p}{K_c} = (RT)^{-0.5} = \frac{1}{\sqrt{RT}} \]

### Final Answer

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

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