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

Options

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

Answer

To solve for 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 \( K_c \) and \( K_p \)

The equilibrium constant \( K_c \) is defined in terms of the concentrations of the reactants and products:

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

The equilibrium constant \( K_p \) is defined in terms of the partial pressures:

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

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

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

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

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

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

To find \( \Delta n \), we need to calculate the difference between the moles of products and the moles of reactants:

- Moles of products = 1 (from CO₂)
- Moles of reactants = 1 (from CO) + 0.5 (from \( \frac{1}{2} O_2 \)) = 1.5

Thus,

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

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

Now we can substitute \( \Delta n \) into the equation relating \( K_p \) and \( K_c \):

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

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

To find \( \frac{K_p}{K_c} \), we rearrange the equation:

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

### Final Result

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

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