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

Options

A. `RT`
B. `(RT)^(-1)`
C. `(RT)^(-1//2)`
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 can follow these steps:

### Step 1: Identify the reaction and its components
The given reaction involves the following:
- Reactants: 1 mole of CO and 0.5 moles of \( O_2 \)
- Product: 1 mole of \( CO_2 \)

### Step 2: Calculate the change in moles of gas (\( \Delta n \))
To find \( \Delta n \), we use the formula:

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

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

Thus,

\[ \Delta n = 1 - 1.5 = -0.5 \]

### Step 3: Use the relationship between \( K_p \) 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

### Step 4: Substitute \( \Delta n \) into the equation
Now, substituting \( \Delta n = -0.5 \) into the equation:

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

This can be rearranged to find \( \frac{K_p}{K_c} \):

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

### Step 5: Simplify the expression
The expression 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:

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