At 1000 K, the value of `K_(p)` for the reaction: `A(g) + 2B(g)hArr3C(g) + D(g)` is `0.05` atmosphere. The value of `K_(c)` in terms of R would be:

Options

A. `20000 R`
B. `0.02 R`
C. `5xx10^(-6)R`
D. `5xx10^(-5)xxR^(-1)`

Explanation

To find the value of \( K_c \) in terms of \( R \) for the reaction \( A(g) + 2B(g) \rightleftharpoons 3C(g) + D(g) \) at 1000 K, we can follow these steps:

### Step 1: Write the balanced chemical equation
The balanced chemical equation is:
\[ A(g) + 2B(g) \rightleftharpoons 3C(g) + D(g) \]

### Step 2: Calculate the change in moles of gas (\( \Delta n \))
To calculate \( \Delta n \), we need to find the difference between the total moles of gaseous products and the total moles of gaseous reactants.

- Total moles of products = 3 (from \( 3C \)) + 1 (from \( D \)) = 4
- Total moles of reactants = 1 (from \( A \)) + 2 (from \( 2B \)) = 3

Now, we calculate \( \Delta n \):
\[ \Delta n = \text{(moles of products)} - \text{(moles of reactants)} = 4 - 3 = 1 \]

### Step 3: Use the relationship between \( K_p \) and \( K_c \)
The relationship between \( K_p \) and \( K_c \) is given by the formula:
\[ K_p = K_c \cdot R^T \cdot (P^{\Delta n}) \]
Where:
- \( R \) is the ideal gas constant
- \( T \) is the temperature in Kelvin
- \( \Delta n \) is the change in moles of gas

Rearranging this formula to find \( K_c \):
\[ K_c = \frac{K_p}{R^T} \]

### Step 4: Substitute the known values
We know:
- \( K_p = 0.05 \, \text{atm} \)
- \( T = 1000 \, \text{K} \)

Substituting these values into the equation:
\[ K_c = \frac{0.05}{R \cdot 1000} \]

### Step 5: Simplify the expression
This can be simplified to:
\[ K_c = \frac{0.05}{1000} \cdot \frac{1}{R} = 5 \times 10^{-5} \cdot R^{-1} \]

### Final Answer
Thus, the value of \( K_c \) in terms of \( R \) is:
\[ K_c = 5 \times 10^{-5} R^{-1} \]