If `K_(p)` for a reaction is 0.1 at `27^(@)C` the `K_(c )` of the reaction will be

Options

A. can not be calculate as `Delta n_(g) ` is not given
B. `(K_(p))/( RT)`
C. `K_(p) RT`
D. `(K_(p))/( ( RT)^(2))`

Answer

To find the value of \( K_c \) from the given \( K_p \) for a reaction, we can use the following relationship:

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

Where:
- \( K_p \) is the equilibrium constant in terms of partial pressures.
- \( K_c \) is the equilibrium constant in terms of concentrations.
- \( R \) is the universal gas constant (0.0821 L·atm/(K·mol)).
- \( T \) is the temperature in Kelvin.
- \( \Delta n \) is the change in the number of moles of gas (moles of gaseous products - moles of gaseous reactants).

### Step-by-Step Solution:

1. **Convert Temperature to Kelvin**:
\[ T = 27^\circ C + 273.15 = 300.15 \, K \]

2. **Identify Given Values**:
\[ K_p = 0.1 \]
\[ R = 0.0821 \, L·atm/(K·mol) \]

3. **Determine \(\Delta n\)**:
For this step, we need to know the balanced chemical equation to find \(\Delta n\). Assuming a general reaction:
\[ aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) \]
Then,
\[ \Delta n = (c + d) - (a + b) \]
For simplicity, let's assume \(\Delta n = 1\) (you can adjust based on the actual reaction).

4. **Use the Relationship**:
Substitute the known values into the equation:
\[ K_p = K_c (RT)^{\Delta n} \]
Rearranging gives:
\[ K_c = \frac{K_p}{(RT)^{\Delta n}} \]

5. **Calculate \(K_c\)**:
\[ K_c = \frac{0.1}{(0.0821 \times 300.15)^1} \]
\[ K_c = \frac{0.1}{24.726} \]
\[ K_c \approx 0.00404 \]

### Final Answer:
\[ K_c \approx 0.00404 \]