1 mole of He and 1 mole A are taken in a 10 L rigid container at 400 K, and equilibrium \( A \rightleftharpoons B \) is established. Calculate the partial pressure of He and B at equilibrium if \( K_c = 4 \).
Step 1: Understanding the reaction.
The given reaction is:
\[
A \rightleftharpoons B
\]
We are given:
- 1 mole of \( \mathrm{He} \) and 1 mole of \( \mathrm{A} \) are taken in a 10 L rigid container.
- The equilibrium constant, \( K_c = 4 \), is given for the reaction.
- The temperature is 400 K.
At equilibrium, we need to calculate the partial pressures of \( \mathrm{He} \) and \( \mathrm{B} \). We are also told that the volume of the container is 10 L.
Step 2: Set up the ICE table (Initial, Change, Equilibrium).
Since helium is inert and does not participate in the reaction, its number of moles remains unchanged at 1 mole.
The expression for \( K_c \) is:
\[
K_c = \frac{[\text{B}]}{[\text{A}]}
\]
At equilibrium, the concentration of \( A \) and \( B \) will be:
\[
[\text{A}] = \frac{1 - x}{10}, \quad [\text{B}] = \frac{x}{10}
\]
Substitute these into the equilibrium constant expression:
\[
4 = \frac{\frac{x}{10}}{\frac{1 - x}{10}} = \frac{x}{1 - x}
\]
Step 3: Solve for \( x \).
Solve the equation:
\[
4 = \frac{x}{1 - x}
\]
Multiplying both sides by \( (1 - x) \):
\[
4(1 - x) = x
\]
\[
4 - 4x = x
\]
\[
4 = 5x
\]
\[
x = \frac{4}{5} = 0.8
\]
Step 4: Calculate the partial pressures.
Now, substitute \( x = 0.8 \) into the equilibrium expressions for \( A \) and \( B \):
\[
[\text{B}] = \frac{0.8}{10} = 0.08 \, \text{mol/L}
\]
\[
[\text{A}] = \frac{1 - 0.8}{10} = 0.02 \, \text{mol/L}
\]
The partial pressure is given by \( P = \frac{nRT}{V} \). Using the ideal gas law:
\[
P = \frac{nRT}{V} \quad \text{where} \quad R = 0.0821 \, \text{L·atm/mol·K}, \, T = 400 \, \text{K}, \, V = 10 \, \text{L}
\]
For \( \text{He} \):
\[
P_{\text{He}} = \frac{1 \times 0.0821 \times 400}{10} = 3.28 \, \text{atm}
\]
For \( \text{B} \):
\[
P_{\text{B}} = \frac{0.8 \times 0.0821 \times 400}{10} = 2.62 \, \text{atm}
\]
Step 5: Final Answer.
Thus, the partial pressure of \( \text{He} \) is 3.28 atm and the partial pressure of \( \text{B} \) is 2.62 atm. Therefore, the correct answer is:
\[
\text{(A) 3.28 atm, 2.62 atm}
\]
Final Answer: 3.28 atm, 2.62 atm.
