To solve the problem, we need to find the equilibrium constant \( K_c \) for the reaction:
\[ X(g) + Y(g) \rightleftharpoons 2Z(g) \]
Given:
- Initial moles of \( X = 2 \)
- Initial moles of \( Y = 1 \)
- Initial moles of \( Z = 1 \)
- Volume of the vessel = 10 L
- Final concentration of \( Z = 0.2 \, M \)
### Step 1: Calculate Initial Concentrations
First, we calculate the initial concentrations of \( X \), \( Y \), and \( Z \):
\[
\text{Concentration of } X = \frac{2 \, \text{moles}}{10 \, \text{L}} = 0.2 \, M
\]
\[
\text{Concentration of } Y = \frac{1 \, \text{mole}}{10 \, \text{L}} = 0.1 \, M
\]
\[
\text{Concentration of } Z = \frac{1 \, \text{mole}}{10 \, \text{L}} = 0.1 \, M
\]
### Step 2: Set Up the Change in Concentration
Let \( a \) be the change in moles of \( X \) and \( Y \) that react to form \( Z \). According to the stoichiometry of the reaction:
- For every 1 mole of \( X \) and \( Y \) that reacts, 2 moles of \( Z \) are produced.
- Therefore, at equilibrium:
- Moles of \( X \) = \( 2 - a \)
- Moles of \( Y \) = \( 1 - a \)
- Moles of \( Z \) = \( 1 + 2a \)
### Step 3: Use the Final Concentration of \( Z \)
We know the final concentration of \( Z \) is \( 0.2 \, M \):
\[
\text{Concentration of } Z = \frac{1 + 2a}{10} = 0.2
\]
Multiplying both sides by 10:
\[
1 + 2a = 2
\]
Solving for \( a \):
\[
2a = 2 - 1 \implies 2a = 1 \implies a = \frac{1}{2}
\]
### Step 4: Calculate Equilibrium Concentrations
Now we can calculate the equilibrium concentrations:
- Concentration of \( X \):
\[
\frac{2 - a}{10} = \frac{2 - \frac{1}{2}}{10} = \frac{1.5}{10} = 0.15 \, M
\]
- Concentration of \( Y \):
\[
\frac{1 - a}{10} = \frac{1 - \frac{1}{2}}{10} = \frac{0.5}{10} = 0.05 \, M
\]
- Concentration of \( Z \):
\[
\frac{1 + 2a}{10} = \frac{1 + 1}{10} = \frac{2}{10} = 0.2 \, M
\]
### Step 5: Calculate \( K_c \)
The equilibrium constant \( K_c \) is given by:
\[
K_c = \frac{[Z]^2}{[X][Y]}
\]
Substituting the equilibrium concentrations:
\[
K_c = \frac{(0.2)^2}{(0.15)(0.05)} = \frac{0.04}{0.0075}
\]
Calculating \( K_c \):
\[
K_c = \frac{0.04}{0.0075} = \frac{4}{0.75} = \frac{4 \times 4}{3} = \frac{16}{3}
\]
### Final Answer
Thus, the equilibrium constant \( K_c \) for the reaction is:
\[
K_c = \frac{16}{3}
\]
---