To solve the problem, we will analyze the given chemical reaction and the equilibrium conditions.
### Step-by-Step Solution:
1. **Write the Reaction and Initial Concentrations**:
The reaction is given as:
\[
P(g) + 2Q(g) \rightleftharpoons R(g) + S(g)
\]
The initial concentrations are:
- \([P] = 2 \, M\)
- \([Q] = 4 \, M\)
- \([R] = 0 \, M\)
- \([S] = 0 \, M\)
2. **Determine the Equilibrium Constant**:
The equilibrium constant \(K_c\) for the reaction is given as:
\[
K_c = 10^{12}
\]
Since \(K_c\) is very large, it indicates that the reaction favors the formation of products \(R\) and \(S\).
3. **Set Up the Change in Concentration**:
Let \(x\) be the change in concentration of \(P\) that reacts. Therefore, at equilibrium:
- \([P] = 2 - x\)
- \([Q] = 4 - 2x\)
- \([R] = x\)
- \([S] = x\)
4. **Express \(K_c\) in Terms of Equilibrium Concentrations**:
The expression for \(K_c\) is:
\[
K_c = \frac{[R][S]}{[P][Q]} = \frac{x \cdot x}{(2 - x)(4 - 2x)} = \frac{x^2}{(2 - x)(4 - 2x)}
\]
Setting this equal to \(10^{12}\):
\[
\frac{x^2}{(2 - x)(4 - 2x)} = 10^{12}
\]
5. **Assume Reaction Goes to Completion**:
Since \(K_c\) is very large, we can assume that the reaction goes nearly to completion. Thus, \(x\) will be close to 2 for \(P\) and close to 4 for \(Q\):
- Therefore, we can assume \(x \approx 2\).
6. **Calculate Equilibrium Concentrations**:
- \([P] \approx 2 - 2 = 0 \, M\)
- \([Q] \approx 4 - 4 = 0 \, M\)
- \([R] \approx 2 \, M\)
- \([S] \approx 2 \, M\)
7. **Final Concentrations**:
Thus, at equilibrium:
- \([P] \approx 0 \, M\)
- \([Q] \approx 0 \, M\)
- \([R] = 2 \, M\)
- \([S] = 2 \, M\)
### Conclusion:
From the analysis, we can conclude:
- The concentration of \(P\) at equilibrium is very close to \(0\).
- The concentration of \(Q\) at equilibrium is also very close to \(0\).
- The concentrations of \(R\) and \(S\) are both \(2 \, M\).
### Correct Statements:
Based on the calculations, the correct statements regarding the equilibrium concentrations are:
- \([P] \approx 0 \, M\)
- \([Q] \approx 0 \, M\)
- \([R] = 2 \, M\)
- \([S] = 2 \, M\)
