At a particular temperature, `PCl_(5)(g)` undergoes 50% dissociation. The equilibrium constant for `PCl_(5)(g) rarr PCl_(3)(g) + Cl_(2)(g)` is 2atm. The pressure of the equilibrium mixture is

Options

A. 2 atm
B. 6 atm
C. 8 atm
D. 5 atm

Explanation

To solve the problem step by step, we will follow the information provided in the question regarding the dissociation of \( PCl_5 \) and the equilibrium constant.

### Step 1: Define Initial Conditions
Let the initial pressure of \( PCl_5 \) be \( P \) atm. At the start, the pressures of \( PCl_3 \) and \( Cl_2 \) are both 0 atm.

### Step 2: Determine Changes at Equilibrium
According to the problem, \( PCl_5 \) undergoes 50% dissociation. This means that at equilibrium, 50% of the initial \( PCl_5 \) has dissociated.

- The amount of \( PCl_5 \) that dissociates = \( 0.5P \)
- The remaining pressure of \( PCl_5 \) at equilibrium = \( P - 0.5P = 0.5P \)
- The pressure of \( PCl_3 \) formed = \( 0.5P \)
- The pressure of \( Cl_2 \) formed = \( 0.5P \)

### Step 3: Write the Expression for the Equilibrium Constant
The equilibrium constant \( K_p \) for the reaction is given by:
\[ K_p = \frac{P_{PCl_3} \cdot P_{Cl_2}}{P_{PCl_5}} \]
Substituting the pressures at equilibrium:
\[ K_p = \frac{(0.5P)(0.5P)}{0.5P} \]

### Step 4: Simplify the Expression
This simplifies to:
\[ K_p = \frac{0.25P^2}{0.5P} = \frac{0.25P}{0.5} = 0.5P \]

### Step 5: Set Up the Equation with Given \( K_p \)
We know from the problem that \( K_p = 2 \) atm. Therefore, we can set up the equation:
\[ 0.5P = 2 \]

### Step 6: Solve for \( P \)
To find \( P \):
\[ P = \frac{2}{0.5} = 4 \text{ atm} \]

### Step 7: Calculate Total Pressure at Equilibrium
Now, we need to find the total pressure of the equilibrium mixture. The total pressure \( P_{total} \) at equilibrium is the sum of the pressures of all species:
\[ P_{total} = P_{PCl_5} + P_{PCl_3} + P_{Cl_2} \]
Substituting the values:
\[ P_{total} = 0.5P + 0.5P + 0.5P = 0.5(4) + 0.5(4) + 0.5(4) = 2 + 2 + 2 = 6 \text{ atm} \]

### Final Answer
The pressure of the equilibrium mixture is **6 atm**.

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