Under what pressure must an equimolar mixture of `PCl_(5)` and `Cl_(2)` be placed at `250^(@)C` in order to obtain `PCl_(5)` at `1` atm? `(K_(p) "for dissociation of" PCl_(5)=1.78)`.

Explanation

To solve the problem, we need to find the pressure at which an equimolar mixture of \( PCl_5 \) and \( Cl_2 \) should be placed at \( 250^\circ C \) in order to obtain \( PCl_5 \) at \( 1 \, atm \). Given that the equilibrium constant \( K_p \) for the dissociation of \( PCl_5 \) is \( 1.78 \), we can follow these steps:

### Step 1: Write the Reaction and Reverse it
The dissociation of \( PCl_5 \) can be represented as:
\[ PCl_5 \rightleftharpoons PCl_3 + Cl_2 \]
The equilibrium constant for this reaction is given as \( K_p = 1.78 \). When we reverse the reaction, it becomes:
\[ PCl_3 + Cl_2 \rightleftharpoons PCl_5 \]
The new equilibrium constant \( K'_p \) for this reversed reaction is:
\[ K'_p = \frac{1}{K_p} = \frac{1}{1.78} \]

### Step 2: Set Up the Equilibrium Expression
Let \( P \) be the total pressure of the system. At equilibrium, we have:
- Partial pressure of \( PCl_5 = 1 \, atm \)
- Partial pressure of \( PCl_3 = P - 1 \, atm \)
- Partial pressure of \( Cl_2 = P - 1 \, atm \)

The equilibrium expression for \( K'_p \) is:
\[ K'_p = \frac{P_{PCl_5}}{P_{PCl_3} \cdot P_{Cl_2}} = \frac{1}{(P - 1)(P - 1)} = \frac{1}{(P - 1)^2} \]

### Step 3: Substitute the Values
Substituting \( K'_p \) into the expression gives:
\[ \frac{1}{1.78} = \frac{1}{(P - 1)^2} \]
This simplifies to:
\[ (P - 1)^2 = 1.78 \]

### Step 4: Solve for \( P - 1 \)
Taking the square root of both sides:
\[ P - 1 = \sqrt{1.78} \approx 1.33 \]

### Step 5: Solve for Total Pressure \( P \)
Now, adding 1 to both sides gives:
\[ P = 1 + 1.33 = 2.33 \, atm \]

### Step 6: Calculate Total Pressure at Equilibrium
The total pressure at equilibrium will be:
\[ P_{total} = P_{PCl_5} + P_{PCl_3} + P_{Cl_2} = 1 + (P - 1) + (P - 1) = 1 + 1.33 + 1.33 = 3.66 \, atm \]

### Final Answer
Thus, the pressure at which the equimolar mixture of \( PCl_5 \) and \( Cl_2 \) must be placed to obtain \( PCl_5 \) at \( 1 \, atm \) is:
\[ \boxed{3.66 \, atm} \]