Solid `A "and" B` are taken in a closed container at a certain temperature. These two solids decompose are following equilibria are established simultaneously
`A(s)hArrX(g)+Y(g) K_(P_(1)=250atm^(2)`
` B(s)hArrY(g)+Z(g) K_(P_(2)=?`
If the total pressure developed over the solid mixture is `50 atm`. Then the value of `K_(P)` for the `2^(nd)` reaction

Options

A. `375`
B. `625`
C. `225`
D. `250`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Equilibria
We have two reactions occurring simultaneously:
1. \( A(s) \rightleftharpoons X(g) + Y(g) \) with \( K_{P1} = 250 \, \text{atm}^2 \)
2. \( B(s) \rightleftharpoons Y(g) + Z(g) \) with \( K_{P2} = ? \)

### Step 2: Define Variables for Partial Pressures
Let:
- \( P_X = P_X \) (partial pressure of gas \( X \))
- \( P_Y = P_Y \) (partial pressure of gas \( Y \))
- \( P_Z = P_Z \) (partial pressure of gas \( Z \))

### Step 3: Total Pressure Equation
The total pressure in the system is given as \( 50 \, \text{atm} \):
\[ P_X + P_Y + P_Z = 50 \, \text{atm} \]

### Step 4: Express \( K_{P1} \) in Terms of Partial Pressures
From the first reaction:
\[ K_{P1} = P_X \cdot P_Y = 250 \, \text{atm}^2 \]

### Step 5: Relate Partial Pressures
From the second reaction:
\[ K_{P2} = P_Y \cdot P_Z \]

### Step 6: Substitute Total Pressure
We know that:
\[ P_Y = P_X + P_Z \]
Substituting this into the total pressure equation gives:
\[ P_X + (P_X + P_Z) + P_Z = 50 \]
This simplifies to:
\[ 2P_X + 2P_Z = 50 \]
Thus,
\[ P_X + P_Z = 25 \quad \text{(1)} \]

### Step 7: Solve for \( P_Y \)
From (1), we can express \( P_Y \):
\[ P_Y = 50 - P_X - P_Z \]

### Step 8: Substitute into \( K_{P1} \)
From \( K_{P1} \):
\[ P_X \cdot P_Y = 250 \]
Substituting \( P_Y \):
\[ P_X \cdot (50 - P_X - P_Z) = 250 \]

### Step 9: Solve for \( P_X \) and \( P_Z \)
Using the relationship \( P_X + P_Z = 25 \), we can express \( P_Z \) in terms of \( P_X \):
\[ P_Z = 25 - P_X \]
Substituting this into the equation:
\[ P_X \cdot (50 - P_X - (25 - P_X)) = 250 \]
This simplifies to:
\[ P_X \cdot 25 = 250 \]
Thus,
\[ P_X = 10 \, \text{atm} \]

### Step 10: Calculate \( P_Z \)
Using \( P_X + P_Z = 25 \):
\[ 10 + P_Z = 25 \implies P_Z = 15 \, \text{atm} \]

### Step 11: Calculate \( P_Y \)
Using the total pressure:
\[ P_Y = 50 - P_X - P_Z = 50 - 10 - 15 = 25 \, \text{atm} \]

### Step 12: Calculate \( K_{P2} \)
Now substituting into \( K_{P2} \):
\[ K_{P2} = P_Y \cdot P_Z = 25 \cdot 15 = 375 \, \text{atm}^2 \]

### Final Answer
Thus, the value of \( K_{P2} \) is:
\[ \boxed{375 \, \text{atm}^2} \]