On decomposition of `NH_(4)HS` , the following equilibrium is estabilished: `NH_(4) HS(s)hArrNH_(3)(g) + H_(2)S(g)` If the total pressure is P atm, then the equilibrium constant `K_(p)` is equal to

Options

A. `P` atm
B. `P^(2)`atm
C. `P^(2)//4 atm^(2)`
D. `2P` atm

Explanation

To find the equilibrium constant \( K_p \) for the decomposition of \( NH_4HS \), we can follow these steps:

### Step 1: Write the balanced chemical equation
The decomposition of ammonium hydrosulfide can be represented as:
\[ NH_4HS(s) \rightleftharpoons NH_3(g) + H_2S(g) \]

### Step 2: Identify the components of the equilibrium
In this equilibrium, \( NH_4HS \) is a solid and does not contribute to the pressure. The gaseous products are \( NH_3 \) and \( H_2S \).

### Step 3: Define the pressures of the gases
Let \( p \) be the partial pressure of \( NH_3 \) and \( H_2S \). Since both gases are produced in equal amounts, we can express the total pressure \( P \) as:
\[ P = p_{NH_3} + p_{H_2S} = p + p = 2p \]

### Step 4: Solve for the partial pressure
From the equation \( P = 2p \), we can solve for \( p \):
\[ p = \frac{P}{2} \]

### Step 5: Write the expression for \( K_p \)
The equilibrium constant \( K_p \) is defined in terms of the partial pressures of the gaseous products:
\[ K_p = p_{NH_3} \cdot p_{H_2S} \]
Substituting the values we have:
\[ K_p = p \cdot p = p^2 \]

### Step 6: Substitute \( p \) in the \( K_p \) expression
Now substituting \( p = \frac{P}{2} \) into the equation for \( K_p \):
\[ K_p = \left(\frac{P}{2}\right)^2 = \frac{P^2}{4} \]

### Step 7: State the final answer
Thus, the equilibrium constant \( K_p \) is:
\[ K_p = \frac{P^2}{4} \text{ atm}^2 \]