`N_(2)+3H(2)hArr2NH_(3)` if in equilibrium mixture by volume `N_(2)` is `40%, H_(2)` is `30%` and rest is `NH_(3)` and total pressure is 5 atm then find out `K_(p)`:-

Options

A. `(1)/(300)`
B. `(1)/(3)`
C. `(1)/(30)`
D. `(2)/(3)`

Answer

To find the equilibrium constant \( K_p \) for the reaction

\[ N_2 + 3H_2 \rightleftharpoons 2NH_3 \]

given the volume percentages and total pressure, we can follow these steps:

### Step 1: Determine the Mole Fractions
From the problem, we know the volume percentages of the gases:
- \( N_2 \) = 40%
- \( H_2 \) = 30%
- \( NH_3 \) = 30%

### Step 2: Assign Moles Based on Total Volume
Let the total volume be represented by \( a \) (which is arbitrary). Thus, we can express the number of moles (or volume) of each gas as:
- Moles of \( N_2 = 0.4a \)
- Moles of \( H_2 = 0.3a \)
- Moles of \( NH_3 = 0.3a \)

### Step 3: Calculate Partial Pressures
The partial pressure of each gas can be calculated using the formula:
\[ P_X = \text{Mole Fraction of } X \times P_{total} \]
Given that the total pressure \( P_{total} = 5 \, \text{atm} \):

- Partial pressure of \( N_2 \):
\[ P_{N_2} = \left( \frac{0.4a}{a} \right) \times 5 = 0.4 \times 5 = 2 \, \text{atm} \]

- Partial pressure of \( H_2 \):
\[ P_{H_2} = \left( \frac{0.3a}{a} \right) \times 5 = 0.3 \times 5 = 1.5 \, \text{atm} \]

- Partial pressure of \( NH_3 \):
\[ P_{NH_3} = \left( \frac{0.3a}{a} \right) \times 5 = 0.3 \times 5 = 1.5 \, \text{atm} \]

### Step 4: Write the Expression for \( K_p \)
The equilibrium constant \( K_p \) for the reaction is given by:
\[ K_p = \frac{(P_{NH_3})^2}{(P_{N_2})(P_{H_2})^3} \]

### Step 5: Substitute the Partial Pressures
Substituting the values we found:
\[ K_p = \frac{(1.5)^2}{(2)(1.5)^3} \]

### Step 6: Calculate \( K_p \)
Calculating the numerator and denominator:
- Numerator: \( (1.5)^2 = 2.25 \)
- Denominator: \( 2 \times (1.5)^3 = 2 \times 3.375 = 6.75 \)

Now substituting these into the equation:
\[ K_p = \frac{2.25}{6.75} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( K_p \) is:
\[ K_p = \frac{1}{3} \]

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