A mixture of `1.57 mol` of `N_(2), 1.92 mol` of `H_(2)` and `8.13 mol` of `NH_(3)` is introduced into a `20 L` reaction vessel at `500 K`. At this temperature, the equilibrium constant `K_(c )` for the reaction `N_(2)(g)+3H_(2)(g) hArr 2NH_(3)(g)` is `1.7xx10^(2)`. Is the reaction mixture at equilibrium? If not, what is the direction of the net reaction?

Options

A. Forward
B. Backward
C. At equilibrium
D. Data is insufficient

Explanation

To determine if the reaction mixture is at equilibrium and the direction of the net reaction, we will follow these steps:

### Step 1: Write the balanced chemical equation
The balanced chemical equation for the reaction is:
\[ \text{N}_2(g) + 3\text{H}_2(g) \rightleftharpoons 2\text{NH}_3(g) \]

### Step 2: Calculate the concentrations of each species
We need to find the concentrations of \( \text{N}_2 \), \( \text{H}_2 \), and \( \text{NH}_3 \) in the 20 L reaction vessel.

- Concentration of \( \text{N}_2 \):
\[ [\text{N}_2] = \frac{1.57 \, \text{mol}}{20 \, \text{L}} = 0.0785 \, \text{mol/L} \]

- Concentration of \( \text{H}_2 \):
\[ [\text{H}_2] = \frac{1.92 \, \text{mol}}{20 \, \text{L}} = 0.096 \, \text{mol/L} \]

- Concentration of \( \text{NH}_3 \):
\[ [\text{NH}_3] = \frac{8.13 \, \text{mol}}{20 \, \text{L}} = 0.4065 \, \text{mol/L} \]

### Step 3: Write the expression for the reaction quotient \( Q_c \)
The expression for the reaction quotient \( Q_c \) is given by:
\[ Q_c = \frac{[\text{NH}_3]^2}{[\text{N}_2][\text{H}_2]^3} \]

### Step 4: Substitute the concentrations into the \( Q_c \) expression
Substituting the calculated concentrations into the \( Q_c \) expression:
\[ Q_c = \frac{(0.4065)^2}{(0.0785)(0.096)^3} \]

Calculating the values:
\[ Q_c = \frac{0.1652}{0.0785 \times 0.000884736} = \frac{0.1652}{0.0000695} \approx 2383.3 \]

### Step 5: Compare \( Q_c \) with \( K_c \)
The equilibrium constant \( K_c \) is given as:
\[ K_c = 1.7 \times 10^2 = 170 \]

Now we compare \( Q_c \) and \( K_c \):
\[ Q_c \approx 2383.3 \quad \text{and} \quad K_c = 170 \]

### Step 6: Determine if the reaction is at equilibrium
Since \( Q_c > K_c \), the reaction is not at equilibrium.

### Step 7: Determine the direction of the net reaction
Since \( Q_c > K_c \), the reaction will shift to the left (backward direction) to reach equilibrium.

### Final Answer
The reaction mixture is not at equilibrium, and the direction of the net reaction is backward.

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