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?

Explanation

To determine whether 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:
\[ N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \]

### Step 2: Calculate the concentrations of the reactants and products
We need to calculate the concentrations of \(N_2\), \(H_2\), and \(NH_3\) using the formula:
\[ \text{Concentration} = \frac{\text{moles}}{\text{volume}} \]

Given:
- Moles of \(N_2 = 1.57 \, \text{mol}\)
- Moles of \(H_2 = 1.92 \, \text{mol}\)
- Moles of \(NH_3 = 8.13 \, \text{mol}\)
- Volume of the vessel = 20 L

Calculating the concentrations:
- \[ [N_2] = \frac{1.57 \, \text{mol}}{20 \, \text{L}} = 0.0785 \, \text{mol/L} \]
- \[ [H_2] = \frac{1.92 \, \text{mol}}{20 \, \text{L}} = 0.096 \, \text{mol/L} \]
- \[ [NH_3] = \frac{8.13 \, \text{mol}}{20 \, \text{L}} = 0.4065 \, \text{mol/L} \]

### Step 3: Calculate the reaction quotient \(Q_c\)
The reaction quotient \(Q_c\) is calculated using the formula:
\[ Q_c = \frac{[NH_3]^2}{[N_2][H_2]^3} \]

Substituting the concentrations:
- \[ Q_c = \frac{(0.4065)^2}{(0.0785)(0.096)^3} \]

Calculating \(Q_c\):
- \[ Q_c = \frac{0.1652}{(0.0785)(0.000884736)} \]
- \[ Q_c = \frac{0.1652}{0.0000695} \approx 2382.61 \]

### Step 4: Compare \(Q_c\) with \(K_c\)
Given \(K_c = 1.7 \times 10^2 = 170\).

Since \(Q_c (2382.61) > K_c (170)\), the reaction is not at equilibrium.

### Step 5: Determine the direction of the net reaction
Since \(Q_c > K_c\), the reaction will shift to the left (towards the reactants) to reach equilibrium.

### Final Answer
1. The reaction mixture is **not at equilibrium**.
2. The direction of the net reaction is **to the left** (towards the reactants).

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