The following equilibria are given : $N _{2}+3 H _{2} \rightleftharpoons 2 NH _{3}\, K _{1}$ $N2 + O2 \rightleftharpoons 2NO\, K_2$ $H_2 + \frac{1}{2} O_2 \rightleftharpoons H_2 O \, K_3$ The equilibrium constant of the reaction $2NH_3 + \frac{5}{2} O_2 \rightleftharpoons 2NO + 3H_2 O$ in terms of $K_1,\, K_2$ and $K_3$ is

Question

For the reaction $ \text{N}_2(g) + 3 \, \text{H_2(g) \rightleftharpoons 2 \, \text{NH}_3(g) $, the equilibrium constant $ K_c $ at a certain temperature is $ 0.5 $. If the initial concentrations of $ \text{N}_2 $, $ \text{H}_2 $, and $ \text{NH}_3 $ are $ 1.0 \, \text{mol/L} $, $ 1.0 \, \text{mol/L} $, and $ 0 \, \text{mol/L} $ respectively, calculate the equilibrium concentrations of all species.}

Answer 1

To determine the equilibrium constant of the reaction 2NH_3 + \frac{5}{2} O_2 \rightleftharpoons 2NO + 3H_2 O in terms of the given equilibrium constants K_1, K_2, and K_3, we will use the equilibrium constants from the given reactions and apply the principles of chemical equilibrium.

The given reactions and their equilibrium constants are:

1. N_2 + 3H_2 \rightleftharpoons 2NH_3, \; K_1

2. N_2 + O_2 \rightleftharpoons 2NO, \; K_2

3. H_2 + \frac{1}{2}O_2 \rightleftharpoons H_2O, \; K_3

The goal is to obtain the equilibrium constant for the reaction: 2NH_3 + \frac{5}{2} O_2 \rightleftharpoons 2NO + 3H_2O.

To solve this, we will use the relationships between these reactions:

- For the reverse reaction of Reaction 1, equilibrium constant \frac{1}{K_1} is used: 2NH_3 \rightleftharpoons N_2 + 3H_2\\left(\frac{1}{K_1}\right)

- Reaction 2 is used as is: N_2 + O_2 \rightleftharpoons 2NO, \; K_2

- For Reaction 3, multiply by 3 for the desired products: 3(H_2 + \frac{1}{2}O_2 \rightleftharpoons H_2O)\(K_3^3)

Combine these reactions:

- Reverse Reaction 1: 2NH_3 \rightleftharpoons N_2 + 3H_2\, equilibrium constant \frac{1}{K_1}

- Add Reaction 2: N_2 + O_2 \rightleftharpoons 2NO, \; K_2

- Adjusted Reaction 3: 3H_2 + \frac{3}{2}O_2 \rightleftharpoons 3H_2O,\; K_3^3

Summing the above reactions gives the target reaction: 2NH_3 + \frac{5}{2}O_2 \rightleftharpoons 2NO + 3H_2O.

Calculate the combined equilibrium constant: \text{Product of the constants} = \frac{1}{K_1} \cdot K_2 \cdot K_3^3 = \frac{K_2 K_3^3}{K_1}

Thus, the equilibrium constant for the reaction 2NH_3 + \frac{5}{2}O_2 \rightleftharpoons 2NO + 3H_2O is: \frac{K_2 K_3^3}{K_1}.

Therefore, the correct answer is: \frac{K_2 K_3^3}{K_1}.

Answer 2