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Exercises · 6.8

Q.Reaction between N2 and O2– takes place as follows: 2N2

(g) + O2
(g) ⇌ 2N2O
(g) If a mixture of 0.482 mol N2 and 0.933 mol of O2 is placed in a 10 L reaction vessel and allowed to form N 2O at a temperature for which Kc= 2.0 × 10⁻³⁷, determine the composition of equilibrium mixture.
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When KcK_c is extremely small (2.0×10−372.0 \times 10^{-37}), the forward reaction is negligible; equilibrium lies almost entirely with the reactants. At equilibrium: [N2]=0.0482 M[\text{N}_2] = 0.0482\,\text{M}, [O2]=0.0933 M[\text{O}_2] = 0.0933\,\text{M}, and [N2O]≈0 M[\text{N}_2\text{O}] \approx 0\,\text{M}.

The equilibrium constant KcK_c tells us the position of equilibrium—how far a reaction proceeds before the forward and reverse rates balance. A value like 2.0×10−372.0 \times 10^{-37} is astronomically small, meaning the reaction barely moves forward at all. The reactants are overwhelmingly favored, and the product concentration will be vanishingly small. This is why nitrogen and oxygen coexist peacefully in our atmosphere at room temperature rather than forming nitrous oxide spontaneously.

Let's set up the ICE table (Initial, Change, Equilibrium) to track concentrations systematically.

Step-by-step solution

1. Calculate initial concentrations

The vessel volume is 10 L10\,\text{L}, so:

[N2]0=0.482 mol10 L=0.0482 M[\text{N}_2]_0 = \frac{0.482\,\text{mol}}{10\,\text{L}} = 0.0482\,\text{M}

[O2]0=0.933 mol10 L=0.0933 M[\text{O}_2]_0 = \frac{0.933\,\text{mol}}{10\,\text{L}} = 0.0933\,\text{M}

[N2O]0=0 M[\text{N}_2\text{O}]_0 = 0\,\text{M}

2. Set up the ICE table

For the reaction 2N2(g)+O2(g)⇌2N2O(g)2\text{N}_2(g) + \text{O}_2(g) \rightleftharpoons 2\text{N}_2\text{O}(g), let xx be the change in [O2][\text{O}_2] that reacts:

SpeciesInitial (M)Change (M)Equilibrium (M)
N2\text{N}_20.04820.0482−2x-2x0.0482−2x0.0482 - 2x
O2\text{O}_20.09330.0933−x-x0.0933−x0.0933 - x
N2O\text{N}_2\text{O}00+2x+2x2x2x

The stoichiometry dictates that for every mole of O2\text{O}_2 consumed, 22 moles of N2\text{N}_2 are consumed and 22 moles of N2O\text{N}_2\text{O} are formed.

3. Write the equilibrium expression

Kc=[N2O]2[N2]2[O2]=2.0×10−37K_c = \frac{[\text{N}_2\text{O}]^2}{[\text{N}_2]^2[\text{O}_2]} = 2.0 \times 10^{-37}

Substituting equilibrium concentrations:

2.0×10−37=(2x)2(0.0482−2x)2(0.0933−x)2.0 \times 10^{-37} = \frac{(2x)^2}{(0.0482 - 2x)^2(0.0933 - x)}

4. Apply the small-xx approximation

Given that KcK_c is extraordinarily small, xx will be negligible compared to the initial concentrations. We can approximate:

0.0482−2x≈0.04820.0482 - 2x \approx 0.0482

0.0933−x≈0.09330.0933 - x \approx 0.0933

Tip

When Kc≪1K_c \ll 1, the reaction proceeds negligibly forward. The approximation x≪[reactants]0x \ll [\text{reactants}]_0 simplifies the algebra dramatically and is justified by checking afterward.

The expression becomes:

2.0×10−37=4x2(0.0482)2(0.0933)2.0 \times 10^{-37} = \frac{4x^2}{(0.0482)^2(0.0933)}

5. Solve for xx

4x2=2.0×10−37×(0.0482)2×0.09334x^2 = 2.0 \times 10^{-37} \times (0.0482)^2 \times 0.0933

4x2=2.0×10−37×2.323×10−3×0.09334x^2 = 2.0 \times 10^{-37} \times 2.323 \times 10^{-3} \times 0.0933 …

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