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Problems · Problem 6.4

Q.The value of K c = 4.24 at 800K for the reaction, CO

(g) + H2O
(g) ⇌ CO2
(g) + H2
(g) Calculate equilibrium concentrations of CO2, H2, CO and H 2O at 800 K, if only CO and H2O are present initially at concentrations of 0.10M each.
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For the reaction CO+H2O⇌CO2+H2\text{CO} + \text{H}_2\text{O} \rightleftharpoons \text{CO}_2 + \text{H}_2 with Kc=4.24K_c = 4.24 at 800 K and initial concentrations of 0.10 M each for CO and H₂O, the equilibrium concentrations are: [CO2]=[H2]=0.067 M[\text{CO}_2] = [\text{H}_2] = 0.067\text{ M} and [CO]=[H2O]=0.033 M[\text{CO}] = [\text{H}_2\text{O}] = 0.033\text{ M}.


The key insight here is that the reaction is perfectly balanced in stoichiometry: one mole of CO reacts with one mole of H₂O to give one mole each of CO₂ and H₂. When you start with equal initial concentrations of the two reactants, the symmetry means that at equilibrium, the concentrations of the two products will be equal, and the concentrations of the two remaining reactants will also be equal. This simplifies the algebra considerably.

Why does this work? Because the equilibrium constant expression is Kc=[CO2][H2][CO][H2O]K_c = \frac{[\text{CO}_2][\text{H}_2]}{[\text{CO}][\text{H}_2\text{O}]}. If the change in concentration for CO is −x-x, then by the 1:1:1:1 stoichiometry, the change for H₂O is also −x-x, and the changes for CO₂ and H₂ are each +x+x. So the equilibrium concentrations become:

  • [CO]=0.10−x[\text{CO}] = 0.10 - x
  • [H2O]=0.10−x[\text{H}_2\text{O}] = 0.10 - x
  • [CO2]=x[\text{CO}_2] = x
  • [H2]=x[\text{H}_2] = x

Substituting into the KcK_c expression gives a clean quadratic — or better yet, a perfect square.


  1. Set up the ICE table (Initial, Change, Equilibrium) in terms of xx, the concentration of CO that reacts.
SpeciesInitial (M)Change (M)Equilibrium (M)
CO0.10−x-x0.10−x0.10 - x
H₂O0.10−x-x0.10−x0.10 - x
CO₂0+x+xxx
H₂0+x+xxx
  1. Write the equilibrium constant expression:

Kc=[CO2][H2][CO][H2O]=(x)(x)(0.10−x)(0.10−x)=x2(0.10−x)2K_c = \frac{[\text{CO}_2][\text{H}_2]}{[\text{CO}][\text{H}_2\text{O}]} = \frac{(x)(x)}{(0.10 - x)(0.10 - x)} = \frac{x^2}{(0.10 - x)^2}

Given Kc=4.24K_c = 4.24.

  1. Take the square root of both sides — this is the neat shortcut. Because both numerator and denominator are perfect squares, we avoid solving a quadratic:

Kc=x0.10−x\sqrt{K_c} = \frac{x}{0.10 - x}

4.24=x0.10−x\sqrt{4.24} = \frac{x}{0.10 - x}

Now 4.24\sqrt{4.24} is not a nice round number, but we can compute it. 2.062=4.24362.06^2 = 4.2436, so 4.24≈2.06\sqrt{4.24} \approx 2.06. Let's use the exact value: 4.24=2.059\sqrt{4.24} = 2.059 (to three decimal places).

2.059=x0.10−x2.059 = \frac{x}{0.10 - x}

  1. Solve for xx:

Multiply both sides: 2.059(0.10−x)=x2.059(0.10 - x) = x

0.2059−2.059x=x0.2059 - 2.059x = x

0.2059=x+2.059x=3.059x0.2059 = x + 2.059x = 3.059x

x=0.20593.059≈0.0673x = \frac{0.2059}{3.059} \approx 0.0673

So x≈0.067 Mx \approx 0.067\text{ M} (to two significant figures, matching the initial concentrations). …

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