Question

Write the equilibrium constant expression for this reaction: 2H+(aq)+CO−23(aq) → H2CO3(aq)

Answer 1

Equilibrium constant expression for
`\rm 2\; H^(+)\, (aq) + {CO_3}^(2-)\, (aq) \rightleftharpoons H_2CO_3\, (aq)`:

`\displaystyle K = \frac{\left(a_{\mathrm{H_2CO_3\, (aq)}}\right)}{\left(a_{\mathrm{H^(+)}}\right)^2\, \left(a_{\mathrm{{CO_3}^(2-)\, (aq)}}\right)} \approx \frac{[\mathrm{H_2CO_3}]}{\left[\mathrm{H^(+)\, (aq)}\right]^(2) \, \left[\mathrm{CO_3}^(2-)\right]}`.

Where

• 
  `a_{\mathrm{H_2CO_3}}`,
  `a_{\mathrm{H^(+)}}`, and
  `a_{\mathrm{CO_3}^(2-)}` denote the activities of the three species, and
• 
  `[\mathrm{H_2CO_3}]`,
  `\left[\mathrm{H^(+)}\right]`, and
  `\left[\mathrm{CO_3}^(2-)\right]` denote the concentrations of the three species.

Step-by-step explanation:

Equilibrium Constant Expression

The equilibrium constant expression of a (reversible) reaction takes the form a fraction.

Multiply the activity of each product of this reaction to get the numerator.
`\rm H_2CO_3\; (aq)` is the only product of this reaction. Besides, its coefficient in the balanced reaction is one. Therefore, the numerator would simply be
`\left(a_{\mathrm{H_2CO_3\, (aq)}}\right)`.

Similarly, multiply the activity of each reactant of this reaction to obtain the denominator. Note the coefficient "
`2`" on the product side of this reaction.
`\rm 2\; H^(+)\, (aq) + {CO_3}^(2-)\, (aq)` is equivalent to
`\rm H^(+)\, (aq) + H^(+)\, (aq) + {CO_3}^(2-)\, (aq)`. The species
`\rm H^(+)\, (aq)` appeared twice among the reactants. Therefore, its activity should also appear twice in the denominator:

`\left(a_{\mathrm{H^(+)}}\right)\cdot \left(a_{\mathrm{H^(+)}}\right)\cdot \, \left(a_{\mathrm{{CO_3}^(2-)\, (aq)}})\right = \left(a_{\mathrm{H^(+)}}\right)^2\, \left(a_{\mathrm{{CO_3}^(2-)\, (aq)}})\right`.

That's where the exponent "
`2`" in this equilibrium constant expression came from.

Combine these two parts to obtain the equilibrium constant expression:

`\displaystyle K = \frac{\left(a_{\mathrm{H_2CO_3\, (aq)}}\right)}{\left(a_{\mathrm{H^(+)}}\right)^2\, \left(a_{\mathrm{{CO_3}^(2-)\, (aq)}}\right)} \quad\begin{matrix}\leftarrow \text{from products} \\[0.5em] \leftarrow \text{from reactants}\end{matrix}`.

Equilibrium Constant of Concentration

In dilute solutions, the equilibrium constant expression can be approximated with the concentrations of the aqueous "
`(\rm aq)`" species. Note that all the three species here are indeed aqueous. Hence, this equilibrium constant expression can be approximated as:

`\displaystyle K = \frac{\left(a_{\mathrm{H_2CO_3\, (aq)}}\right)}{\left(a_{\mathrm{H^(+)}}\right)^2\, \left(a_{\mathrm{{CO_3}^(2-)\, (aq)}}\right)} \approx \frac{\left[\mathrm{H_2CO_3\, (aq)}\right]}{\left[\mathrm{H^(+)\, (aq)}\right]^2\cdot \left[\mathrm{{CO_3}^(2-)\, (aq)}\right]}`.