Respuesta :
Answer:
Equilibrium constant expression for [tex]\rm 2\; H^{+}\, (aq) + {CO_3}^{2-}\, (aq) \rightleftharpoons H_2CO_3\, (aq)[/tex]:
[tex]\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]}[/tex].
Where
- [tex]a_{\mathrm{H_2CO_3}}[/tex], [tex]a_{\mathrm{H^{+}}}[/tex], and [tex]a_{\mathrm{CO_3}^{2-}}[/tex] denote the activities of the three species, and
- [tex][\mathrm{H_2CO_3}][/tex], [tex]\left[\mathrm{H^{+}}\right][/tex], and [tex]\left[\mathrm{CO_3}^{2-}\right][/tex] denote the concentrations of the three species.
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.[tex]\rm H_2CO_3\; (aq)[/tex] is the only product of this reaction. Besides, its coefficient in the balanced reaction is one. Therefore, the numerator would simply be [tex]\left(a_{\mathrm{H_2CO_3\, (aq)}}\right)[/tex].
Similarly, multiply the activity of each reactant of this reaction to obtain the denominator. Note the coefficient "[tex]2[/tex]" on the product side of this reaction. [tex]\rm 2\; H^{+}\, (aq) + {CO_3}^{2-}\, (aq)[/tex] is equivalent to [tex]\rm H^{+}\, (aq) + H^{+}\, (aq) + {CO_3}^{2-}\, (aq)[/tex]. The species [tex]\rm H^{+}\, (aq)[/tex] appeared twice among the reactants. Therefore, its activity should also appear twice in the denominator:
[tex]\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[/tex].
That's where the exponent "[tex]2[/tex]" in this equilibrium constant expression came from.
Combine these two parts to obtain the equilibrium constant expression:
[tex]\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}[/tex].
Equilibrium Constant of Concentration
In dilute solutions, the equilibrium constant expression can be approximated with the concentrations of the aqueous "[tex](\rm aq)[/tex]" species. Note that all the three species here are indeed aqueous. Hence, this equilibrium constant expression can be approximated as:
[tex]\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]}[/tex].
