To determine the equilibrium constant, [tex]\( K_a \)[/tex], for the dissociation of hydrocyanic acid ([tex]\( HCN \)[/tex]) into hydrogen ions ([tex]\( H^+ \)[/tex]) and cyanide ions ([tex]\( CN^- \)[/tex]), we start by recalling the general form of the equilibrium constant expression for a weak acid dissociation reaction. The given dissociation reaction is:
[tex]\[ HCN(aq) \rightleftharpoons H^+(aq) + CN^-(aq) \][/tex]
The equilibrium constant expression, [tex]\( K_a \)[/tex], for this reaction is defined as the ratio of the concentrations of the products to the concentration of the reactant, each raised to the power of their respective coefficients in the balanced chemical equation.
For the reaction [tex]\( HCN(aq) \rightleftharpoons H^+(aq) + CN^-(aq) \)[/tex], the equilibrium expression is:
[tex]\[ K_a = \frac{[H^+][CN^-]}{[HCN]} \][/tex]
Where:
- [tex]\([H^+]\)[/tex] is the concentration of hydrogen ions at equilibrium.
- [tex]\([CN^-]\)[/tex] is the concentration of cyanide ions at equilibrium.
- [tex]\([HCN]\)[/tex] is the concentration of hydrocyanic acid at equilibrium.
Now, let's examine the given options:
A. [tex]\( K_a = \frac{[HCN]}{[H^+][CN^-]} \)[/tex]
B. [tex]\( K_3 = [H^+][CN^-] \)[/tex]
C. [tex]\( K_a = \frac{[H^+][CN^-]}{[HCN]} \)[/tex]
D. [tex]\( K_a = [HCN][H^+][CN^-] \)[/tex]
From the description above, the correct expression that matches our derived equilibrium constant expression is:
[tex]\[ C. K_a = \frac{[H^+][CN^-]}{[HCN]} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{C} \][/tex]
