The equilibrium constants for the two reactions are known: M^m+ + 4L^- [ML₄]^m^-⁴ K_a
HL H⁺ + L^- K_b
What must be the value of the equilibrium constant, K_overall for the following overall reaction?
M^m+ + 4HL [ML₄]^m^-⁴ + 4H⁺ K_overall

Question

The equilibrium constants for the two reactions are known: M^m+ + 4L^-

The equilibrium constants for the two reactions are known: M<sup>m+</sup> + 4L<sup>-</sup> [ML<sub>4</sub>]<sup>m</sup><sup>-</sup><sup>4</sup> K<sub>a</sub> HL H<sup>+</sup> + L<sup>-</sup> K<sub>b</sub> What must be the value of the equilibrium constant, K<sub>overall</sub> for the following overall reaction? M<sup>m+</sup> + 4HL [ML<sub>4</sub>]<sup>m</sup><sup>-</sup><sup>4</sup> + 4H<sup>+</sup> K<sub>overall</sub> A) K<sub>a</sub>K<sub>b</sub> B) K<sub>a</sub> + 4K<sub>b</sub> C) K<sub>a</sub> + K<sub>b</sub><sup>4</sup> D) K<sub>a</sub>K<sub>b</sub><sup>4</sup> E) K<sub>a</sub>K<sub>b</sub><sup>1/4</sup>

[ML₄]^m^-⁴ K_a
HL

H⁺ + L^- K_b
What must be the value of the equilibrium constant, K_overall for the following overall reaction?
M^m+ + 4HL

[ML₄]^m^-⁴ + 4H⁺ K_overall

Quizplus · Accepted Answer

The overall reaction can be separated into two reactions: 
1. MS1U1P1m + S1S1P0 + 4HL -> [MLS1U1B14S1U1B0]S1U1P1mS1S1P0S1U1P1-S1S1P0S1U1P14S1S1P0 + 4HS1U1P1 + S1S1P0
2. [MLS1U1B14S1U1B0]S1U1P1mS1S1P0S1U1P1-S1S1P0S1U1P14S1S1P0 + 4HS1U1P1 + S1S1P0 -> MS1U1P1m + S1S1P0 + 4LS1U1P1-S1S1P0

The equilibrium constant for reaction 1 is:
KS1U1B1overallS1U1B0 = (MLS1U1B14S1U1B0)(HS1U1P1)^4(S1S1P0)/(MS1U1P1m)(S1S1P0)(HL)^4
The equilibrium constant for reaction 2 is:
1/KS1U1B1overallS1U1B0 = (MS1U1P1m)(S1S1P0)(HS1U1P1)^4/(MLS1U1B14S1U1B0)(S1U1P1-S1S1P0)(S1S1P0)(LS1U1P1-S1S1P0)^4

Multiplying the two equations, we get:
KS1U1B1overallS1U1B0 * (1/KS1U1B1overallS1U1B0) = (MLS1U1B14S1U1B0)(HS1U1P1)^4/(MS1U1P1m)(LS1U1P1-S1S1P0)^4(S1U1P1-S1S1P0)
Canceling out the common terms, we are left with:
KS1U1B1overallS1U1B0 = (MLS1U1B14S1U1B0)(HS1U1P1)^4/(MS1U1P1m)(LS1U1P1-S1S1P0)^4(S1U1P1-S1S1P0)
which is equivalent to option D.

The Equilibrium Constants for the Two Reactions Are Known: Mm+

The Equilibrium Constants for the Two Reactions Are Known: M^m+