Final answer:
To calculate the Ka of protonated caffeine from a solution with a pH of 12.300, we determine the concentration of OH- ions, calculate the Kb for caffeine, and then use the relation Ka = Kw / Kb to find the desired Ka value for the conjugate acid of caffeine.
Step-by-step explanation:
To calculate the Ka of protonated caffeine, we start from the provided pH of 12.300 for the 1.00 M solution of caffeine. Since caffeine is a weak base, the pH tells us about the concentration of hydroxide ions (OH-) rather than hydrogen ions (H+).
We use the relationship pOH = 14 - pH to calculate the pOH, which in this case is 14 - 12.300 = 1.700. The concentration of OH- (in mol/L) can be determined using [OH-] = 10^(-pOH), which gives us [OH-] = 10^(-1.700) ≈ 2.00 x 10^-2 M.
Caffeine, being a base, accepts a proton (H+) to form its conjugate acid during the reaction with water. The equilibrium expression for the ionization of caffeine (C8H10N4O2) in water to form protonated caffeine (C8H10N4O2H+) and hydroxide ions is:
- C8H10N4O2 + H2O ⇌ C8H10N4O2H+ + OH-
The equilibrium constant for this reaction, Kb, can be expressed as:
- Kb = [C8H10N4O2H+][OH-] / [C8H10N4O2]
Given that our solution is 1.00 M and assuming a 1:1 stoichiometry for the reaction, the concentration of protonated caffeine [C8H10N4O2H+] is equal to the [OH-], which is 2.00 x 10^-2 M.
Therefore, the Kb value can be calculated using the initial concentration of the caffeine and the concentration of OH- ions. As we started with a 1.00 M solution, and the concentration of OH- is 2.00 x 10^-2 M, the new concentration of caffeine [C8H10N4O2] is 1.00 M - 2.00 x 10^-2 M ≈ 0.98 M. Hence, Kb ≈ (2.00 x 10^-2 M)^2 / 0.98 M. However, since we need to find the Ka for the conjugate acid, we must use the relationship between Kb for a base and Ka for its conjugate acid given by Kw = Ka * Kb, where Kw is the ionization constant of water (1.00 x 10^-14 at 25°C).
By rearranging this equation, we get Ka = Kw / Kb. Once we've calculated Kb, we can find the Ka for protonated caffeine. If further calculations are needed, the exact value of Kb must be determined first and substituted back into the Ka equation.