Captain Kirk of the Starship Enterprise has been told by his superiors that only

Question

Captain Kirk of the Starship Enterprise has been told by his superiors that only a chemist can be trusted with the combination of a safe containing the dilithium crystals that power the ship.

            The combination is the pH of solution A, described below, followed by the pH of solution C, also described below. (Example: If the pH of solution A were 3.47 and the pH of solution C were 8.15, the combination would be 3-47-8-15.)

            Determine the combination of the safe from the information below:

Solution A:     is a 0.100 M solution of a weak monoprotic acid HX.

Solution B:     is a 0.0500 M solution of NaX, the sodium salt of this acid. It has a pH of 10.02.

Solution C:     is made by adding 15.0 mL of 0.2500 M KOH to 50.0 mL of Solution A.

What is the combination to the safe?

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Explanation / Answer

solution B
NaX is a strong salt => Na+ + X-

X- + H2O <-----> HX + OH-
K = Kw/ Ka
pH = 10.02
pOH = 14 - 10.02 = 3.98
[OH-] = 10^-3.98 = 0.000105 M

K = Kw/ Ka = ( 0.000105)^2 / 0.0500 - 0.000105 = 2.21 x 10^-7
Ka = 10^-14 / 2.21 x 10^-7 =4.52 x 10^-8

Solution A
HX <------> H+ + X-
4.52 x 10^-8 = x^2 / 0.100 -x
x = [H+] = 0.0000673 M
pH = 4.17

Solution C
Moles HX = 0.0500 L x 0.100 = 0.00500
Moles KOH = 0.0150 L x 0.250 M = 0.00375
HX + OH- => X- + H2O
moles HX = 0.00500 - 0.00375 = 0.00125
moles X- = 0.00375
Total volume = 0.065 L
[HX] = 0.00125 / 0.065 = 0.0192 M
[X-] = 0.00375 / 0.065 = 0.0577 M
pKa = - log 4.52 x 10^-8 = 7.34
pH = 7.34 + log 0.0577 / 0.0192 = 7.82

combination to safe is 4 - 17 - 7 - 82

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