Puzzler in pH calculations

[Morx]

Hey!

 

I just wanted to pose a question that I hope someone can find the time to answer.

 

It's regarding part of my homework - don't worry, I already have the answer. I was just confused about some of the methods.

The question is part of a larger one:

 

"Calculate the pOH of a 0.0500 mol dm^-3 aqueous solution of methylamine. State any assumptions made in your calculation."

We are given the K_b value at 4.37*10^-4 mol dm^-3

The assumption is referring to the fact that I can disregard the concentration value in the denominator of this equation [H⁺]²/C-[H⁺]=K_a

_{I could see two ways of calculating the answer. The first one gave me the right answer.}

If [H⁺]²/C=K_a then [OH^-]²/C=K

We can isolate for the [OH^-] value (0.004674), and take the negative logarithm in base 10 of it, producing a pOH of 2.33

 

I wanted to be thorough and double check my answer, so I ran through the next method too:

 

pOH = 14 - pH

pH = -log[H⁺]

[H⁺] = sqrt(K_a*C)

K_{a = 10}^-pKa

pK_{a = 14 - pKb}

pK_b = -log(K_b)

_{Putting in values and collecting equations I get: pOH = 14 - (-log(sqrt(10}^{-(14-(-log0.000437))}*0.05)))

This gives me a pOH value of 8.03 which is ridiculous, since methylamine is a weak base.

 

My question: what am I doing wrong in the second method? I know it's an unnecessary way around it, but from what I've read all the constants are mathematically connected to their conjugates. In theory, shouldn't it work?

 

Thank you!