Heres the problem:
Calcaulte hte pH of a .20M aqueous solution of (CH₃)₂NH₂C₂H₃O₂, dimethylamine acetate.

Here's what I did:

(CH₃)₂NH₂C₂H₃O₂ --> (CH₃)₂NH₂⁺ + C₂H₃O^2-.

Since acetate doesn't do anything to the pH I take it out of the equation so I have

(CH₃)₂NH₂C₂H₃O₂ --> (CH₃)₂NH₂ + H⁺

I   .20                            0               0
C -x                              +x           +x
F .20 - x                         x             x

x²/(.20-x) = 1.0 x 10^-14/5.9x10^-4 = 1.7 x 10^-11

x=1.8 x 10^-6
.log(x) = 5.74

What am I doing wrong?? Thanks for your help

Since acetate doesn't do anything to the pH I take it out of the equation

There are two hydrolysis processes here. One of acetic ion, one of dimethylamine. The dissociation constants are too close for any of the equlibriums to be neglected, so it is not possible to do any simple approximations. I will take a look at the problem, right now all I know is that pH is 7.72 (but it was calculated using BATE, so no fancy thinking on my side).

 :Lighten:

Symmetry does the trick!

We have a base and an acid - lets call them HA and BOH.

Ka = [A][H]/[HA]
Kb = [ B][OH]/[BOH]

so

[H] = Ka[HA]/0.2
[OH]=Kb[BOH]/0.2

Why 0.2? Because the hydrolysis doesn't go too far and we can safely assume that the dissociated form concentration is almost the same as the concentration of salt. Besides, that's only assumption - we can later check if it was OK, and if not, start again.

Now, first fancy move:

[H][OH] = Ka[HA]/0.2 * Kb[BOH]/0.2 = Kw

after rearranging:

[HA][BOH] = 0.2^2*Kw/(Ka*Kb)

And now... SYMMETRY DOES THE TRICK!

Ka and Kb have very similar values, so lets assume that [HA] = [BOH]. If so

[HA]^2 = 0.2^2*Kw/(Ka*Kb)

and [HA] = 0.00022

Getting back to [H] and acid dissociation constant:

[H] = Ka [HA] / 0.2 = 10^(-4.75) * 0.00022 / 0.2 = 1.96E-8

and pH = 7.71

That's the same value as the one calculated with BATE. To be OK we now should check if our assumption was right, putting known values of [H], [OH], [HA] and [BOH] into all equations and checking if they hold - but I can assure you they do.

OK, I am going before my wife filles for divorce....