|[HnA-(N-n)]||=||Kn+1||*||Kn+2||* ... *||KN|
|[HnA-(N-n)] =||[HnA-(N-n)] / [HNA]||* [B]|
|sum [HnA-(N-n)] / [HNA]|
The relative ratios can be substituted into this equation to obtain each concentration as a function of the dissociation constants, [B], and pH.To determine what ingredients to put in the buffer to obtain these concentrations, the script uses the facts that positive ions must balance negative ions, and that water dissociates with equilibrium constant
[OH-] = Kw/[H3O+].For NaOH and MN-aHaA (M being a monovalent cation distinguishable from Na+) the charge balance is:
[H3O+] + [Na+] + (N-a)[B] = [OH-] + sum (N-n) [HnA-(N-n)]Note that [M+] = (N-a)[B]. The amount of NaOH added should result in the sodium ion concentration [Na+] and the amount of buffering anions should result in a concentration equal to [B].
For MN-bHbA and HCl (A being an anion distinguishable from Cl-) the charge balance is:
[H3O+] + (N-b)[B] = [OH-] + [Cl-] + sum (N-n) [HnA-(N-n)]The amount of HCl added should equal [Cl-] and the amount of base should be [B].
For MN-bHbA and MN-aHaA :
[H3O+] + x(N-b)[B] + (1-x)(N-a)[B] = [OH-] + sum (N-n) [HnA-(N-n)]where x[B] is the amount of acid to add and (1-x)[B] the amount of base.
The sum in these equations is the number of bound protons:
[H+]bound = sum n[HnA-(N-n)]subtracted from the total number of protons that can be bound, the buffer capacity C:
[C] = sum N[HnA-(N-n)] = N[B]Note: this is different from the minimum slope of the titration curve, as others define "buffer capacity".
|L =||[H3O+] + [H+]bound - [OH-]|
If only HnA is added to water, the buffer load is n/N. By mixing different MN-nHnA's, intermediate buffer loads can be obtained. The buffer load can also be adjusted by adding a strong acid or base. The above equations can be rearranged to give simple formulas that show how much.
For NaOH and MN-aHaA, the concentrations should be:
(a/N - L)[C] and [B]; this buffer works if L < a/N.For MN-bHbA / HCl, the concentrations should be:
[B] and (L - b/N)[C]; this buffer works if L > b/N.For MN-bHbA / MN-aHaA, the concentrations should be:
|(a/N - L)[C]||and||(L - b/N)[C]||; this buffer works if b/N < L < a/N.|
Here is a table of pKw for various temperatures:
By Dave Robinson,
source, cross platform scripting language for the web.
The graphs are produced by
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