# Chemistry - Am I understanding buffering capacity against strong acid/base correctly?

## Solution 1:

Further to the answer by Poutnik the buffer capacity $\beta$ of a weak acid - conjugate base buffer is defined as the number of moles of strong acid or base $C_B$ needed to change the $p$H by $\pm1$ unit, where

$$\displaystyle \beta=\frac{d[B]}{d\text{pH}}$$

and the equilibrium concentration base present is $\displaystyle \mathrm{[B]}=\frac{k_w}{\mathrm{[H^+]}} - \mathrm{[H^+]}+\frac{C_BK_a}{\mathrm{[H^+]}+K_a} $.

(See answer How to set up equation for buffer reaction?) where $K_w$ is the water ionization equilibrium constant $K_w =\mathrm{ [H^+][OH^-]} = 10^{-14}$, $K_a$ is the acid dissociation constant, and $C_B$ the total concentration of buffer.

After performing the calculation $\displaystyle \beta=2.303\left[\frac{K_w}{[\text{H}^+]}+[\text{H}^+]+\frac{C_BK_a[\text{H}^+]}{ ([\text{H}^+]+K_a)^2 } \right]$

This can be analysed to find its maximum, but can be simplified first by ignoring the first two terms, because $K_w$ is tiny as is $[H^+]$ compared to the last term close to $K_a$. Plotting this last term with $C_B=0.02$ produces the following curves

where the max buffering is seen to be very close to the $pK_A$. You can see that buffering will be ok only when the $p$H changes by $\approx \pm 1$ and when the $pK_A$ is in the range from about $pK_A $ from $\approx 4 \to 9$. The red curve is the last term in the equation, the blue curve the full equation.

To answer your first question the $p$H should be close to the $pK_A$ and as the total concentration of base ($C_B$) increases the maximum just increases but does not shift.

To find the maximum differentiating the last term only with $[H^+]\equiv x$ produces, after a little algebra, $\displaystyle \frac{d\beta}{dx}=C_BK_a\frac{K_a-x}{(K_a+x)^3}=0$ where the maximum is found when $[H^+]=K_a$

## Solution 2:

User porphyrin is correct that buffer capacity is defined as the number of moles of strong acid or strong base needed to change the pH of 1 liter of solution by ±1 unit.

- The buffer capacity is a dimensionless number.
- The buffer capacity is a somewhat fuzzy number in that the number of moles of base to cause + 1 pH change may not be the same as the number of moles of acid to cause a -1 pH change.
- Any dilution of the starting solution is typically ignored.

Typically a buffer is supposed to protect against the solution becoming more acidic or more basic. Assuming a reasonable concentration of the buffer, at a pH of 4.74 the concentration of acetate anion will equal the concentration of acetic acid. Hence that pH, which is equal to the pKa, will have the *optimal buffer capacity* against acid or base. (Note that this is a bit fuzzy since there is no universal definition of "optimal buffer capacity.")

However the problem also asks about only acid or only base being added. So:

- At a pH of 5.60 there is more of the acetate anion to react with HCl. (starting pH = 5.60, final pH = 4.60) However this buffer would poorly buffer the addition of a strong base since there is very little acetic acid.
- At a pH of 4.00 there is more acetic acid to react with NaOH. (starting pH = 4.00, final pH = 5.00) However this buffer would poorly buffer the addition of a strong acid since there is little acetate which isn't protonated to acetic acid.

## Solution 3:

The purpose of buffers is to keep $\mathrm{pH}$, with the differential buffering capacity $\frac { \mathrm{d[B]}}{ \mathrm{d(pH)}}$

If you are interested in the integral buffer capacity across $\mathrm{pH}$ range, than optimal is the buffer with the maximum capacity in the middle of the range. But the useful range for a single pair buffers is usually just about 2-2.5.

The rest of the answer depends on if more important is the initial $\mathrm{pH}$, the differential or integral capacity, or just the amount of strong acid/base needed for the solution to become strongly acidic/alkalic.

As for buffering against , we would want our to be as high as possible, right?

No, we would want $\mathrm{pH}$ to have the desired value, otherwise we do not speak about $\mathrm{pH}$ buffers.

Is this generalization correct: if my buffer $\mathrm{pH} <<< \mathrm{p}K_\mathrm{a}$, then does that mean it is optimal against strong bases (and vice versa)?

It would not be a buffer, but just a weak acid. If $\mathrm{pH}$ is high enough, it would have higher neutralization capacity than a strong acid, as higher molar concentration is needed for given $\mathrm{pH}$. But its initial buffer capacity would be very low.