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

I just want to make sure I'm understanding buffering capacity correctly.


I've been working on this problem:



> 
> Suppose you have an acetic acid buffer, $\mathrm{p}K\_\mathrm{a} = 4.74$, at the following $\mathrm{pH}$:
> 
> 
> 1) $\mathrm{pH} = 4.00$
> 
> 
> 2) $\mathrm{pH} = 4.35$
> 
> 
> 3) $\mathrm{pH} = 4.70$
> 
> 
> 4) $\mathrm{pH} = 5.00$
> 
> 
> 5) $\mathrm{pH} = 5.40$
> 
> 
> 6) $\mathrm{pH} = 5.60$
> 
> 
> Which buffer solution will have the highest buffering capacity against $\ce{HCl}$? Against $\ce{NaOH}$ ? Which is the best optimal buffer?
> 
> 
> 


From what I understand, optimal buffers are where $\mathrm{pH} = \mathrm{p}K\_\mathrm{a}$, since that is when the ratio between conjugate base and acid is equal to $1$. This means that there is just enough acid/base relative to each other to minimize $\mathrm{pH}$ changes whether a strong acid or base is added.


As for buffering against $\ce{HCl}$, we would want our $\mathrm{pH}$ to be as high as possible, right?


We want as much conjugate base as possible in our buffer to react with the $\ce{HCl}$, while minimizing the changes to the acid, which means we want $\frac{[\ce{A-}]}{[\ce{HA}]}$ to be as high as possible. Maximizing $\frac{[\ce{A-}]}{[\ce{HA}]}$ means $\log \left(\frac{[\ce{A-}]}{[\ce{HA}]}\right) \rightarrow \infty$, which means $\mathrm{pH} \rightarrow 14$, since $\mathrm{pH} = \mathrm{p}K\_\mathrm{a} + \log \left(\frac{[\ce{A-}]}{[\ce{HA}]}\right)$.


The explanation for $\ce{NaOH}$ would be the exact opposite. You want as much acid compared to conjugate base as possible so that the acid reacts with $\ce{NaOH}$ while minimizing changes to the conjugate base, which means $\frac{[\ce{A-}]}{[\ce{HA}]} \rightarrow 0$, $\log \left(\frac{[\ce{A-}]}{[\ce{HA}]}\right) \rightarrow -\infty$, and $\mathrm{pH} \rightarrow 0$. 


Is my understanding correct? The thing that's tripping me up is that I was told the exact amount of conjugate base and acid is more important than the ratio (*e.g.*, $0.5 / 0.9$ is better than $0.05 / 0.09$), but since there's no mention of exact concentrations in the problem, I'm guessing we just assume that we have sufficient amounts of $\ce{A-}$ and $\ce{HA}$.


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)?


Thank you for your help.




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?](https://chemistry.stackexchange.com/questions/60068/how-to-set-up-equation-for-buffer-reaction/95566)) 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


[![buffer](https://i.stack.imgur.com/p3yFp.png)](https://i.stack.imgur.com/p3yFp.png)


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 $[H^+]=K\_a$


 8

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.


 6

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.


 4