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By this I mean what is the rigorous definition of a buffer solution?

I know that a buffer is a solution able to resist changes of $\mathrm{pH}$, but how much resistance to $\mathrm{pH}$ does a solution need to be considered a buffer?

For example I was told that one cannot make an acetic acid buffer $\left(K_\mathrm{a} = 1.8 \times 10^{-5}\right)$ at a desired $\mathrm{pH}$ of 6.5. Is this true?

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  • $\begingroup$ en.wikipedia.org/wiki/Buffer_solution#Buffer_capacity $\endgroup$ – Mithoron Jun 17 '20 at 0:31
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    $\begingroup$ @Blade: See how I manipulate $K_\mathrm{a}$ using $K_\mathrm{a}$ formatting. Rate constants and equilibrium constants should be itallics such as other variables (e.g., $V, T, R, P, x, y, z,$ etc.). $\endgroup$ – Mathew Mahindaratne Jun 17 '20 at 1:28
  • $\begingroup$ Who "told you", instead of teaching you how buffers work? $\endgroup$ – Karl Jun 17 '20 at 7:33
  • $\begingroup$ Buffer solution Is considered efficient, if it meets your criteria for pH stabilization. Note the buffers are defined qualitatively, not quantitatively. 1 M acetic acid buffer at pH 6.5 is roughly comparable in buffer capacity to 0.01 M acetic acid buffer at ideal pH 4.75. So it is inefficient buffer, but still a buffer. $\endgroup$ – Poutnik Jun 17 '20 at 10:12
  • $\begingroup$ Well, I´m not blaming you, but he seems to have left out a few important points in his teaching. A buffer is obviously most efficient when it is not too dilute and has not already used up part of its buffer capacity in one direction, i.e. it´s fresh. A fresh buffer solution is an equimolar mixture of an acid and its conjugate base. That means you have no choice in the pH. It´s a property of the compound pair. $\endgroup$ – Karl Jun 17 '20 at 18:52
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Efficiency of a buffer is not an absolute parameter, but it is conditional and case dependent.

If buffer differencial capacity ( $\frac{\mathrm{d}n}{\mathrm{dpH}}\cdot \frac{1}{V}$, where $n$ is molar amount of an added base/acid and $V$ is buffer volume) or integral capacity ($\frac{\Delta n}{\Delta \mathrm{pH}}\cdot \frac{1}{V}$ per $\Delta pH = \pm 1$) meet criteria required by the solution of interest, the buffer is efficient and vice versa. The same buffer may be efficient for one application, while not for another.

As $\mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log \frac{\ce{[A-]}}{c-\ce{[A-]}}=\mathrm{p}K_\mathrm{a} + \log {\ce{[A-]}} - \log{(c-\ce{[A-]})}$

then relative differential capacity wrt the maximal capacity at $\mathrm{pH}=\mathrm{p}K_\mathrm{a}$ is $\frac{4}{ \frac{c}{\ce{[A-]}} + \frac{c}{c-\ce{[A-]}}} $, where $c$ is the total molar concentration of the buffer conjugate acid + base pair.

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