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I am having a difficult time understanding what makes a buffer a buffer.

Buffers in my textbook are defined as a solution of a weak acid or base and their conjugate acid/base. So if I were to just dissolve acetic acid (a weak acid) in water why would this not be defined as a buffer. Acetic acid incompletely dissociates into an acetate ion (the conjugate base) and a Hydrogen ion. Why is it that for a base to be made the acetic acid would have to be mixed with something like Sodium acetate if acetic acid incompletely dissociates into a conjugate base in the first place? Could someone please help me understand why this is the case?

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The term "buffer" implies a purpose of resisting changes in pH when small amounts of acid or base are added. The buffer will be most effective when the pH equals the pKa of the acid. This is when there are equal amounts of acid and conjugate base. For each pH unit the system deviates from pH=pKa, the buffer is about a factor 10 less effective.

If you just dissolve acetic acid in water, yes it is true that there will be a small fraction that ionizes to yield the conjugate base, acetate. However, the pH will be far from the pKa of acetic acid, and the system will not be very effective as a buffer.

In summary, "buffer" is not a very precise term, but it implies that there are significant amounts of both the acid and the conjugate base.

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I totally agree with @DavePhD and I want to show you how the pH dependence of the degree of acid-base ionisation provides the background of the buffering mechanism. Titration curves effectively illustrate the influence of the pH on the ionisation of weak acids and bases. The Figure below shows the change in the pH of a 0.1 M acetic acid solution on its titration with a strong base. If you just dissolve acetic acid in water, yes it is true that there will be a small fraction that ionizes to yield the conjugate base, acetate. However if you add the slightest amount of a base, for example, it will result into a steep increase in the pH (look at the beginning of the titration curve) and the solution is not a buffer.

That initial steep increase in the pH slows down quickly, and a pH plateau appears in the range of 4.0-5.8. As in this range $[\ce{CH3COOH}]$ is comparable to $[\ce{CH3COO-}]$ (i.e. their ratio is within a tenfold range), we will have a two-component system containing both a weak acid and a weak base—i.e. a buffer. The centre of the plateau is at 50 % titration where $[\ce{CH3COOH}]=[\ce{CH3COO-}]$. Therefore, according to the equation $\mathrm{pH}=\mathrm{p}K_\mathrm{a}+\log{\frac{[\ce{CH3COO-}]}{[\ce{CH3COOH}]}}$, the $\mathrm{pH}$ will just equal the $\mathrm{p}K_\mathrm{a}$. Following the plateau (upon the exhaustion of the buffer capacity), the pH will again increase steeply, and the top of the titration step will be reached, i.e. the point of neutralization at which the degree of titration of the acetic acid will be 100 % enter image description here

From the figure above, the region marked inside the rectangle is a buffer. Because it can provide the most efficient pH stabilization against both acidic and basic shifts. In this region, the ratio of the two components (acid and conjugated base) is one or close to one. Accordingly, a buffer is effective in the $\mathrm{pH}$ range of its $\mathrm{p}K_\mathrm{a} \pm 1$ $\mathrm{pH}$ unit

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