Can you make an acetic acid/sodium acetate buffer at pH 3? What is the lower pH limit?

I will answer by saying that you can make a CH3COOH/CH3COONa solution with pH = 3.00, but I doubt if this will be a good buffer solution. In general you should make the buffer with pH within 1 pH unit of the pKa of the weak acid ( or base). The pKa of CH3COOH is 4.74 , so a buffer with pH 3 is outside this limit. The solution will have pH = 3, but the buffer capacity will be low

How do you do this?

Use the H-H equation:

pH = pKa + log ( mol salt/mol acid)

3.00 = 4.74 + log ( mol salt/mol acid)

log ( mol salt/mol acid) = -1.74

( mol salt/mol acid) = 10^-1.74

( mol salt/mol acid) = 0.0182

You require 0.0182 mol CH3COONa to 1 mol CH3COOH . Or any convenient sub-multiple of this ratio. As you see you have essentially a CH3COOH solution with a small addition of CH3COONa.

The effective buffer pH range of an acid/salt system is generally estimated at 1 pH unit above and below the acid's pKa (-logKa).

The Ka of acetic acid is 1.8x10^-5, so its pKa = 4.74. This would suggest that acetic acid/sodium acetate could be used for buffers ranging from pH 3.7 to 5.7.

Of course, the ideal buffer pH for an acid is its pKa. This is because a buffer with a pH=pKa is produced when equal concentrations of acid and salt (conjugate base) are conmbined. For example, an acetic acid/sodium acetate buffer with pH 4.74 can be produced with 0.10 M concentrations of both components (or any other equal concentrations). This is ideal, as this buffer has an equal capacity for neutralizing excess acid or base.

As the pH of the buffer moves away from pKa, the concentrations of acid and conjugate base must differ. For example, with an acetic acid/sodium acetate system, to create a buffer with a pH lower than 4.74, the concentration of the acid must be greater than that of the salt (base). The reverse is true if the buffer pH exceeds pKa. For the acetic acid/sodium acetate buffer to have a pH greater than 4.74 the concentration of the salt(base) must be greater than that of the acetic acid.

This is the logic behind the +/-1pH unit around pKa as an effective buffer range for the acid/salt system. Consider again the acetic acid/sodium acetate system. To reach a pH of 1 pH unit less than pKa, the acid concentration must be 10x higher than the salt concentration. So, an acetic acid/sodium acetate buffer of pH 3.74 could be produced with 0.10 M acetic acid and 0.010 M sodium acetate. Similarly, for a buffer with a pH of 1 pH unit higher than pKa the concentration of the salt (base) must be 10x higher than that of the acid. So, an acetic acid/sodium acetate buffer with pH 5.74 could be produced with 0.010 M acetic acid and 0.10M sodium acetate.

What you can see is that as the buffer pH gets farther away from pKa, the buffering capacity for excess acid and base are no longer equal, the buffer becomes capable of counteracting the effect of one stress more than the other (ie; able to counteract the effect excess acid more than excess base).

So, to your question, is it possible to use acetic acid/sodium acetate to create a buffer with a pH = 3.0. Well, this pH could be achieved with 0.10 M acetic acid / 0.0018 M sodium acetate. Here the the salt (base) concentration is ~56x lower than the acid. So, not the best buffer, as it would quickly lose it capacity to neutralize excess acid compared to excess base.