# Does a weak buffer solution still maintain a mostly steady pH if the acid or base load it has to buffer is small?

I need to buffer the $\mathrm{pH}$ of a solution to $3.0$ using a citrate buffer. I found a recipe to make a buffer solution that would use $82\%$ of $0.1\,\mathrm M$ citric acid (monohydrate) and $18\%$ $0.1\,\mathrm M$ trisodium citrate dihydrate to give a citrate buffer with a $\mathrm{pH}$ of $3.0$. The purpose of buffering the $\mathrm{pH}$ to $3$ is that this solution contains an active ingredient that has a very short shelf-life at a more neutral $\mathrm{pH}$. The buffer solution will make up about two-thirds of the solution by weight, and isopropyl alcohol will make up about $30\%$ of the solution. Assume all of the remaining ingredients are non-dissociable in the relevant $\mathrm{pH}$ range, except for the active ingredient which has the following properties:

• Concentration of $0.03\%$ by weight
• A single weakly basic moiety with a $\mathrm pK_\mathrm a$ of $11.45$
• Molecular mass of approximately $420\,\mathrm{g\over mol}$

Since this will be applied to skin, I'd like to use the weakest buffer solution that would still maintain the $\mathrm{pH}$ at $3.0$. My question is, if I use $0.01\,\mathrm M$ or $0.001\,\mathrm M$ solutions to make the buffer, would it still be able to buffer the $\mathrm{pH}$ at $3.0$?

## 1 Answer

First, a disclaimer: This answer addresses only the question of the chemical buffering capacity of the questioner's formulation, given the information provided. Neither I nor Stack Exchange / Stack Overflow can take any responsibility for the safety or efficacy of the proposed formulation, for the effectiveness of the buffer concentration indicated below for maintaining the formulation $\mathrm{pH}$, or for any other hazard, harm, or adverse event, without limitation, arising from the use of the information provided below.

To answer the general question of the title: Yes.

You haven't given enough information to answer your question in the particulars, though.

For your particular application:

Worst-case, buffer-wise, as long as the following assumptions are accurate, the maximum amount of buffering capacity you'd need can be estimated from the total number of moles of active ingredient:

• None of the other ingredients have appreciable acid-base character (appears to be the case here)
• Degradation of the other ingredients into acidic/alkaline species can be neglected
• No acidity or alkalinity is likely to enter from the environment

At $0.03\%$ w/w, assuming the overall density of the formulation is approximately $1\,\mathrm{g\over cm^3}$ (probably reasonable, with $67\%$ w/w composed of aqueous buffer) a species with a molecular weight of $420\,\mathrm{g\over mol}$ will have a mole concentration of $0.714\,\mathrm{mM}$. In general, I would probably want the buffering species to have a concentration about five to ten times my expected net acid/base load. So, I would choose no less than about $3.6\,\mathrm{mM}$ for the total concentration of all forms of citrate, perhaps going as high as $7.2\,\mathrm{mM}$ to be on the safe side.

Note: The $\mathrm pK_\mathrm a$ of the ionizable group on the active ingredient is sufficiently far from your target $\mathrm{pH}$ that it will be fully protonated in your formulation. Thus, if it is possible to add this ingredient in its protonated form, the required buffer concentration is probably somewhat lower than the above.

General Principles

As long as the number of moles of acid or base to be neutralized is small compared to the total number of moles of the relevant buffering species in the solution, and as long as the target $\mathrm{pH}$ is within about one unit of the nearest $\mathrm pK_\mathrm a$, the $\mathrm{pH}$ should hold steady.

As the number of moles of acid/base to be buffered against approaches the number of moles of the buffering species, the target $\mathrm{pH}$ needs to be closer and closer to the $\mathrm pK_\mathrm a$ in order to hold the $\mathrm{pH}$ steady.

If the number of moles of acid/base to be buffered against exceeds the number of moles of buffering species, the buffer will in general fail to hold the target $\mathrm{pH}$. The main exceptions would be polyprotic species with multiple $\mathrm pK_\mathrm a$ values in close proximity (polyphosphates come to mind).

• I think the general answer should work for me, but I'd appreciate it if you could guide me to give you the information that I need to give to get the specific answer. I tried to provide that information in the question, but I'm not an expert here. The buffer solution is about 67%(w/w) of the total product and the isopropyl alcohol is 30%, 1.5% is a non-ionic thickener, and 1% is another alcohol. The active ingredient is .03% with a pKa of 11.45 and a molecular mass of around 420g/mol. Is this enough information? Feb 9, 2016 at 6:24
• @Itsme2003 Are you concerned about keeping the formulation buffered against any acidity/alkalinity generated by the skin, or just buffering the formulation itself? Feb 9, 2016 at 11:57
• My concern is buffering the formulation itself in order to preserve the active ingredient. I apologize if the following description is imprecise but it's been a long time since I took a chemistry course. Once it's on the skin I don't care if the pH does change, and in fact I think it's probably best if the pH does change because I don't really understand the potential harm if something this acidic is applied to the skin. My thoughts are that if it has only a very tiny amount of acidity that the acidity would be "neutralized" by the skin without causing noticeable harm to the skin. Feb 9, 2016 at 14:17