Is this stoichiometric question answerable? (unknown variable)

Question

This question kind of borders on the fields of both semantics and math, as well as chemistry, but I still believe this is the right place to ask.

On a test I took, a question was posed as such:

Via titration we are adding a solution of 25 ml, with 0.5 M NaOH, to 25 ml of HCl. What will the concentration of the acid be?

Now, this question seemed a bit strange to be, and it was definitely new. I'm used to calculating concentrations after dilution and other stoichiometric calculations, but this question seemed like a misworded version of those other questions I've grown familiar to.

So, I asked my teacher if she could clarify. First I asked whether the 25 ml of HCl was pure HCl, or a water solution with HCl in it. She said it was pure HCl. Then I asked if she was looking for the concentration of the acid before ANY reaction between the NaOH and the HCl had occured. To this, she also said yes.

The options were:

0.25 M

0.5 M

0.75 M

1 M

I answered 1 M, my reasoning being that 25 ml of HCl is the same volume as 25 ml of water with NaOH. Obviously, this reasoning is quite flawed, but I didn't have anything better. The answer was wrong. However, when my teacher told me what was meant by the question afterwards, her explanation did not align with the question, nor her previous clarifications. She said that this was a simple case of:

$$C_1 \times V_1 = C_2 \times V_2$$

Where $C_2$ was the unknown, since the concentration of HCl was not specified. I don't see how this makes sense. All the question is saying, is that there is a 25 ml container, with HCl in it. It doesn't even say that this 25 ml consists of a water solution with HCl in it, but a more knowledgeable chemist would've known that HCl cannot exist outside of a water solution (that's what my teacher said at least).

Then the the question says that 25 ml of water, with a concentration of 0.5 M of NaOH is added. And from those two values, I'm supposed to find the concentration of HCl. That basically means:

$0.025$ l of H$_2$O$ \times 0.5$ M NaOH(aq) $= 0.025$ l of water $\times x$ M HCl (aq)

Which would then make the answer 0.5 M. Maybe she wanted me two add the two volumes, which would make sense:

$0.025$ l of H$_2$O$ \times 0.5$ M NaOH(aq) $= 0.05$ l of water $\times x$ M HCl (aq)

Which would then make the answer 0.025, which wasn't even an option.

But even if it was, it still wouldn't make sense to me. Just because you have a solution with a given volume and an unknown concentration, adding another solution with a known volume AND concentration won't just magically let you know what the unknown concentration in the original solution is, now in this new solution. Or am I wrong?

Answer 1

With a lot of assumptions, 0.5 M is a reasonable answer.

Here are the holes you have to fill in.

1. If it is a titration of NaOH and HCl, it is probably an acid/base titration where we stop the titration when the pH is neutral.

2. Titrations are often performed to figure out the concentration of a solution. You would titrate a smaller sample of the solution to figure out the concentration. Technically, the concentration of HCl after the titration is zero because all of it is supposed to react as you reach the endpoint of the titration. So it makes sense to ask about the concentration in the original solution.

3. The stoichiometry of the neutralization of HCl and NaOH is one-to-one: $$\ce{NaOH(aq) + HCl(aq) -> H2O(l) + Na+(aq) + Cl-(aq)}$$

Of course, it is silly not to give any context when posing such a question on a quiz. I won't comment on the statements you attribute to your teacher other than to say that hydrochloric acid always contains water (pure HCl is a gas under ambient conditions).

Answer 2

The formula $$C_1 \times V_1 = C_2 \times V_2$$ should never ever be used for titrations calculations. It fails miserably in most of the cases (e.g. titration of sulfuric acid with NaOH). This formula only works when the concentrations are in normality, which is considered an obsolete unit in modern chemistry. The proper use of this relation is for calculating dilutions because it is simply a mass balance equation and it works perfectly.

The test question as you wrote is rather poorly worded but basically it is asking you to calculate the concentration of the 25 mL HCl sample if 25 mL of 0.5 M NaOH was required to complete the titration.

All you have to do is to write a balance equation.

From the balanced equation determine the stoichiometric coefficients, i.e., the mole ratio between NaOH and HCl.

Calculate the moles of NaOH = Molarity of NaOH x Volume of NaOH used (L).

Apply the right ratio, and if you have written the balanced equation correctly, you will see the mole ratio is 1:1.

So, m= the moles of HCl = moles of NaOH,

Molarity of HCl = m/Volume of the original HCl sample in L.

You do not care about mixing of volumes until and unless the titration is incomplete, but that would be a separate story.

The correct answer is listed in the choices.