Help me answer this!!!

Question

You have found that 13.85mL of 0.112M sulfuric acid reacts with exactly 20.00mL of barium hydroxide solution of unknown concentration. What is the molarity of the barium hydroxide solution? Round your answer to 3 decimal places and include units.

Answer 1

Short answer: 0.078 M (the unrounded value is 0.07756 M).

1. The balanced equation, and why the ratio is 1:1

Sulfuric acid gives up two protons and barium hydroxide gives up two hydroxides, so the two 2s cancel and the reaction is one-to-one:

H2SO4(aq) + Ba(OH)2(aq) -> BaSO4(s) + 2 H2O(l)

  2 H+   +  2 OH-   ->  2 H2O
  Ba2+   +  SO4 2-  ->  BaSO4(s)

This is the one trap in the problem. Students who notice that sulfuric acid is diprotic often multiply by 2 (giving 0.155 M) or divide by 2 (giving 0.0388 M). Neither is right, because the base is dibasic too, so the extra proton is matched by an extra hydroxide. Take the ratio from the coefficients in the balanced equation, never from the number of ionizable hydrogens on its own.

2. Moles of acid

n(H2SO4) = M x V
         = 0.112 mol/L x 0.01385 L
         = 1.5512e-3 mol

3. Moles of base, using the 1:1 ratio

n(Ba(OH)2) = 1.5512e-3 mol x (1 mol base / 1 mol acid)
           = 1.5512e-3 mol

4. Molarity of the base

M = n / V
  = 1.5512e-3 mol / 0.02000 L
  = 0.07756 mol/L

Rounded to three decimal places: 0.078 M

5. About the rounding, which is what this thread got stuck on

Three decimal places and three significant figures are two different instructions, and on this number they disagree:

0.07756 M  ->  three significant figures  ->  0.0776 M
0.07756 M  ->  three decimal places       ->  0.078 M

Decimal places are counted from the decimal point; significant figures start at the first nonzero digit. Because this answer has a zero in the hundredths place, rounding to three decimal places actually discards a digit you legitimately measured. All three of your data values (0.112, 13.85, 20.00) carry at least three significant figures, so 0.0776 M is the chemically honest report. Your assignment asked for three decimal places, so enter 0.078 M, but the two rules are not the same thing, and on a lab report you would write 0.0776 M.

6. A free sanity check

Because the ratio is 1:1, the concentrations have to sit in the inverse ratio of the volumes:

M(base) = M(acid) x V(acid) / V(base)
        = 0.112 x (13.85 / 20.00)
        = 0.112 x 0.6925
        = 0.0776 M

It took less acid volume than base volume, so the base must be the more dilute of the two, and it is, by exactly that 0.6925 factor. If your answer ever comes out larger than 0.112 M on a problem shaped like this, you have inverted the ratio.

Worth knowing: this particular pair is the classic conductometric titration. Both products leave the solution, since barium sulfate precipitates (Ksp about 1.1e-10) and the protons and hydroxides become water, so nearly every ion is consumed at the equivalence point. The conductivity of the solution falls almost to zero there and climbs again once you overshoot, which means you can find the endpoint with a conductivity meter and no indicator at all. That is not true of most acid-base titrations, and it is why this exact pair shows up in textbooks.