# What is the pH at both equivalence points of titration between diprotic tartaric acid and NaOH?

Here's all the info I have and my attempt in solving the question:

The chemical equation for the titration:

$$\ce{C4H4O6 + 2NaOH -> Na2C4H4O6 + 2H2O}$$

Here are the 1st and 2nd $$\mathrm{Ka}$$ value for tartaric acid:

$$K_\mathrm{a1} = 9.2 \times 10^{-4}, K_\mathrm{a2} = 4.3 \times 10^{-5}$$

Molarity of tartaric acid = $$5 \times 10^-4$$

Standard solution of $$\ce{NaOH }$$ was made using $$\pu{0.98 g}$$ of solid $$\ce{NaOH}$$ dissolved in $$\pu{250 ml}$$ of water. The molar mass of $$\ce{NaOH}$$ is $$\pu{39.997 g/mol}$$.

This is what I have done so far.

n (C4H4O6) = cv = (5 × 10-4 M) (25ml ÷ 1000L) = 0.02 mol

n (NaOH) = m ÷ M = (0.98g) ÷ (39.997 g/mol) = 0.0245 mol (4dp).

C (NaOH) = n ÷ v = (0.0245 mol) ÷ (250ml ÷ 1000L) = 0.098M

N (NaOH) at equivalence point = cv = (0.098M) (0.204L) = 0.02 mol

C4H4O6(aq)+ 2NaOH(aq) → Na2C4H4O6(aq) + 2H2O(l)

I 0.02 0.02 0 - C - 0.02 - 0.02 + 0.02 - E 0 0 0.02 -

C (Na2C4H4O6) = n ÷ v = 0.02 ÷ (0.025L + 0.25L) = 0.0727M (4dp)

Na2C4H4O6(aq) + H2O(l) → C4H4O4Na2^+(aq) + OH^-(aq) below is ice table, stackexchange won't allow me to seperate the values.

I 0.0727 - 0 0 C - x - + x + x E 0.0727 – x - x x

Kb = Kw ÷ Ka = (1 × 10-14) ÷ (9.2 × 10-4) = 1.087 × 10-11

Kb = [C4H4O4Na2+] [OH-] ÷ [Na2C4H4O6] 1.087 × 10-11 = x2 ÷ 0.0727 x = √ ((1.087 × 10-11) (0.0727)) x = 8.89 × 10-7

pOH = - log [OH-] = - log (8.89 × 10-7) = 1.9

pH = 14 – pOH = 14 – 1.9 = 12.1

• The question is based on the wrong terminology. Rather than the endpoint, the question would seemingly be asking about the equivalence point. – MaxW May 26 '19 at 16:56
• Thank you, you are right. I've just changed it. Would you by any chance know how to approach this question, as I still don't quite understand it – Lisa May 26 '19 at 16:59

For the 1st equivalence point, the tartaric acid $$\ce{H2A}$$ is titrated to $$\ce{NaHA}$$

For the solutions of acid salts, we can with advantage use the formula

$$\mathrm{pH}=\frac{\mathrm{p}K_\mathrm{a1}+\mathrm{p}K_\mathrm{a2}}{2}$$

that can be derived from equilibrium

$$\ce{2 HA- <=> H2A + A^2-}$$

For the 2nd equivalence point there is $$\ce{Na2A}$$ solution, which is hydrolyzed with equilibrium

$$\ce{A^2- + H2O <=> HA- + OH-}$$

With neglecting OH- from water hydrolysis, $$[\ce{HA^-}] = [\ce{OH^-}]$$

\begin{align} K_\mathrm{a2}&=\frac{[\ce{H+}][\ce{A^2-}]}{[\ce{HA-}]}\\ \mathrm{p}K_\mathrm{a2}&=\mathrm{pH} - \log{[\ce{A^2-}]} +\log {[\ce{HA-}]}\\ &=\mathrm{pH} - \log{[\ce{A^2-}]} +\log {[\ce{OH-}]}\\ &=\mathrm{pH} - \log{[\ce{A^2-}]} -\mathrm{pOH}\\ &=\mathrm{pH} - \log{[\ce{A^2-}]} -14 + \mathrm{pH}\\ \mathrm{pH}&=7+\frac12\left(\mathrm{p}K_\mathrm{a2} +\log{ c_\mathrm{\ce{Na2A}}}\right)\\ \end{align}

The molarity of $$\ce{NaOH}$$ solution $$c_{\ce{NaOH}}=\frac{m}{M\times V}=\pu{0.09801 mol/L}$$

The volume of $$\ce{NaOH}$$ solution for given volume of tartaric acid solution

$$V_{\ce{NaOH}}=V_{\ce{H2A}}\times \frac{2\times c_{\ce{H2A}}}{c_{\ce{NaOH}}}\\=0.0102\times V_{\ce{H2A}}$$

The final concentration

\begin{align} [\ce{Na2A}]&=\frac{[\ce{H2A_\mathrm{init}}]\cdot V_{\ce{H2A}}}{V_{\ce{H2A}} +V_{\ce{NaOH}}}\\ &=\frac{[\ce{H2A_\mathrm{init}}]\cdot V_{\ce{H2A}}}{V_{\ce{H2A}} + 0.0102 \cdot V_{\ce{H2A}}}\\ &=\frac{[\ce{H2A_\mathrm{init}}]}{1+ 0.0102}\\ &=5\times 10^{-4}/1.0102\\ &=\pu{4.9495\times10^{-4} mol/L}\\ &\overset{\mathrm{rounding}}=\pu{4.95\times 10^{-4} mol/L}\\ \end{align}

• Thanks so much. I understand now. Just two questions though. For the volume, wouldn't it be 0.0102 * VH2A. Also, why is it 1.0103 in the formula with final concentration? – Lisa May 27 '19 at 11:38
• It was, perhaps unluckily autowrapped. The acid solution is diluted by NaOH addition. Plus, perhaps rounding issue. Anyway 5 valid digits is excessive precision. – Poutnik May 27 '19 at 12:36
• Hello, sorry just a little confused. So volume of NaOH is meant to be 0.0102 instead of 0.0103? – Lisa May 27 '19 at 12:46
• and why is 1 added to 0.0103? Lastly, where is the last equation for final concentration derived from? – Lisa May 27 '19 at 12:47
• @Lisa As I have said, rounding error. See the fixed and updated answer. – Poutnik May 27 '19 at 14:15