100 mL of 0.1 M HCl is taken in a beaker and to it 100 mL 0.1 M NaOH of is added in steps of 2 mL and the pH is continuously measured. Which of the following graphs correctly depicts the change in pH?

Initially the beaker contains 100 mL of $$0.1\,$$M HCl.

Number of moles of the strong acid present $$n_{\text{acid}} = 0.1 \times \frac{100}{1000} = 0.01\;{\text{mol}}$$ Initial pH: $$[\text H^+] = 0.1\;{\text M}\;\;\Rightarrow\;\;{\text{pH}} = 1$$

A $$0.1\,$$M NaOH solution is now added in 2 mL steps. After the addition of $$V\;({\text{mL}})$$ of base

• moles of base added $$n_{\text{base}} = 0.1 \times \frac{V}{1000} = 0.0001\,V\;{\text{mol}}$$

• Remaining moles of $$\text H^+$$ (as long as $$V \lt 100\;{\text{mL}}$$) $$n_{\text{H}^+}^{\text{left}} = 0.01 - 0.0001\,V$$

• Total volume of the solution $$V_{\text{tot}} = \frac{100 + V}{1000}\;{\text L}$$

• Concentration of the excess $$\text H^+$$ $$[\text H^+] = \frac{0.01 - 0.0001\,V}{(100+V)\,/\,1000}$$

• Hence pH (till $$V = 100\;{\text{mL}}$$) $$\text{pH} = -\log\!\left[\frac{0.01 - 0.0001\,V}{(100+V)/1000}\right]$$

Representative values obtained from the expression:

$$\begin{array}{|c|c|c|} \hline V\;(\text{mL}) & [\text H^+]\;(\text M) & \text{pH}\\ \hline 0 & 0.10 & 1.00\\ 60 & 0.025 & 1.60\\ 80 & 0.011 & 1.95\\ 90 & 0.0053 & 2.28\\ 98 & 0.0010 & 3.00\\ 99 & 5.0\times10^{-4} & 3.30\\ \hline \end{array}$$

Thus, for most of the titration the pH rises very slowly from 1 to about 3.3. However, within the last few millilitres before the equivalence point the residual $$\text H^+$$ becomes extremely small, and the pH shoots up almost vertically.

At exactly $$V = 100\;{\text{mL}}$$ (equivalence) the amounts of $$\text HCl$$ and $$\text{NaOH}$$ are equal, the solution contains only $$\text{NaCl}$$ in water and therefore $$\text{pH} \approx 7.$$ (The addition stops here, so no alkaline region is produced.)

Hence the correct titration curve must

• start at pH 1, • rise very gently up to about pH 3-4, • show a near-vertical jump that ends at pH 7, • not proceed into the basic region.

Among the four given sketches, only Option C exhibits exactly this behaviour.

Therefore, the correct choice is:
Option C which is: the graph showing a slow rise from pH 1 followed by an abrupt jump ending at pH 7.