Ionic Equilibria in Aqueous Systems

The well water in an area is "hard" because it is in equilibrium with ${\mathbf{CaCO}}_{\mathbf{3}}$ in the surrounding rocks. What is the concentration of ${\mathbf{Ca}}^{\mathbf{2}\mathbf{+}}$in the well water (assuming the water's  is such that the $\mathbf{C}{{\mathbf{O}}_{\mathbf{3}}}^{\mathbf{2}\mathbf{-}}$ion is not significantly protonated)? (See Appendix $\mathbf{C}$ for  ${\mathbf{K}}_{\mathbf{sp}}$of ${\mathbf{CaCO}}_{\mathbf{3}}$.)

What is the purpose of an acid-base buffer?

How do the acid and base components of a buffer function? Why are they often a conjugate acid-base pair of a weak acid?

What is the common-ion effect? How is it related to Le Châtelier’s principle? Explain with equations that include HF and NaF

The scenes below depict solutions of the same HA/A buffer (with counterions and water molecules omitted for clarity). 

(a) Which solution has the greatest buffer capacity?

 (b) Explain how the pH ranges of the buffers compare. 

(c) Which solution can react with the largest amount of added strong acid?

What is the difference between buffers with high and low capacities? Will adding 0.01 mol of HCl produce a greater pH change in a buffer with a high or a low capacity? Explain.

Which of these factors influence buffer capacity? How? (a) Conjugate acid-base pair (b) pH of the buffer (c) Concentration of buffer components (d) Buffer range (e) ${\mathbf{\text{pK}}}_{\mathbf{\text{a}}}$ of the acid component.

State and explain the relative change in the pH and in the buffer-component concentration ratio, [NaA]/[HA], for each of the following additions: 

(a) Add 0.1 M NaOH to the buffer 

(b) Add 0.1 M HCl to the buffer 

(c) Dissolve pure NaA in the buffer

 (d) Dissolve pure HA in the buffer.

Does the pH increase or decrease, and does it do so to a large or small extent, with each of the following additions? 

(a) 5 drops of 0.1 M NaOH to 100 mL of 0.5 M acetate buffer 

(b) 5 drops of 0.1 M HCl to 100 mL of 0.5 M acetate buffer 

(c) 5 drops of 0.1 M NaOH to 100 mL of 0.5 M HCl 

(d) 5 drops of 0.1 M NaOH to distilled water.

A buffer containing $\mathbf{0}\mathbf{.2000}\mathbf{\text{M}}$ of acid, $\mathbf{\text{HA}}$, and  $\mathbf{0}\mathbf{.1500}\mathbf{\text{ M}}$ of its conjugate base, ${\mathbf{\text{A}}}^{\mathbf{\text{ - }}}$, has a pH of 3.35. What is the pH after 0.0015 mol of NaoH is added to $\mathbf{0}\mathbf{.5000}\mathbf{\text{ L}}$ of this solution?

An industrial chemist studying bleaching and sterilizing prepares several hypochlorite buffers. Find the pH of (a) $\mathbf{\text{0.100 M HClO}}$ and $\mathbf{\text{0.100 M NaClO}}$; (b) $\mathbf{\text{0.100 M HClO}}$ and $\mathbf{\text{0.150 M NaClO}}$; (c) $\mathbf{\text{0.150 M HClO}}$ and $\mathbf{\text{0.100 M NaClO}}$; (d) $\mathbf{\text{1.0 L}}$ of the solution in part (a) after $\mathbf{\text{0.0050}}$ mol of $\mathbf{\text{NaOH}}$ has been added.?

A buffer is prepared by mixing $\mathbf{\text{204 mL}}$ of $\mathbf{\text{0.452 M HCl}}$ and $\mathbf{\text{0.500 L}}$ of $\mathbf{\text{0.400 M}}$ sodium acetate. (See Appendix C.) (a) What is the pH? (b) How many grams of $\mathbf{\text{KOH}}$ must be added to $\mathbf{\text{0.500 L}}$ of the buffer to change the pH by $\mathbf{\text{0.15}}$ units?

Choose specific acid-base conjugate pairs to make the following buffers: (a) $\mathbf{\text{pH}}\mathbf{\approx }\mathbf{\text{4.5}}$; (b)$\mathbf{\text{pH}}\mathbf{\approx }\mathbf{\text{4.5}}$ . (See Appendix C.)

How can you estimate the pH range of an indicator’s color change? Why do some indicators have two separate pH ranges?

What is the component concentration ratio, $\left[{\text{CH}}_{\text{3}}{\text{COO}}^{\text{ - }}\right]\mathbf{\text{/}}\left[{\text{CH}}_{\text{3}}\text{COOH}\right]$ , of a buffer that has a   pH of 4.39  ( ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$of $\left[{\text{CH}}_{\text{3}}\text{COOH}\right]\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$)?

A buffer is prepared by mixing $\mathbf{\text{50.0 mL}}$ of $\mathbf{\text{0.050 M}}$ sodium bicarbonate and $\mathbf{\text{10.7 mL}}$ of  $\mathbf{\text{0.10 M NaOH}}$. (See Appendix C.) (a) What is the pH? (b) How many grams of HCL must be added to $\mathbf{\text{25.0 mL}}$ of the buffer to change the pH by $\mathbf{\text{0.07}}$ units?

Choose specific acid-base conjugate pairs to make the following buffers: 

(a) $\left[{\text{H}}_3{O}^{+}\right]\mathbf{\approx }\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{9}}\mathbf{\text{ M}}$ ; 

(b) $\left[O{H}^{-}\right]\mathbf{\approx }\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{\text{ M}}$ . (See Appendix C.)

Choose specific acid-base conjugate pairs to make the following buffers: (a) $\mathbf{\text{pH}}\mathbf{\approx }\mathbf{\text{3.5}}$ ; (b) $\mathbf{\text{pH}}\mathbf{\approx }\mathbf{5}\mathbf{\text{.5}}$
. (See Appendix C.)

Choose specific acid-base conjugate pairs to make the following buffers: (a)  $\left[O{H}^{-}\right]\mathbf{\approx }\mathbf{1}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{\text{ M}}$; (b) $\left[H_3{O}^{+}\right]\mathbf{\approx }\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathbf{\text{ M}}$. (See Appendix C.)

A buffer that contains $\mathbf{\text{0.40 M}}$ base, B, and $\mathbf{\text{0.25 M}}$ of its conjugate acid, ${\mathbf{\text{BH}}}^{\mathbf{\text{ + }}}$, has a pH of $\mathbf{\text{8.88}}$. What is the pH after $\mathbf{\text{0.0020}}$ mol of HCL is added to $\mathbf{\text{0.25 L}}$ of this solution?

A buffer that contains $\mathbf{0}\mathbf{.}\mathbf{110} \mathbf{M} \mathbf{HY}$ and ${\mathbf{\text{0.220 M Y}}}^{\mathbf{\text{ - }}}$ has a pH of $\mathbf{\text{8.77}}$. What is the pH after $\mathbf{\text{0.0015}}$ mol of ${\mathbf{\text{Ba(OH)}}}_{\mathbf{\text{2}}}$ is added to $\mathbf{\text{0.350 L}}$ of this solution?

A buffer that contains $\mathbf{\text{1.05 M B}}$ and ${\mathbf{\text{0.750 M BH}}}^{\mathbf{\text{ + }}}$ has a pH of  $\mathbf{\text{9.50}}$. What is the pH after $\mathbf{\text{0.0050}}$ mol of HCL is added to $\mathbf{\text{0.500 L}}$ of this solution?

Oxoanions of phosphorus are buffer components in blood. For a ${\mathbf{\text{KH}}}_{\mathbf{\text{2}}}{\mathbf{\text{PO}}}_{\mathbf{\text{4}}}{\mathbf{\text{/Na}}}_{\mathbf{\text{2}}}{\mathbf{\text{HPO}}}_{\mathbf{\text{4}}}$ solution with $\mathbf{\text{pH = 7.40}}$ (pH of normal arterial blood), what is the buffer-component concentration ratio?

Why does the color change of an indicator take place over a range of about 2 pH units?

What is the component concentration ratio, $\mathbf{\left[}\mathit{B}\mathit{r}{\mathit{O}}^{\mathbf{-}}\mathbf{\right]}\mathbf{/}\mathbf{\left[}\mathit{H}\mathit{B}\mathit{r}\mathit{O}\mathbf{\right]}$, of a buffer that has a pH of  7.95( ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$of $\mathit{H}\mathit{B}\mathit{r}\mathit{O}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{9}}$ )?

What is the component concentration ratio, $\mathbf{\left[}\mathit{N}{\mathit{O}}_{\mathbf{2}}^{\mathbf{-}}\mathbf{\right]}\mathbf{/}\mathbf{\left[}\mathit{H}\mathit{N}{\mathit{O}}_{\mathbf{2}}\mathbf{\right]}$, of a buffer that has a pH of 2.95 ( ${\mathit{K}}_{\mathbf{a}}$of ${\mathbf{\text{HNO}}}_{\mathbf{\text{2}}}{\mathbf{\text{ = 7.1×10}}}^{\mathbf{-}\mathbf{4}}$ )?

What is the component concentration ratio,${\mathbf{\text{[Pr}}}^{\mathbf{\text{ - }}}\mathbf{\text{]/[HPr]}}$, of a buffer that has a pH of  5.44 ( ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$of $\mathbf{\text{HPr}}\mathbf{ }\mathbf{\text{ = }}\mathbf{ }{\mathbf{\text{1.3×10}}}^{\mathbf{-}\mathbf{5}}$ )?

A buffer consists of 0.50 M ${\mathbf{\text{NaH}}}_{\mathbf{\text{2}}}{\mathbf{\text{PO}}}_{\mathbf{\text{4}}}$ and 0.40 M ${\mathbf{\text{Na}}}_{\mathbf{\text{2}}}{\mathbf{\text{HPO}}}_{\mathbf{\text{4}}}$. Phosphoric acid is a triprotic acid (${\mathbf{\text{K}}}_{\mathbf{\text{a1}}}{\mathbf{\text{ = 7.2×10}}}^{\mathbf{-}\mathbf{3}}{\mathbf{\text{,K}}}_{\mathbf{\text{a2}}}{\mathbf{\text{ = 6.3×10}}}^{\mathbf{-}\mathbf{8}}$ and ${\mathbf{\text{K}}}_{\mathbf{\text{a3}}}{\mathbf{\text{ = 4.2×10}}}^{\mathbf{-}\mathbf{1}}$ ).

(a) Which ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$ value is most important to this buffer? 

(b) What is the buffer pH?

A buffer consists of 0.22 M ${\mathbf{\text{KHCO}}}_{\mathbf{\text{3}}}$ and 0.37 M ${\mathbf{\text{K}}}_{\mathbf{\text{2}}}{\mathbf{\text{CO}}}_{\mathbf{\text{3}}}$. Carbonic acid is a diprotic acid with ${\mathbf{\text{K}}}_{\mathbf{\text{a1}}}{\mathbf{\text{ = 4.5×10}}}^{\mathbf{-}\mathbf{7}}$ and ${\mathbf{\text{K}}}_{\mathbf{\text{a2}}}{\mathbf{\text{ = 4.7×10}}}^{\mathbf{-}\mathbf{11}}$. (a) Which  value is more important to this buffer? (b) What is the buffer pH?

Find the pH of a buffer that consists of 0.50 M methylamine (${\mathbf{\text{CH}}}_{\mathbf{\text{3}}}{\mathbf{\text{NH}}}_{\mathbf{\text{2}}}$) and 0.60 M ${\mathbf{\text{CH}}}_{\mathbf{\text{3}}}{\mathbf{\text{NH}}}_{\mathbf{\text{3}}}\mathbf{\text{Cl}}$ ( ${\mathbf{\text{pK}}}_{\mathbf{\text{b}}}$of ${\mathbf{\text{CH}}}_{\mathbf{\text{3}}}{\mathbf{\text{NH}}}_{\mathbf{\text{2}}}\mathbf{\text{ = 3.35}}$)?

Find the pH of a buffer that consists of 0.25 M ${\mathbf{\text{NH}}}_{\mathbf{\text{3}}}$ and 0.15 M ${\mathbf{\text{NH}}}_{\mathbf{\text{4}}}\mathbf{\text{Cl}}$ ( ${\mathbf{\text{pK}}}_{\mathbf{\text{b}}}$of ${\mathbf{\text{NH}}}_{\mathbf{\text{3}}}\mathbf{\text{ = 4.75}}$)?

Find the pH of a buffer that consists of 0.12 M boric acid (${\mathbf{\text{H}}}_{\mathbf{\text{3}}}{\mathbf{\text{BO}}}_{\mathbf{\text{3}}}$) and 0.82 M sodium borate(${\mathbf{\text{NaH}}}_{\mathbf{\text{2}}}{\mathbf{\text{BO}}}_{\mathbf{\text{3}}}$ ) ( ${\mathbf{\text{pK}}}_{\mathbf{\text{a}}}$of boric acid=9.24)?

Find the pH of a buffer that consists of 1.3 M sodium phenolate (${\mathbf{\text{C}}}_{\mathbf{\text{6}}}{\mathbf{\text{H}}}_{\mathbf{\text{5}}}\mathbf{\text{ONa}}$) and 1.2 M phenol (${\mathbf{\text{C}}}_{\mathbf{\text{6}}}{\mathbf{\text{H}}}_{\mathbf{\text{5}}}\mathbf{\text{OH}}$) ( ${\mathbf{\text{pK}}}_{\mathbf{\text{a}}}$of phenol=10.00)?

Find the pH of a buffer that consists of 0.95 M HBrO and 0.68 M KBrO ( ${\mathbf{\text{pK}}}_{\mathbf{\text{a}}}$of HBrO=8.64).

Find the pH of a buffer that consists of 0.45 M HCOOH and  0.63 M HCOONa ( ${\mathbf{\text{pK}}}_{\mathbf{\text{a}}}$of HCOOH=3.74)?

What are the $\mathbf{[}{\mathbf{\text{H}}}_{\mathbf{\text{3}}}{\mathbf{\text{O}}}^{\mathbf{\text{ + }}}\mathbf{\text{]}}$ and the pH of a buffer that consists of  0.20 M HF and  0.25 M KF ( ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$ of $\mathit{H}\mathit{F}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ )?

What are the  $\left[{\text{H}}_{\text{3}}{\text{O}}^{\text{ + }}\right]$ and the pH of a buffer that consists of  0.55 M ${\mathbf{\text{HNO}}}_{\mathbf{\text{2}}}$ and  0.75 M  ${\mathbf{\text{KNO}}}_{\mathbf{\text{2}}}$ ( ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$ of ${\mathbf{\text{HNO}}}_{\mathbf{\text{2}}}{\mathbf{\text{ = 7.1×10}}}^{\mathbf{-}\mathbf{4}}$ )?

What are the ${\mathbf{\text{[H}}}_{\mathbf{\text{3}}}{\mathbf{\text{O}}}^{\mathbf{\text{ + }}}\mathbf{\text{]}}$ and the pH of a benzoic acid–benzoate buffer that consists of 0.33 M  ${\mathbf{\text{C}}}_{\mathbf{\text{6}}}{\mathbf{\text{H}}}_{\mathbf{\text{5}}}\mathbf{\text{COOH}}$ and 0.28 M  ${\mathbf{\text{C}}}_{\mathbf{\text{6}}}{\mathbf{\text{H}}}_{\mathbf{\text{5}}}\mathbf{\text{COONa}}$ (  ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$of benzoic acid=$\mathbf{6}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$) ?

What are the $\mathbf{\left[}{\mathbf{\text{H}}}_{\mathbf{\text{3}}}{\mathbf{\text{O}}}^{\mathbf{\text{ + }}}\mathbf{\right]}$ and the pH of a propanoic acid– propanoate buffer that consists of  0.35 M   ${\mathbf{\text{CH}}}_{\mathbf{\text{3}}}{\mathbf{\text{CH}}}_{\mathbf{\text{2}}}\mathbf{\text{COONa }}$and 0.15 M ${\mathbf{\text{CH}}}_{\mathbf{\text{3}}}{\mathbf{\text{CH}}}_{\mathbf{\text{2}}}\mathbf{\text{COOH}}$ ( ${\mathbf{\text{K}}}_{\mathbf{\text{a}}}$ of propanoic acid=$\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$)?

A chemist needs a pH 3.5 buffer. Should she use NaOH with formic acid (${\mathbf{\text{K}}}_{\mathbf{\text{a}}}{\mathbf{\text{ = 1.8×10}}}^{\mathbf{-}\mathbf{4}}$) or with acetic acid (${\mathbf{\text{K}}}_{\mathbf{\text{a}}}{\mathbf{\text{ = 1.8×10}}}^{\mathbf{-}\mathbf{5}}$ )? Why? What is the disadvantage of choosing the other acid? What is the role of the NaOH?

What is the relationship between the buffer range and the buffer-component concentration ratio?

Write the ion-product expressions for  silver carbonate;

 barium fluoride;  copper (II) sulphide.

Write the ion-product expressions for

$\mathbf{\left(}\mathbf{\text{a}}\mathbf{\right)}\mathbf{\text{ iron}}\mathbf{\left(}\mathbf{\text{III}}\mathbf{\right)}\mathbf{\text{ hydroxide; }}\mathbf{\left(}\mathbf{\text{b}}\mathbf{\right)}\mathbf{\text{ barium phosphate; }}\mathbf{\left(}\mathbf{\text{c}}\mathbf{\right)}\mathbf{\text{ tin}}\mathbf{\left(}\mathbf{\text{II}}\mathbf{\right)}\mathbf{\text{ sulfide.}}$

Write the ion-product expressions for

$\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{ }\mathbf{calcium}\mathbf{ }\mathbf{chromate}\mathbf{;}\mathbf{ }\mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{ }\mathbf{silver}\mathbf{ }\mathbf{cyanide}\mathbf{;}\mathbf{ }\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{ }\mathbf{nickel}\mathbf{\left(}\mathbf{II}\mathbf{\right)}\mathbf{sulfide}$

Calculate the  during the titration of $\mathbf{30}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{mL}}$ of $\mathbf{0}\mathbf{.}\mathbf{1000}\mathbf{ }\mathbf{\text{M }}\mathbf{ }\mathbf{ }\mathbf{\text{KOH}}$ with $\mathbf{0}\mathbf{.}\mathbf{1000}\mathbf{\text{M}}\mathbf{ }\mathbf{ }\mathbf{\text{HBr}}$ solution after the following additions of acid:

(a) 0 mL (b) 15.00 mL  (c) 29.00 mL (d) 29.90 mL 

(e) 30.00 mL (f) 30.10 mL  (g) 40.00 mL

Use Figure 19.5 to find an indicator for these titrations:

(a) $\mathbf{0}\mathbf{.25}{\mathbf{\text{M\hspace{0.17em}\hspace{0.17em}C}}}_{\mathbf{6}}{\mathbf{\text{H}}}_{\mathbf{5}}\mathbf{\text{COOH}}$  (Appendix C) with  $\mathbf{0.25}\mathbf{\text{M\hspace{0.17em}\hspace{0.17em}KOH}}$.

(b) $\mathbf{0.50}{\mathbf{\text{M\hspace{0.17em}\hspace{0.17em}NHNH}}}_{\mathbf{4}}\mathbf{\text{Cl}}$  (Appendix  $\mathbf{C}$  ) with $\mathbf{0.50}\mathbf{\text{M\hspace{0.17em}\hspace{0.17em}NaOH}}$ .

Calculate the $\mathit{p}\mathit{H}$ during the titration of $\mathbf{40}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{mL}}$ of $\mathbf{\text{HCl}}$with  solution after the following additions of base:

(a) $\mathbf{0}\mathbf{ }\mathbf{\text{mL}}$.

(b) $\mathbf{25}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{mL}}$.

(c) $\mathbf{39}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{mL}}$.

(d) $\mathbf{39}\mathbf{.}\mathbf{90}\mathbf{ }\mathbf{\text{mL}}$.

(e) $\mathbf{40}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{mL}}$.

(f) $\mathbf{40}\mathbf{.}\mathbf{10}\mathbf{ }\mathbf{\text{mL}}$.

(g) $\mathbf{50}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{\text{mL}}$.

Find the $\mathbf{\text{pH}}$ during the titration of $\mathbf{\text{20.00 mL}}$ of $\mathbf{\text{0.1000M}}$butanoic acid, ${\mathbf{\text{CH}}}_{\mathbf{\text{3}}}{\mathbf{\text{CH}}}_{\mathbf{\text{2}}}{\mathbf{\text{CH}}}_{\mathbf{\text{2}}}\mathbf{\text{COOH}}\mathbf{\left(}{\mathbf{\text{K}}}_{\mathbf{\text{a}}}{\mathbf{\text{=1.54×10}}}^{\mathbf{\text{-5}}}\mathbf{\right)}$ with $\mathbf{\text{1000MNaOH}}$  solution after the following additions of titrant:

(a)  $\mathbf{\text{0 mL}}$

(b)  $\mathbf{\text{10.00 mL}}$

(c)  $\mathbf{\text{15.00 mL}}$

(d)  $\mathbf{\text{19.00 mL}}$

(c)  $\mathbf{\text{19.95 mL}}$

(f)  $\mathbf{\text{20.00 mL}}$

(g)  $\mathbf{\text{20.05 mL}}$

(h)  $\mathbf{\text{25.00 mL}}$

Find the pH during the titration of $\mathbf{\text{20.00 mL}}$ of $\mathbf{\text{0.1000M}}$triethylamine,  ${\mathbf{\left(}{\mathbf{\text{CH}}}_{\mathbf{\text{3}}}{\mathbf{\text{CH}}}_{\mathbf{\text{2}}}\mathbf{\right)}}_{\mathbf{\text{3}}}\mathbf{\text{ N}}\mathbf{\left(}{\mathbf{\text{K}}}_{\mathbf{\text{b}}}{\mathbf{\text{=5.2×10}}}^{\mathbf{\text{-4}}}\mathbf{\right)}$with  $\mathbf{\text{1000 MNaOH}}$ solution after the following additions of titrant:

(a)  $\mathbf{\text{0 mL}}$

(b)  $\mathbf{\text{10.00 mL}}$

(c)  $\mathbf{\text{15.00 mL}}$

(d)  $\mathbf{\text{19.00 mL}}$

(c)  $\mathbf{\text{19.95 mL}}$

(f)  $\mathbf{\text{20.00 mL}}$

(g)  $\mathbf{\text{20.05 mL}}$

(h)  $\mathbf{\text{25.00 mL}}$

Find the pH of the equivalence point(s) and the volume (mL) of $\mathbf{0}\mathbf{.0372}\mathbf{\text{\hspace{0.17em}M\hspace{0.17em}\hspace{0.17em}NaOH}}$ needed to reach it in titrations of

(a) $\mathbf{42.2}\mathbf{ }\mathbf{\text{mL}}$  of  $\mathbf{0.0520}{\mathbf{\text{M\hspace{0.17em}\hspace{0.17em}CH}}}_{\mathbf{3}}\mathbf{\text{COOH}}$.

(b)  $\mathbf{28.9}\mathbf{ }\mathbf{\text{mL}}$of  $\mathbf{0.0850}{\mathbf{\text{\hspace{0.17em}M\hspace{0.17em}\hspace{0.17em}H}}}_{\mathbf{2}}{\mathbf{\text{SO}}}_{\mathbf{3}}$(two equivalence points)