How can I choose an indicator for determining the concentration of a base using its pKb values?

I have a base $\ce{M(OH)_{3}}$ having 3 $\mathrm{p}K_{\mathrm{b}}$ values as, $$\mathrm{p}K_{\mathrm{b}_{1}} = 0.5\\ \mathrm{p}K_{\mathrm{b}_{2}} = 3.7\\ \mathrm{p}K_{\mathrm{b}_{3}} = 7.7\\$$

If a strong acid is given as the titrant and phenolphthalein and methyl orange are given as possible indicators, how can I choose an indicator using above $\mathrm{p}K_{\mathrm{b}}$ values ?

Since: \begin{align} K_\mathrm{w} &=K_\mathrm{a} \cdot K_\mathrm{b}\\ K_\mathrm{a} &=\frac{K_\mathrm{w}}{K_\mathrm{b}}\\ \log{K_\mathrm{a}} &=\log K_\mathrm{w}-\log K_\mathrm{b}\\ K_\mathrm{w} &=1 \times 10^{-14}\\ \log{K_\mathrm{a}} &= -14-\log K_\mathrm{b}\\ -\log{K_\mathrm{a}} &=14+\log K_\mathrm{b}\\ -\log{K_\mathrm{a}} &=\mathrm{p}K_\mathrm{a}\\ -\log{K_\mathrm{b}} &=\mathrm{p}K_\mathrm{b}\\ \mathrm{p}K_\mathrm{a} &=14-\mathrm{p}K_\mathrm{b} \end{align}

Using the above formula, the three $\mathrm{p}K_\mathrm{a}$ for the conjugate acid of the conjugate base are: $13.5~(\mathrm{p}K_\mathrm{a_1})$, $10.3~(\mathrm{p}K_\mathrm{a_2})$, and $6.3~(\mathrm{p}K_\mathrm{a_3})$, where $\mathrm{p}K_\mathrm{a_1}=14-\mathrm{p}K_\mathrm{b_1}$, etc.

An important equation to note is that the $\mathrm{pH}$ at the equivalence point is given by the equation: $$\mathrm{pH_\mathrm{eq}}=\frac{\mathrm{p}K_\mathrm{a1}+\mathrm{p}K_\mathrm{a2}}{2}$$

This means that two of the equivalence points exist at $$\mathrm{pH_\mathrm{eq_1}}=\frac{13.5+10.3}{2}=11.9$$ and $$\mathrm{pH_\mathrm{eq_2}}=\frac{10.3+6.3}{2}=8.3$$

The $\mathrm{pH}$ range for methyl orange is $3.0~-~4.4$ and the $\mathrm{pH}$ range for phenolphthalein is $8.2~-~10.0$. Phenolphthalein would be useful for detecting the second equivalence point $(\mathrm{pH_\mathrm{eq_2}})$, since this shift occurs within phenolphthalein's range.

The final equivalence point cannot be calculated from the information given, but given that methyl orange's color transition occurs in an acidic medium, and also the relatively basic $\mathrm{p}K_\mathrm{a_1}$, methyl orange can be eliminated as a possible indicator.

• @LDC3 the only $\mathrm{p}K_\mathrm{a}$ close to being within the useful range of one of the indicators is $\mathrm{p}K_\mathrm{a_2}$ to phenolphthalein's range, but it is still not actually within the range. Unless I am misunderstanding how an indicators work (which I guess could be possible) then I don't see how either indicator is useful. – ringo May 12 '15 at 6:06
• @ringo You want the indicator to change color after all the base has been converted (distant from a $\ce {pK_{b}}$), not when it is being converted (near a $\ce {pK_{b}}$). So the indicators you would choose would not be near any of the $\ce {pK_{b}}$ of the base. – LDC3 May 12 '15 at 6:10
• I'll add that the reason to convert to pKa values is that tables for acid-base indicators are usually give in terms of pH not pOH. en.wikipedia.org/wiki/PH_indicator – MaxW Apr 23 '16 at 2:53

The first step is to determine the $\mathrm{p}K_{\mathrm{a}}$ values for the indicators and convert them to $\mathrm{p}K_{\mathrm{b}}$ values. Then determine how you would titrate the solution using the indicator(s).

• Do you mean that the $pK_{b}$ values of $M(OH)_{3}$ have no part to play in choosing the indicator ? – chemkatku May 12 '15 at 5:04
• Yes it does, but you need to know when the indicators change colors. – LDC3 May 12 '15 at 5:05
• OK. Say I have methyl orange with pH range of 3.0 - 4.4. How do I proceed ? – chemkatku May 12 '15 at 5:42