An indicator BPB (Brophenol Blue) is used and dissolved in solutions of various known $\mathrm{pH}$. Then, the absorbance of each solution at 590 nm was measured and the question asks for molar absorption coefficient of the $\ce{HIn}$ form and $\ce{In-}$ form at the 590 nm wavelength.

Derive the relation of absorptivity coefficient $\varepsilon(\ce{HIn})$ and $\varepsilon(\ce{In-})$ to the total concentration of indicator and also the $K_\mathrm{a}$ for the indicator.

Hint: plot the value of $A$ vs $\mathrm{pH}$.

I don't know how to interpret the resulting graph. I tried to derive a relation, but it is not linear with respect to $\mathrm{pH}$. Here I'm determining the relations of absorbance to $\mathrm{pH}$, trying to interpret the graph of $A$ vs $\mathrm{pH}$, but the relations are not linear.

$$[\ce{HIn}] + [\ce{In-}] = c(\text{BPB})$$

$$K_\mathrm{a} = \frac{[\ce{H+}][\ce{In-}]}{[\ce{HIn}]} \quad\to\quad [\ce{In-}] = K_\mathrm{a}\frac{[\ce{HIn}]}{[\ce{H+}]}$$

$$ \begin{align} A &= \varepsilon(\ce{HIn})\cdot c \\ &= b\cdot\left(\varepsilon(\ce{HIn})\cdot [\ce{HIn}] + \varepsilon(\ce{In-})\cdot [\ce{In-}]\right) \\ &= b\cdot\left(\varepsilon(\ce{HIn})\cdot[\ce{HIn}] + \varepsilon(\ce{In-})\cdot K_\mathrm{a}\frac{[\ce{HIn}]}{[\ce{H+}]}\right) \\ &= b\cdot[\ce{HIn}]\left(\varepsilon(\ce{HIn}) + \varepsilon(\ce{In-})\cdot\frac{K_\mathrm{a}}{[\ce{H+}]}\right) \\ &= \frac{b\cdot c(\text{BPB})}{1 + \frac{K_\mathrm{a}}{[\ce{H+}]}}\left(\varepsilon(\ce{HIn}) + \varepsilon(\ce{In-})\cdot\frac{K_\mathrm{a}}{[\ce{H+}]}\right) \end{align} $$


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