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Chapter 17: Problem 83

Derive an equation similar to the Henderson-Hasselbalch equation relating the pOH of a buffer to the \(\mathrm{pK}_{b}\) of its base component.

Short Answer

Expert verified
The derived equation relating pOH to the \(\mathrm{pK}_{b}\) of a buffer's base component is: \[ pOH = pK_{b} + \log{\frac{[B]}{[BH^+]}} \]

Step by step solution

01

Write the general equation for a weak base reacting with water

For a weak base B, the general equation for its reaction with water can be written as: B + H2O <=> BH+ + OH-
02

Write the expression for the base ionization constant (Kb)

The base ionization constant can be described as the equilibrium constant for the ionization of B: \[ K_{b} = \frac{[BH^+][OH^-]}{[B]}\]
03

Find the relationship between OH- and the weak base

The pOH is related to the concentration of OH- through the equation: \[ pOH = -\log [OH^-]\] We can rewrite this expression to find the concentration of OH- in terms of pOH: \[ [OH^-] = 10^{-pOH}\]
04

Write the expression for the pKb

The relationship between the base ionization constant (Kb) and pKb is given by: \[ pK_{b} = -\log K_{b}\] Rewrite the expression to find Kb in terms of pKb: \[ K_{b} = 10^{-pK_{b}}\]
05

Combine the previous expressions to derive the required equation

Substitute the expressions for [OH-] and Kb from Steps 3 and 4 into the Kb expression derived in step 2: \begin{align*} 10^{-pK_{b}} &= \frac{[BH^+][10^{-pOH}]}{[B]} \end{align*} Now, we solve for pOH: \begin{align*}pOH &= pK_{b} + \log{\frac{[B]}{[BH^+]}}\end{align*} This is the required equation, which is similar to the Henderson-Hasselbalch equation, but relates the pOH of a buffer to the \(\mathrm{pK}_{b}\) of its base component.

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Most popular questions from this chapter

Fluoridation of drinking water is employed in many places to aid in the prevention of tooth decay. Typically. the \(\mathrm{F}^{-}\) ion concentration is adjusted to about \(1 \mathrm{ppm}\). Some water supplies are also "hard"; that is, they contain certain cations such as \(\mathrm{Ca}^{2+}\) that interfere with the action of soap. Consider a case where the concentration of \(\mathrm{Ca}^{2+}\) is \(8 \mathrm{ppm}\). Could a precipitate of \(\mathrm{CaF}_{2}\) form under these conditions? (Make any necessary approximations.)

(a) Calculate the pH of a buffer that is \(0.125 \mathrm{M}\) in \(\mathrm{NaHCO}_{3}\) and \(0.095 \mathrm{M}\) in $\mathrm{Na}_{2} \mathrm{CO}_{3} .\( (b) Calculate the pH of a solution formed by mixing \)25 \mathrm{~mL}\( of \)0.25 \mathrm{M} \mathrm{NaHCO}_{3}\( with \)75 \mathrm{~mL}$ of \(0.15 \mathrm{M} \mathrm{Na}_{2} \mathrm{CO}_{3}\)

A solution of \(\mathrm{Na}_{2} \mathrm{SO}_{4}\) is added dropwise to a solution that is \(0.010 \mathrm{M}\) in \(\mathrm{Ba}^{2+}(a q)\) and $0.010 \mathrm{M}\( in \)\mathrm{Sr}^{2+}(a q) .(\mathbf{a}) \mathrm{What}$ concentration of \(\mathrm{SO}_{4}^{2-}\) is necessary to begin precipitation? (Neglect volume changes. $\mathrm{BaSO}_{4}: K_{s p}=1.1 \times 10^{-10} ; \mathrm{SrSO}_{4}:\( \)K_{s p}=3.2 \times 10^{-7} .$ ) (b) Which cation precipitates first? (c) What is the concentration of $\mathrm{SO}_{4}^{2-}(a q)$ when the second cation begins to precipitate?

A 10.0-mL sample of \(0.250 \mathrm{M}\) acetic acid $\left(\mathrm{CH}_{3} \mathrm{COOH}\right)\( is titrated with \)0.100 \mathrm{M}$ KOH solution. Calculate the pH after the following volumes of base have been added: (a) \(0 \mathrm{~mL},\) (b) \(12.5 \mathrm{~mL}\) (c) \(24.5 \mathrm{~mL}\) (d) \(25.0 \mathrm{~mL}\) (e) \(25.5 \mathrm{~mL}\) (f) \(30.0 \mathrm{~mL}\).

A \(1.0 \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\) solution is slowly added to \(10.0 \mathrm{~mL}\) of a solution that is \(0.20 M\) in \(\mathrm{Ca}^{2+}\) and \(0.30 \mathrm{M}\) in \(\mathrm{Ag}^{+} .\) (a) Which compound will precipitate first: \(\operatorname{CaSO}_{4}\left(K_{s p}=2.4 \times 10^{-5}\right)\) or $\mathrm{Ag}_{2} \mathrm{SO}_{4}\left(K_{s p}=1.5 \times 10^{-5}\right) ?(\mathbf{b})\( How much \)\mathrm{Na}_{2} \mathrm{SO}_{4}$ solution must be added to initiate the precipitation?
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