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Chapter 17: Problem 75

Derive an equation similar to the HendersonHasselbalch equation relating the pOH of a buffer to the \(\mathrm{p} K_{b}\) of its base component.

Short Answer

Expert verified
The derived equation relating pOH to the \(pK_b\) of a weak base in a buffer solution is: \[ pOH = \log{\left( \frac{[B]}{[HB^+]} \right)} + pK_b \] This equation can be used to determine the pOH of a buffer system given the concentration of the base and its conjugate acid, and the \(pK_b\) of the base component.

Step by step solution

01

Write down the generic reaction with a weak base and water.

For a buffer solution composed of a weak base, B, and its conjugate acid, HB+, the following equilibrium reaction takes place when the base is mixed with water: \( B + H_2O \leftrightarrows HB^+ + OH^- \) Here, B is the weak base, \(H_2O\) is water, \(HB^+\) is the conjugate acid of B, and \(OH^-\) is hydroxide ion.
02

Write down the expression for the base ionization constant.

The base ionization constant, \(K_b\), can be defined as follows. \[ K_b = \frac{[HB^+][OH^-]}{[B]} \] This equation represents the equilibrium between the base (B), the conjugate acid (HB+), and the hydroxide (OH-) ions.
03

Define pOH and pKb.

The pOH is a measure of the acidity of a solution and is calculated as the negative logarithm of the hydroxide ion concentration. \[ pOH = -\log{[OH^-]} \] Similarly, \(pK_b\) is a measure of the basicity of a substance and is calculated as the negative logarithm of the base ionization constant. \[ pK_b = -\log{K_b} \]
04

Rewrite the expression for the base ionization constant in terms of pOH and pKb.

We can rewrite the expression for the base ionization constant from Step 2 as follows. \[ 10^{pOH} = \frac{[HB^+]}{[B]} \cdot \frac{1}{K_b} \] Take the negative logarithm of both sides of the equation. \[ -\log{10^{pOH}} = -\log{\left( \frac{[HB^+]}{[B]} \cdot \frac{1}{K_b} \right)} \] Using the properties of logarithms, we can simplify the equation to: \[ pOH = \log{\left( \frac{[B]}{[HB^+]} \right)} + \log{K_b} \] But, since \(pK_b = -\log{K_b}\), we can substitute it into the equation.
05

Derive the final equation relating pOH and pKb.

Replacing the appropriate terms in the equation, we get the final equation relating pOH and \(pK_b\): \[ pOH = \log{\left( \frac{[B]}{[HB^+]} \right)} + pK_b \] This equation can be used to determine the pOH of a buffer system given the concentration of the base and its conjugate acid, and the \(pK_b\) of the base component.

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Most popular questions from this chapter

Assume that \(30.0 \mathrm{~mL}\) of a \(0.10 \mathrm{M}\) solution of a weak base \(B\) that accepts one proton is titrated with a \(0.10 \mathrm{M}\) solution of the monoprotic strong acid HX. (a) How many moles of \(\mathrm{HX}\) have been added at the equivalence point? (b) What is the predominant form of \(B\) at the equivalence point? (c) What factor determines the \(\mathrm{pH}\) at the equivalence point? (d) Which indicator, phenolphthalein or methyl red, is likely to be the better choice for this titration?

A buffer contains a weak acid, \(\mathrm{HX}\), and its conjugate base. The weak acid has a \(\mathrm{pK}_{a}\) of \(4.5\), and the buffer solution has a \(\mathrm{pH}\) of \(4.3\). Without doing a calculation, predict whether \([\mathrm{HX}]=\left[\mathrm{X}^{-}\right],[\mathrm{HX}]>\left[\mathrm{X}^{-}\right]\), or \([\mathrm{HX}]<\left[\mathrm{X}^{-}\right]\) Explain. [Section 17.2]

A sample of \(0.2140 \mathrm{~g}\) of an unlenown monoprotic acid was dissolved in \(25.0 \mathrm{~mL}\) of water and titrated with \(0.0950 \mathrm{M} \mathrm{NaOH}\). The acid required \(27.4 \mathrm{~mL}\) of base to reach the equivalence point. (a) What is the molar mass of the acid? (b) After \(15.0 \mathrm{~mL}\) of base had been added in the titration, the \(\mathrm{pH}\) was found to be \(6.50 .\) What is the \(K_{a}\) for the unknown acid?

How many milliliters of \(0.0850 \mathrm{M} \mathrm{NaOH}\) are required to titrate each of the following solutions to the equivalence point: (a) \(40.0 \mathrm{~mL}\) of \(0.0900 \mathrm{M} \mathrm{HNO}_{3}\), (b) \(35.0 \mathrm{~mL}\) of \(0.0850 \mathrm{MCH}_{3} \mathrm{COOH}\), (c) \(50.0 \mathrm{~mL}\) of a solution that contains \(1.85 \mathrm{~g}\) of \(\mathrm{HCl}\) per liter?

The acid-base indicator bromcresol green is a weak acid. The yellow acid and blue base forms of the indicator are present in equal concentrations in a solution when the \(\mathrm{pH}\) is \(4.68 .\) What is the \(\mathrm{pK}_{a}\) for bromcresol green?
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