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Chapter 14: Problem 26

Derive an expression for the relationship between \(\mathrm{p} K_{\mathrm{a}}\) and \(\mathrm{p} K_{\mathrm{b}}\) for a conjugate acid-base pair. \((\mathrm{p} K=-\log K .)\)

Short Answer

Expert verified
The relationship between the acidity constant \(pK_a\) and the basicity constant \(pK_b\) for a conjugate acid-base pair is: \(pK_a + pK_b = pK_w\).

Step by step solution

01

Write expression for \(K_a\) and \(K_b\)

For the acid dissociation reaction: \[HA + H_2O \rightleftharpoons H_3O^+ + A^-\] The acidity constant, \(K_a\), is defined as: \[K_a = \frac{[H_3O^+][A^-]}{[HA]}\] Similarly, for the base dissociation reaction: \[A^- + H_2O \rightleftharpoons HA + OH^-\] The basicity constant, \(K_b\), is defined as: \[K_b = \frac{[HA][OH^-]}{[A^-]}\] Step 2: Write the expression for \(pK_a\) and \(pK_b\)
02

Write expression for \(pK_a\) and \(pK_b\)

The \(pK_a\) and \(pK_b\) are the negative logarithm of \(K_a\) and \(K_b\) respectively. We have: \[pK_a = -\log{K_a}\] \[pK_b = -\log{K_b}\] Step 3: Use the relationship between \(K_a\), \(K_b\), and \(K_w\)
03

Use the relationship between \(K_a\), \(K_b\), and \(K_w\)

The relationship between \(K_a\), \(K_b\), and the auto-ionization constant of water (\(K_w = [H_3O^+][OH^-]\)) is given by: \[K_a \cdot K_b = K_w\] Step 4: Write the relationship between \(pK_a\) and \(pK_b\)
04

Write the relationship between \(pK_a\) and \(pK_b\)

Taking the logarithm of the relation \(K_a \cdot K_b = K_w\), we get: \[\log{K_a} + \log{K_b} = \log{K_w}\] Now, multiply the whole equation by -1: \[-\log{K_a} - \log{K_b} = -\log{K_w}\] Finally, substituting the expressions for \(pK_a\) and \(pK_b\): \[pK_a + pK_b = pK_w\] Therefore, the relationship between the acidity constant \(pK_a\) and the basicity constant \(pK_b\) for a conjugate acid-base pair is: \[pK_a + pK_b = pK_w\]

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Most popular questions from this chapter

A solution is prepared by adding \(50.0 \mathrm{~mL}\) concentrated hydrochloric acid and \(20.0 \mathrm{~mL}\) concentrated nitric acid to \(300 \mathrm{~mL}\) water. More water is added until the final volume is \(1.00 \mathrm{~L}\). Calculate \(\left[\mathrm{H}^{+}\right],\left[\mathrm{OH}^{-}\right]\), and the \(\mathrm{pH}\) for this solution. [Hint: Concentrated HCl is \(38 \%\) HCl (by mass) and has a density of \(1.19\) \(\mathrm{g} / \mathrm{mL} ;\) concentrated \(\mathrm{HNO}_{3}\) is \(70 . \% \mathrm{HNO}_{3}\) (by mass) and has a density of \(1.42 \mathrm{~g} / \mathrm{mL} .\) ]

Calculate \(\left[\mathrm{OH}^{-}\right], \mathrm{pOH}\), and \(\mathrm{pH}\) for each of the following. a. \(0.00040 \mathrm{M} \mathrm{Ca}(\mathrm{OH})_{2}\) b. a solution containing \(25 \mathrm{~g}\) KOH per liter c. a solution containing \(150.0 \mathrm{~g} \mathrm{NaOH}\) per liter

Hemoglobin (abbreviated \(\mathrm{Hb}\) ) is a protein that is responsible for the transport of oxygen in the blood of mammals. Each hemoglobin molecule contains four iron atoms that are the binding sites for \(\mathrm{O}_{2}\) molecules. The oxygen binding is pH-dependent. The relevant equilibrium reaction is $$\mathrm{HbH}_{4}^{4+}(a q)+4 \mathrm{O}_{2}(g) \rightleftharpoons \mathrm{Hb}\left(\mathrm{O}_{2}\right)_{4}(a q)+4 \mathrm{H}^{+}(a q)$$ Use Le Châtelier's principle to answer the following. a. What form of hemoglobin, \(\mathrm{HbH}_{4}{ }^{4+}\) or \(\mathrm{Hb}\left(\mathrm{O}_{2}\right)_{4}\), is favored in the lungs? What form is favored in the cells? b. When a person hyperventilates, the concentration of \(\mathrm{CO}_{2}\) in the blood is decreased. How does this affect the oxygenbinding equilibrium? How does breathing into a paper bag help to counteract this effect? (See Exercise 148.) c. When a person has suffered a cardiac arrest, injection of a sodium bicarbonate solution is given. Why is this necessary?

Papaverine hydrochloride (abbreviated papH \(^{+} \mathrm{Cl}^{-} ;\) molar mass \(=\) \(378.85 \mathrm{~g} / \mathrm{mol}\) ) is a drug that belongs to a group of medicines called vasodilators, which cause blood vessels to expand, thereby increasing blood flow. This drug is the conjugate acid of the weak base papaverine (abbreviated pap; \(K_{\mathrm{b}}=8.33 \times 10^{-9}\) at \(35.0^{\circ} \mathrm{C}\) ). Calculate the \(\mathrm{pH}\) of a \(30.0 \mathrm{mg} / \mathrm{mL}\) aqueous dose of papH \(^{+} \mathrm{Cl}^{-}\) prepared at \(35.0^{\circ} \mathrm{C} . K_{\mathrm{w}}\) at \(35.0^{\circ} \mathrm{C}\) is \(2.1 \times 10^{-14}\).

Will the following oxides give acidic, basic, or neutral solutions when dissolved in water? Write reactions to justify your answers. a. \(\mathrm{Li}_{2} \mathrm{O}\) b. \(\mathrm{CO}_{2}\) c. \(\mathrm{SrO}\)
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