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Chapter 45: Problem 4

Provide an expression relating \(K_{\mathrm{w}}\) to \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\) of a conjugate acid-base pair.

Short Answer

Expert verified
The relationship between the ion product of water (\(K_{w}\)), the acid dissociation constant (\(K_{a}\)), and the base dissociation constant (\(K_{b}\)) for a conjugate acid-base pair is given by: \(K_{w} = K_{a} * K_{b}\)

Step by step solution

01

Understanding Conjugate Acid-Base Pairs

In an acid/base reaction, when the acid donates a proton, the remaining portion becomes its conjugate base. Similarly, when a base accepts a proton, it becomes a conjugate acid. For a given acid and its conjugate base or a given base and its conjugate acid, \(K_{a}\) expresses the degree to which the acid dissociates to form its conjugate base and a free proton (H+), whereas \(K_{b}\) expresses the degree to which the base dissociates to form its conjugate acid and a free hydroxide ion (OH−).
02

Understanding Ion Product of Water (\(K_{w}\))

The product of the concentrations of the hydrogen ions (H+) and hydroxide ions (OH−) in water at a given temperature is known as the ion product of water (\(K_{w}\)). At 25°C, \(K_{w}\) is 1.0*10^-14.
03

Relating \(K_{w}\), \(K_{a}\) and \(K_{b}\)

Given that acid and base dissociation processes produce hydrogen and hydroxide ions, we can relate \(K_{w}\), \(K_{a}\), and \(K_{b}\) as follows: \(K_{w} = K_{a} * K_{b}\). This equation states that for a given acidic solution at 25°C, the product of \(K_{a}\) for the acid and \(K_{b}\) for its conjugate base equals the ion product of water (\(10^{-14}\)).

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Most popular questions from this chapter

The nitrate ion is the conjugate base of nitric acid, \(\mathrm{HNO}_{3}\). Explain why \(\mathrm{NO}_{3}^{-}\) does not produce a basic solution when dissolved in water.

Consider two acids, HA and HX, with HA being a stronger acid than HX. a) Which acid has a larger value of \(K_{\mathrm{a}}\) ? b) Which conjugate base, \(\mathrm{A}^{-}\) or \(\mathrm{X}^{-}\), has a larger value of \(K_{\mathrm{b}}\) ? c) Provide a qualitative description of the relationship between the relative strength of an acid and the relative strength of its conjugate base. d) Assume that \(K_{\mathrm{a}}\) of \(\mathrm{HA}\) is \(1.0 \times 10^{-5}\) and \(K_{\mathrm{a}}\) of \(\mathrm{HX}\) is \(3.5 \times 10^{-7}\). Calculate the \(K_{\mathrm{b}}\) for \(\mathrm{A}^{-}\) and for \(\mathrm{X}^{-}\) and confirm that your answers to parts \(\mathrm{b}\) and \(\mathrm{c}\) are correct.

To 2 significant figures, predict the \(\mathrm{pH}\) of a \(1.00\) molar solution of \(\mathrm{NaBr}\). Explain your reasoning carefully.

\(\mathrm{HNO}_{3}\) is considered to be the weakest of the strong acids. Explain why the conjugate bases of all strong acids do not produce basic solutions when dissolved in water.

Indicate which of the following are potential acids, potential bases, or both (potential acid and potential base). a) \(\mathrm{Al}^{3+}\) b) \(\mathrm{CH}_{3} \mathrm{NH}_{3}^{+}\) c) \(\mathrm{HPO}_{4}{ }^{2-}\) d) \(\quad \mathrm{F}^{-}\) e) \(\mathrm{NH}_{4}^{+}\) f) \(\mathrm{H}_{2} \mathrm{O}\)
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