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Chapter 17: Problem 85

Consider a solution containing two weak monoprotic acids with dissociation constants \(K_{\mathrm{HA}}\) and \(K_{\mathrm{HB}}\). Find the charge balance equation for this system, and use it to derive an expression that gives the concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) as a function of the concentrations of \(\mathrm{HA}\) and HB and the various constants.

Short Answer

Expert verified
The concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) in the solution, which is equivalent to the concentration of \([H^+]\), can be expressed as \([H^+] = \sqrt{K_{\mathrm{HA}}[HA] + K_{\mathrm{HB}}[HB]}\). This is derived by setting up the dissociation reactions, writing the charge balance equation, expressing the species in terms of equilibrium constants and substituting back into the charge balance equation.

Step by step solution

01

Defining the Acid Dissociation Reactions

The acid dissociation of the weak monoprotic acids can be represented by the following reactions: \(HA \rightleftharpoons H^{+} + A^-\) and \(HB \rightleftharpoons H^{+} + B^- \). Both of these reactions have their respective equilibrium constants, \(K_{\mathrm{HA}}\) and \(K_{\mathrm{HB}}\), defined by concentration of products over concentration of reactants.
02

Formulate the Charge Balance Equation

In any given aqueous solution at equilibrium, the total positive charge must equal the total negative charge. Thus, the charge balance equation for this system can be written as: \([H^+] = [A^-] + [B^-]\) which indicates the amount of \(H^+\) ions equals the amount of \(A^-\) and \(B^-\) ions.
03

Define Concentrations Using Equilibrium Constants

We can use the equilibrium constants to express each species in terms of the others. The equilibrium expressions for \(HA\) and \(HB\) can be respectively written as \(K_{\mathrm{HA}} = \frac{[H^+][A^-]}{[HA]}\) and \(K_{\mathrm{HB}} = \frac{[H^+][B^-]}{[HB]}\). From these, we can derive expressions for the concentrations of \(A^-\) and \(B^-\). Thus, \([A^-] = \frac{K_{\mathrm{HA}}[HA]}{[H^+]}\) and \([B^-] = \frac{K_{\mathrm{HB}}[HB]}{[H^+]}\).
04

Substitute the Concentrations in the Charge Balance Equation

Substitute the derived expressions for \([A^-]\) and \([B^-]\) in the charge balance equation to obtain an expression for \([H^+]\) as a function of the concentrations of \(HA\) and \(HB\) and the various constants: \([H^+] = \frac{K_{\mathrm{HA}}[HA]}{[H^+]}+ \frac{K_{\mathrm{HB}}[HB]}{[H^+]}\). This equation can be rearranged to give \([H^+] = \sqrt{K_{\mathrm{HA}}[HA] + K_{\mathrm{HB}}[HB]}\).

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Most popular questions from this chapter

A very common buffer agent used in the study of biochemical processes is the weak base TRIS, \(\left(\mathrm{HOCH}_{2}\right)_{3} \mathrm{CNH}_{2},\) which has a \(\mathrm{pK}_{\mathrm{b}}\) of 5.91 at \(25^{\circ} \mathrm{C} . \mathrm{A}\) student is given a sample of the hydrochloride of TRIS together with standard solutions of \(10 \mathrm{M}\) NaOH and HCl. (a) Using TRIS, how might the student prepare 1 L of a buffer of \(\mathrm{pH}=7.79 ?\) (b) In one experiment, 30 mmol of protons are released into \(500 \mathrm{mL}\) of the buffer prepared in part (a). Is the capacity of the buffer sufficient? What is the resulting pH? (c) Another student accidentally adds \(20 \mathrm{mL}\) of \(10 \mathrm{M}\) HCl to 500 mL of the buffer solution prepared in part (a). Is the buffer ruined? If so, how could the buffer be regenerated?

Phenol red indicator changes from yellow to red in the pH range from 6.6 to \(8.0 .\) Without making detailed calculations, state what color the indicator will assume in each of the following solutions: (a) \(0.10 \mathrm{M} \mathrm{KOH}\) (b) \(0.10 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOH} ;\) (c) \(0.10 \mathrm{M} \mathrm{NH}_{4} \mathrm{NO}_{3} ;\) (d) \(0.10 \mathrm{M}\) HBr; (e) \(0.10 \mathrm{M} \mathrm{NaCN} ;\) (f) \(0.10 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOH}-0.10 \mathrm{M}\) \(\mathrm{NaCH}_{3} \mathrm{COO}\).

A \(25.00 \mathrm{mL}\) sample of \(\mathrm{H}_{3} \mathrm{PO}_{4}(\text { aq) requires } 31.15 \mathrm{mL}\) of \(0.2420 \mathrm{M}\) KOH for titration to the second equivalence point. What is the molarity of the \(\mathrm{H}_{3} \mathrm{PO}_{4}(\mathrm{aq}) ?\)

A solution is prepared that is \(0.150 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOH}\) and \(0.250 \mathrm{M} \mathrm{NaHCOO}\) (a) Show that this is a buffer solution. (b) Calculate the pH of this buffer solution. (c) What is the final pH if 1.00 L of 0.100 M HCl is added to \(1.00 \mathrm{L}\) of this buffer solution?

Piperazine is a diprotic weak base used as a corrosion inhibitor and an insecticide. Its ionization is described by the following equations. \(\mathrm{HN}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\) \(\left[\mathrm{HN}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}_{2}\right]^{+}+\mathrm{OH}^{-} \quad \mathrm{p} K_{\mathrm{b}_{1}}=4.22\) \(\left[\mathrm{HN}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}_{2}\right]^{+}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\) \(\left[\mathrm{H}_{2} \mathrm{N}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}_{2}\right]^{2+}+\mathrm{OH}^{-} \quad \mathrm{p} K_{\mathrm{b}_{2}}=8.67\) . The piperazine used commercially is a hexahydrate, \(\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{N}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O} .\) A \(1.00-\mathrm{g}\) sample of this hexahydrate is dissolved in \(100.0 \mathrm{mL}\) of water and titrated with 0.500 M HCl. Sketch a titration curve for this titration, indicating (a) the initial \(\mathrm{pH} ;\) (b) the pH at the halfneutralization point of the first neutralization; (c) the volume of \(\mathrm{HCl}(\text { aq })\) required to reach the first equivalence point; (d) the pH at the first equivalence point; (e) the \(\mathrm{pH}\) at the point at which the second step of the neutralization is half-completed; (f) the volume of \(0.500 \mathrm{M} \mathrm{HCl}(\) aq) required to reach the second equivalence point of the titration; (g) the pH at the second equivalence point.
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