Chapter 17

A buffer contains a weak acid, HX, and its conjugate base. The weak acid has a \(\mathrm{p} K_{a}\) of \(4.5,\) and the buffer 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) What is the common-ion effect? (b) Give an example of a salt that can decrease the ionization of \(\mathrm{HNO}_{2}\) in solution.

(a) Consider the equilibrium \(\mathrm{B}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons\) \(\mathrm{HB}^{+}(a q)+\mathrm{OH}^{-}(a q) .\) Using Le Châtelier's principle, explain the effect of the presence of a salt of \(\mathrm{HB}^{+}\) on the ionization of B. (b) Give an example of a salt that can decrease the ionization of \(\mathrm{NH}_{3}\) in solution.

Use information from Appendix \(D\) to calculate the pH of (a) a solution that is \(0.250 \mathrm{M}\) in sodium formate \((\mathrm{HCOONa})\) and \(0.100 M\) in formic acid \((\mathrm{HCOOH}) ;\) (b) a solution that is \(0.510 \mathrm{M}\) in pyridine \(\left(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{~N}\right)\) and \(0.450 \mathrm{M}\) in pyridinium chloride \(\left(\mathrm{C}_{5} \mathrm{H}_{5} \mathrm{NHCl}\right) ;\) (c) a solution that is made by combining \(55 \mathrm{~mL}\) of \(0.050 \mathrm{M}\) hydrofluoric acid with \(125 \mathrm{~mL}\) of \(0.10 \mathrm{M}\) sodium fluoride.

(a) Calculate the percent ionization of \(0.0075 \mathrm{M}\) butanoic acid \(\left(K_{a}=1.5 \times 10^{-5}\right) .\) (b) Calculate the percent ionization of \(0.0075 \mathrm{M}\) butanoic acid in a solution containing \(0.085 \mathrm{M}\) sodium butanoate.

(a) Calculate the percent ionization of \(0.125 \mathrm{M}\) lactic acid \(\left(K_{a}=1.4 \times 10^{-4}\right)\). (b) Calculate the percent ionization of \(0.125 \mathrm{M}\) lactic acid in a solution containing \(0.0075 \mathrm{M}\) sodium lactate.

Explain why a mixture of \(\mathrm{CH}_{3} \mathrm{COOH}\) and \(\mathrm{CH}_{3} \mathrm{COONa}\) can act as a buffer while a mixture of \(\mathrm{HCl}\) and \(\mathrm{NaCl}\) cannot.

Explain why a mixture formed by mixing \(100 \mathrm{~mL}\) of \(0.100 \mathrm{M}\) \(\mathrm{CH}_{3} \mathrm{COOH}\) and \(50 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{NaOH}\) will act as a buffer.

(a) Calculate the pH of a buffer that is \(0.12 \mathrm{M}\) in lactic acid and \(0.11 M\) in sodium lactate. (b) Calculate the pH of a buffer formed by mixing \(85 \mathrm{~mL}\) of \(0.13 \mathrm{M}\) lactic acid with \(95 \mathrm{~mL}\) of \(0.15 \mathrm{M}\) sodium lactate.

(a) Calculate the pH of a buffer that is \(0.105 \mathrm{M}\) in \(\mathrm{NaHCO}_{3}\) and \(0.125 \mathrm{M}\) in \(\mathrm{Na}_{2} \mathrm{CO}_{3}\). (b) Calculate the \(\mathrm{pH}\) of a solution formed by mixing \(65 \mathrm{~mL}\) of \(0.20 \mathrm{M} \mathrm{NaHCO}_{3}\) with \(75 \mathrm{~mL}\) of \(0.15 \mathrm{M} \mathrm{Na}_{2} \mathrm{CO}_{3}\)

A buffer is prepared by adding \(20.0 \mathrm{~g}\) of sodium acetate \(\left(\mathrm{CH}_{3} \mathrm{COONa}\right)\) to \(500 \mathrm{~mL}\) of a \(0.150 \mathrm{M}\) acetic acid \(\left(\mathrm{CH}_{3} \mathrm{COOH}\right)\) solution. (a) Determine the \(\mathrm{pH}\) of the buffer. (b) Write the complete ionic equation for the reaction that occurs when a few drops of hydrochloric acid are added to the buffer. (c) Write the complete ionic equation for the reaction that occurs when a few drops of sodium hydroxide solution are added to the buffer.

A buffer is prepared by adding \(10.0 \mathrm{~g}\) of ammonium chloride \(\left(\mathrm{NH}_{4} \mathrm{Cl}\right)\) to \(250 \mathrm{~mL}\) of \(1.00 \mathrm{M} \mathrm{NH}_{3}\) solution. (a) What is the \(\mathrm{pH}\) of this buffer? (b) Write the complete ionic equation for the reaction that occurs when a few drops of nitric acid are added to the buffer. (c) Write the complete ionic equation for the reaction that occurs when a few drops of potassium hydroxide solution are added to the buffer.

You are asked to prepare a \(\mathrm{pH}=3.00\) buffer solution starting from \(1.25 \mathrm{~L}\) of a \(1.00 \mathrm{M}\) solution of hydrofluoric acid (HF) and an excess of sodium fluoride (NaF). (a) What is the \(\mathrm{pH}\) of the hydrofluoric acid solution prior to adding sodium fluoride? (b) How many grams of sodium fluoride should be added to prepare the buffer solution? Neglect the small volume change that occurs when the sodium fluoride is added.

(a) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in blood of \(\mathrm{pH} 7.4\) ? (b) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in an exhausted marathon runner whose blood \(\mathrm{pH}\) is \(7.1 ?\)

You have to prepare a pH 5.00 buffer, and you have the following \(0.10 \mathrm{M}\) solutions available: \(\mathrm{HCOOH}, \mathrm{HCOONa}, \mathrm{CH}_{3} \mathrm{COOH},\) \(\mathrm{CH}_{3} \mathrm{COONa}, \mathrm{HCN},\) and \(\mathrm{NaCN}\). Which solutions would you use? How many milliliters of each solution would you use to make approximately a liter of the buffer?

How does titration of a strong, monoprotic acid with a strong base differ from titration of a weak, monoprotic acid with a strong base with respect to the following: (a) quantity of base required to reach the equivalence point, (b) \(\mathrm{pH}\) at the beginning of the titration, \((\mathbf{c}) \mathrm{pH}\) at the equivalence point, \((\mathbf{d}) \mathrm{pH}\) after addition of a slight excess of base, (e) choice of indicator for determining the equivalence point?

Predict whether the equivalence point of each of the following titrations is below, above, or at \(\mathrm{pH}\) 7: (a) \(\mathrm{NaHCO}_{3}\) titrated with \(\mathrm{NaOH},\) (b) \(\mathrm{NH}_{3}\) titrated with \(\mathrm{HCl}\) (c) KOH titrated with HBr.

Predict whether the equivalence point of each of the following titrations is below, above, or at \(\mathrm{pH} 7:\) (a) formic acid titrated with \(\mathrm{NaOH},\) (b) calcium hydroxide titrated with perchloric acid, (c) pyridine titrated with nitric acid.

Assume that \(30.0 \mathrm{~mL}\) of a \(0.10 \mathrm{M}\) solution of a weak base \(\mathrm{B}\) 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 \(\mathrm{B}\) 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?

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 \(0.0850 \mathrm{M}\) \(\mathrm{CH}_{3} \mathrm{COOH},\) (c) \(50.0 \mathrm{~mL}\) of a solution that contains \(1.85 \mathrm{~g}\) of HCl per liter?

How many milliliters of \(0.105 \mathrm{M} \mathrm{HCl}\) are needed to titrate each of the following solutions to the equivalence point: (a) \(45.0 \mathrm{~mL}\) of \(0.0950 \mathrm{M} \mathrm{NaOH},\) (b) \(22.5 \mathrm{~mL}\) of \(0.118 \mathrm{M} \mathrm{NH}_{3},(\mathrm{c})\) \(125.0 \mathrm{~mL}\) of a solution that contains \(1.35 \mathrm{~g}\) of \(\mathrm{NaOH}\) per liter?

A 20.0-mL sample of \(0.200 \mathrm{M}\) HBr solution is titrated with \(0.200 \mathrm{M} \mathrm{NaOH}\) solution. Calculate the \(\mathrm{pH}\) of the solution after the following volumes of base have been added: (a) \(15.0 \mathrm{~mL}\) (b) \(19.9 \mathrm{~mL}\) (c) \(20.0 \mathrm{~mL},\) (d) \(20.1 \mathrm{~mL},\) (e) \(35.0 \mathrm{~mL}\)

A 20.0 -mL sample of \(0.150 \mathrm{M} \mathrm{KOH}\) is titrated with \(0.125 \mathrm{M}\) \(\mathrm{HClO}_{4}\) solution. Calculate the \(\mathrm{pH}\) after the following volumes of acid have been added: (a) \(20.0 \mathrm{~mL},\) (b) \(23.0 \mathrm{~mL},\) (c) \(24.0 \mathrm{~mL}\), (d) \(25.0 \mathrm{~mL},(\mathrm{e}) 30.0 \mathrm{~mL}\).

A \(35.0-\mathrm{mL}\) sample of \(0.150 \mathrm{M}\) acetic acid \(\left(\mathrm{CH}_{3} \mathrm{COOH}\right)\) is titrated with \(0.150 \mathrm{M} \mathrm{NaOH}\) solution. Calculate the \(\mathrm{pH}\) after the following volumes of base have been added: (a) \(0 \mathrm{~mL},(\mathbf{b})\) \(17.5 \mathrm{~mL},(\mathrm{c}) 34.5 \mathrm{~mL},(\) d) \(35.0 \mathrm{~mL},\) (e) \(35.5 \mathrm{~mL},\) (f) \(50.0 \mathrm{~mL} .\)

Consider the titration of \(30.0 \mathrm{~mL}\) of \(0.050 \mathrm{M} \mathrm{NH}_{3}\) with \(0.025 \mathrm{M}\) HCl. Calculate the pH after the following volumes of titrant have been added: (a) \(0 \mathrm{~mL},(\mathbf{b}) 20.0 \mathrm{~mL},(\mathbf{c}) 59.0 \mathrm{~mL},(\mathrm{~d}) 60.0 \mathrm{~mL},\) (e) \(61.0 \mathrm{~mL}\) (f) \(65.0 \mathrm{~mL}\).

Calculate the pH at the equivalence point for titrating \(0.200 \mathrm{M}\) solutions of each of the following bases with \(0.200 \mathrm{M} \mathrm{HBr}\) : (a) sodium hydroxide \((\mathrm{NaOH}),(\mathbf{b})\) hydroxylamine \(\left(\mathrm{NH}_{2} \mathrm{OH}\right),(\mathbf{c})\) aniline \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NH}_{2}\right)\).

Calculate the \(\mathrm{pH}\) at the equivalence point in titrating \(0.100 \mathrm{M}\) solutions of each of the following with \(0.080 \mathrm{M} \mathrm{NaOH}\) : (a) hydrobromic acid (HBr), (b) chlorous acid (HClO \(_{2}\) ), (c) benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\).

(a) Why is the concentration of undissolved solid not explicitly included in the expression for the solubility-product constant? (b) Write the expression for the solubility-product constant for each of the following strong electrolytes: AgI, \(\mathrm{SrSO}_{4}, \mathrm{Fe}(\mathrm{OH})_{2},\) and \(\mathrm{Hg}_{2} \mathrm{Br}_{2}\)

(a) Explain the difference between solubility and solubilityproduct constant. (b) Write the expression for the solubility-product constant for each of the following ionic compounds: \(\mathrm{MnCO}_{3}, \mathrm{Hg}(\mathrm{OH})_{2},\) and \(\mathrm{Cu}_{3}\left(\mathrm{PO}_{4}\right)_{2}\).

\(\begin{array}{llll}& \text { (a) If the molar solubility of } & \mathrm{CaF}_{2} & \text { at } & 35^{\circ} \mathrm{C} & \text { is }\end{array}\) \(1.24 \times 10^{-3} \mathrm{~mol} / \mathrm{L},\) what is \(K_{s p}\) at this temperature? (b) It is found that \(1.1 \times 10^{-2}\) of \(\mathrm{SrF}_{2}\) dissolves per \(100 \mathrm{~mL}\) of aqueous solution at \(25^{\circ} \mathrm{C} .\) Calculate the solubility product for \(\mathrm{SrF}_{2}\). (c) The \(K_{s p}\) of \(\mathrm{Ba}\left(\mathrm{IO}_{3}\right)_{2}\) at \(25^{\circ} \mathrm{C}\) is \(6.0 \times 10^{-10} .\) What is the molar solubility of \(\mathrm{Ba}\left(\mathrm{IO}_{3}\right)_{2}\) ?

\(\begin{array}{llll} \text { (a) The molar solubility of } \mathrm{PbBr}_{2} & \text { at } 25^{\circ} \mathrm{C} & \text { is }\end{array}\) \(1.0 \times 10^{-2} \mathrm{~mol} / \mathrm{L} .\) Calculate \(K_{s p} .(\mathbf{b})\) If \(0.0490 \mathrm{~g}\) of \(\mathrm{AgIO}_{3}\) dis- solves per liter of solution, calculate the solubility-product constant. (c) Using the appropriate \(K_{s p}\) value from Appendix \(\mathrm{D},\) calculate the \(\mathrm{pH}\) of a saturated solution of \(\mathrm{Ca}(\mathrm{OH})_{2}\)

A \(1.00-\mathrm{L}\) solution saturated at \(25^{\circ} \mathrm{C}\) with calcium oxalate \(\left(\mathrm{CaC}_{2} \mathrm{O}_{4}\right)\) contains \(0.0061 \mathrm{~g}\) of \(\mathrm{CaC}_{2} \mathrm{O}_{4} .\) Calculate the solubilityproduct constant for this salt at \(25^{\circ} \mathrm{C}\).

A \(1.00-\mathrm{L}\) solution saturated at \(25^{\circ} \mathrm{C}\) with lead(II) iodide contains \(0.54 \mathrm{~g}\) of \(\mathrm{PbI}_{2} .\) Calculate the solubility-product constant for this salt at \(25^{\circ} \mathrm{C}\).

Consider a beaker containing a saturated solution of \(\mathrm{CaF}_{2}\) in equilibrium with undissolved \(\mathrm{CaF}_{2}(s) .\) (a) If solid \(\mathrm{CaCl}_{2}\) is added to this solution, will the amount of solid \(\mathrm{CaF}_{2}\) at the bottom of the beaker increase, decrease, or remain the same? (b) Will the concentration of \(\mathrm{Ca}^{2+}\) ions in solution increase or decrease? (c) Will the concentration of \(\mathrm{F}^{-}\) ions in solution increase or decrease?

Consider a beaker containing a saturated solution of \(\mathrm{Pbl}_{2}\) in equilibrium with undissolved \(\mathrm{Pbl}_{2}(s) .\) (a) If solid KI is added to this solution, will the amount of solid \(\mathrm{PbI}_{2}\) at the bottom of the beaker increase, decrease, or remain the same? (b) Will the concentration of \(\mathrm{Pb}^{2+}\) ions in solution increase or decrease? (c) Will the concentration of \(\mathrm{I}^{-}\) ions in solution increase or decrease?

Calculate the molar solubility of \(\mathrm{Ni}(\mathrm{OH})_{2}\) when buffered at \(\mathrm{pH}(\) a) \(8.0,(\mathbf{b}) 10.0,(\mathrm{c}) 12.0\)

Which of the following salts will be substantially more soluble in acidic solution than in pure water: (a) \(\mathrm{ZnCO}_{3},(\mathbf{b}) \mathrm{ZnS}\), (c) \(\mathrm{BiI}_{3}\) (d) \(\mathrm{AgCN},(\mathrm{e}) \mathrm{Ba}_{3}\left(\mathrm{PO}_{4}\right)_{2} ?\)

For each of the following slightly soluble salts, write the net ionic equation, if any, for reaction with acid: (a) MnS, (b) \(\mathrm{PbF}_{2},\) (c) \(\mathrm{AuCl}_{3}\) (e) CuBr. (d) \(\mathrm{Hg}_{2} \mathrm{C}_{2} \mathrm{O}_{4}\)

From the value of \(K_{f}\) listed in Table \(17.1,\) calculate the concentration of \(\mathrm{Ni}^{2+}\) in \(1.0 \mathrm{~L}\) of a solution that contains a total of \(1 \times 10^{-3} \mathrm{~mol}\) of nickel(II) ion and that is \(0.20 \mathrm{M}\) in \(\mathrm{NH}_{3}\)

To what final concentration of \(\mathrm{NH}_{3}\) must a solution be adjusted to just dissolve \(0.020 \mathrm{~mol}\) of \(\mathrm{NiC}_{2} \mathrm{O}_{4}\left(K_{s p}=4 \times 10^{-10}\right)\) in \(1.0 \mathrm{~L}\) of solution? (Hint: You can neglect the hydrolysis of \(\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\) because the solution will be quite basic.)

Using the value of \(K_{s p}\) for \(\mathrm{Ag}_{2} \mathrm{~S}, K_{a 1}\) and \(K_{a 2}\) for \(\mathrm{H}_{2} \mathrm{~S},\) and \(K_{f}=1.1 \times 10^{5}\) for \(\mathrm{AgCl}_{2}^{-},\) calculate the equilibrium constant for the following reaction: \(\mathrm{Ag}_{2} \mathrm{~S}(s)+4 \mathrm{Cl}^{-}(a q)+2 \mathrm{H}^{+}(a q) \rightleftharpoons 2 \mathrm{AgCl}_{2}^{-}(a q)+\mathrm{H}_{2} \mathrm{~S}(a q)\)

(a) Will \(\mathrm{Ca}(\mathrm{OH})_{2}\) precipitate from solution if the \(\mathrm{pH}\) of a \(0.050 \mathrm{M}\) solution of \(\mathrm{CaCl}_{2}\) is adjusted to \(8.0 ?\) (b) Will \(\mathrm{Ag}_{2} \mathrm{SO}_{4}\) precipitate when \(100 \mathrm{~mL}\) of \(0.050 \mathrm{M} \mathrm{AgNO}_{3}\) is mixed with \(10 \mathrm{~mL}\) of \(5.0 \times 10^{-2} \mathrm{M} \mathrm{Na}_{2} \mathrm{SO}_{4}\) solution?

(a) Will \(\mathrm{Co}(\mathrm{OH})_{2}\) precipitate from solution if the \(\mathrm{pH}\) of a \(0.020 \mathrm{M}\) solution of \(\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}\) is adjusted to \(8.5 ?\) (b) Will \(\mathrm{AgIO}_{3}\) precipitate when \(20 \mathrm{~mL}\) of \(0.010 \mathrm{M} \mathrm{AgNO}_{3}\) is mixed with \(10 \mathrm{~mL}\) of \(0.015 \mathrm{M} \mathrm{NaIO}_{3} ?\left(K_{s p}\right.\) of \(\mathrm{AgIO}_{3}\) is \(\left.3.1 \times 10^{-8} .\right)\)

A solution contains \(2.0 \times 10^{-4} \mathrm{MAg}^{+}\) and \(1.5 \times 10^{-3} \mathrm{M}\) \(\mathrm{Pb}^{2+}\). If NaI is added, will AgI \(\left(K_{s p}=8.3 \times 10^{-17}\right)\) or \(\mathrm{PbI}_{2}\) \(\left(K_{s p}=7.9 \times 10^{-9}\right)\) precipitate first? Specify the concentration of \(1^{-}\) needed to begin precipitation.

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

A solution contains three anions with the following concentrations: \(0.20 \mathrm{M} \mathrm{CrO}_{4}^{2-}, 0.10 \mathrm{M} \mathrm{CO}_{3}^{2-},\) and \(0.010 \mathrm{M} \mathrm{Cl}^{-}\). If a dilute \(\mathrm{AgNO}_{3}\) solution is slowly added to the solution, what is the first compound to precipitate: \(\mathrm{Ag}_{2} \mathrm{CrO}_{4}\) \(\left(K_{s p}=1.2 \times 10^{-12}\right), \mathrm{Ag}_{2} \mathrm{CO}_{3}\left(K_{s p}=8.1 \times 10^{-12}\right),\) or \(\mathrm{AgCl}\) \(\left(K_{s p}=1.8 \times 10^{-10}\right) ?\)

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

In the course of various qualitative analysis procedures, the following mixtures are encountered: (a) \(\mathrm{Zn}^{2+}\) and \(\mathrm{Cd}^{2+}\) (b) \(\mathrm{Cr}(\mathrm{OH})_{3}\) and \(\mathrm{Fe}(\mathrm{OH})_{3}\) (c) \(\mathrm{Mg}^{2+}\) and \(\mathrm{K}^{+},\) (d) \(\mathrm{Ag}^{+}\) and \(\mathrm{Mn}^{2+}\). Suggest how each mixture might be separated.

Suggest how the cations in each of the following solution mixtures can be separated: (a) \(\mathrm{Na}^{+}\) and \(\mathrm{Cd}^{2+},(\mathbf{b}) \mathrm{Cu}^{2+}\) and \(\mathrm{Mg}^{2+}\), (c) \(\mathrm{Pb}^{2+}\) and \(\mathrm{Al}^{3+},(\mathbf{d}) \mathrm{Ag}^{+}\) and \(\mathrm{Hg}^{2+}\).

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