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Chapter 19: Problem 136

A biochemist needs a medium for acid-producing bacteria. The \(\mathrm{pH}\) of the medium must not change by more than \(0.05 \mathrm{pH}\) units for every \(0.0010 \mathrm{~mol}\) of \(\mathrm{H}_{3} \mathrm{O}^{+}\) generated by the organisms per liter of medium. A buffer consisting of \(0.10 \mathrm{M}\) HA and \(0.10 \mathrm{M} \mathrm{A}^{-}\) is included in the medium to control its \(\mathrm{pH}\). What volume of this buffer must be included in \(1.0 \mathrm{~L}\) of medium?

Short Answer

Expert verified
Use a buffer volume of 1.0 L to maintain pH stability for generated \( \text{H}_3 \text{O}^+ \)

Step by step solution

01

- Understand the problem

The problem requires determining the volume of buffer needed to maintain the pH of 1.0 L of medium within a specific range when a certain amount of \(\text{H}_3\text{O}^+\) is generated.
02

- Define the buffer capacity

Buffer capacity \(\beta\) is defined as the amount of acid or base that can be added to a buffer solution before a significant change in pH occurs. Here, \(\beta\) should not exceed \(\frac{0.05 \text{ pH units}}{0.001 \text{ mol H}_3\text{O}^+} \). Hence, \(\beta = 50 \frac{\text{pH units}}{\text{mol H}_3\text{O}^+}} \)
03

- Calculate the buffer capacity of the given buffer solution

Use the equation for buffer capacity: \[ \beta = 2.3c \frac{K_{\text{a}}[\text{A}^-]}{([H_3O^+] + K_{\text{a}})^2} \] where \(c\) is the concentration of the buffer components, and \(K_{\text{a}}\) is the acid dissociation constant of \(HA\). Since \( [\text{HA}] = [\text{A}^-] = 0.10\text{ M} \), \( [H_3O^+] = 10^{-\text{pH}} \)
04

- Insert the known values

Let's assume \( \text{pH} = 7 \), \( K_{\text{a}} = \text{unknown} \), \(\beta_{\text{required}} = 50 \frac{\text{pH units}}{\text{mol H}_3\text{O}^+} \) and solve for the unknown volume of buffer needed. Use the relationship: \[ V = \frac{\text{required buffer capacity}}{\text{buffer capacity per liter}} \]
05

- Calculate the volume of buffer needed

If one liter of buffer solution has a capacity, \(\beta_1 \times V\) must equal the required capacity. Hence, \(\beta = 0.10 \text{M} \times 10 \text{L} \) to maintain pH...

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buffer Solution
A buffer solution is a mixture of a weak acid and its conjugate base or a weak base and its conjugate acid. It helps maintain a stable pH in a solution even when small amounts of acid or base are added.
Practical examples include the use of buffer solutions in biochemical experiments and industrial processes where specific pH ranges are critical.
  • In our example, the buffer comprises 0.10 M HA (weak acid) and 0.10 M A- (conjugate base)
  • It counteracts minor changes in H3O+ concentration, maintaining pH stability.
Buffer solutions are vital in biological systems, such as human blood, which uses a bicarbonate buffer to maintain pH within a narrow range essential for physiological functions.
Acid Dissociation Constant
The acid dissociation constant, or Ka, is a measure of the strength of an acid in solution. It is represented as:
\( K_a = \frac{[H_3O^+][A^-]}{[HA]} \)
Where:
  • [H3O+] = concentration of hydronium ions
  • [A-] = concentration of conjugate base
  • [HA] = concentration of the weak acid
Higher Ka values indicate stronger acids because they dissociate more in water.
In our example, knowing the pH helps estimate the concentration of H3O+, which simplifies the computation of buffer capacity using the given concentrations of HA and A-.
By doing this, we can quantify the buffer's ability to neutralize added acids/bases, thus maintaining pH stability.
pH Stability
pH stability refers to the ability of a solution to resist changes in pH when acids or bases are added. Buffers are pivotal in achieving this.
  • The pH scale measures how acidic or basic a solution is, with 7 being neutral.
  • Biological processes often require a tightly controlled pH range to function optimally, like enzymatic activities in cells.
In the exercise, the buffer must keep the pH stable against changes caused by bacteria in the medium producing 0.0010 mol/L of H3O+.
For our buffer, the required stability was defined: the pH should not change more than 0.05 units. This was tied to the buffer capacity, ensuring the correct volume maintains this balance. By calculating how much of our buffer solution is needed per liter of the medium, we make sure that pH adjustments stay within the desired limits, thereby ensuring the stability of the medium for the bacteria.

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Most popular questions from this chapter

An ecobotanist separates the components of a tropical bark extract by chromatography. She discovers a large proportion of quinidine, a dextrorotatory isomer of quinine used for control of arrhythmic heartbeat. Quinidine has two basic nitrogens \(\left(K_{\mathrm{b} 1}=4.0 \times 10^{-6}\right.\) and \(\left.K_{\mathrm{b} 2}=1.0 \times 10^{-\mathrm{i} 0}\right) .\) To measure the concentration of quinidine, she carries out a titration. Because of the low solubility of quinidine, she first protonates both nitrogens with excess \(\mathrm{HCl}\) and titrates the acidified solution with standardized base. A 33.85-mg sample of quinidine \((\mathscr{M}=324.41 \mathrm{~g} / \mathrm{mol})\) is acidified with \(6.55 \mathrm{~mL}\) of \(0.150 \mathrm{M} \mathrm{HCl}\) (a) How many milliliters of \(0.0133 \mathrm{M} \mathrm{NaOH}\) are needed to titrate the excess HCl? (b) How many additional milliliters of titrant are needed to reach the first equivalence point of quinidine dihydrochloride? (c) What is the \(\mathrm{pH}\) at the first equivalence point?

Human blood contains one buffer system based on phosphate species and one based on carbonate species. Assuming that blood has a normal \(\mathrm{pH}\) of \(7.4,\) what are the principal phosphate and carbonate species present? What is the ratio of the two phosphate species? (In the presence of the dissolved ions and other species in blood, \(K_{\mathrm{a} 1}\) of \(\mathrm{H}_{3} \mathrm{PO}_{4}=1.3 \times 10^{-2}, K_{\mathrm{a} 2}=2.3 \times 10^{-7},\) and \(K_{\mathrm{a} 3}=6 \times 10^{-12} ; K_{\mathrm{a} 1}\) of \(\mathrm{H}_{2} \mathrm{CO}_{3}=8 \times 10^{-7}\) and \(\left.K_{\mathrm{a} 2}=1.6 \times 10^{-10} .\right)\)

A \(50.0-\mathrm{mL}\) volume of \(0.50 M \mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{3}\) is mixed with \(125 \mathrm{~mL}\) of \(0.25 M \mathrm{Cd}\left(\mathrm{NO}_{3}\right)_{2}\) (a) If aqueous \(\mathrm{NaOH}\) is added, which ion precipitates first? (See Appendix C.) (b) Describe how the metal ions can be separated using \(\mathrm{NaOH}\). (c) Calculate the \(\left[\mathrm{OH}^{-}\right]\) that will accomplish the separation.

The solubility of zinc oxalate is \(7.9 \times 10^{-3} M\) at \(18^{\circ} \mathrm{C}\). Calculate its \(K_{\mathrm{sp}}\)

Find the molar solubility of \(\mathrm{BaCrO}_{4}\left(K_{\mathrm{sp}}=2.1 \times 10^{-10}\right) \mathrm{in}\) (a) pure water and (b) \(1.5 \times 10^{-3} M \mathrm{Na}_{2} \mathrm{CrO}_{4}\)
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