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Chapter 15: Problem 89

Hippuric acid \(\left(\mathrm{HC}_{g} \mathrm{H}_{8} \mathrm{NO}_{3}\right)\), found in horse urine, has \(\mathrm{p} K_{\mathrm{a}}=3.62\) (a) Calculate the \(\mathrm{pH}\) and the concentrations of all species \(\left(\mathrm{H}_{3} \mathrm{O}^{+}, \mathrm{C}_{9} \mathrm{H}_{8} \mathrm{NO}_{3}^{-}, \mathrm{HC}_{9} \mathrm{H}_{3} \mathrm{NO}_{3}\right.\) and \(\left.\mathrm{OH}^{-}\right)\) in \(0.100 \mathrm{M}\) hip- puric acid. (b) Calculate the percent dissociation in \(0.0750 \mathrm{M}\) hippuric acid.

Short Answer

Expert verified
(a) pH 鈮� 2.50. (b) % dissociation 鈮� 3.23%.

Step by step solution

01

Step 1: Understanding the Problem

We are given the pK_a of hippuric acid and need to calculate the pH and concentrations of species in a 0.100 M solution as well as the percent dissociation in a 0.0750 M solution.
02

Relationship with pKa

First, we recognize the relationship between pK_a and K. For a general acid, HA, its dissociation in aqueous solution is HA 鈫� H^+ + A^-. The equilibrium expression is K_a = \([H^+][A^-]/[HA]\). The given pK_a is 3.62, thus K_a = 10^{-3.62}.
03

Setup ICE Table for Dissociation

For the dissociation of hippuric acid, use an ICE table:Initial concentrations: [HC鈧塇鈧圢O鈧僝 = 0.100 M, [C鈧塇鈧圢O鈧冣伝] = [H鈧僌鈦篯 = 0 M.Change: [HC鈧塇鈧圢O鈧僝 = -x, [C鈧塇鈧圢O鈧冣伝] = [H鈧僌鈦篯 = +x.Equilibrium: [HC鈧塇鈧圢O鈧僝 = 0.100-x, [C鈧塇] = \( x \).
04

Solving for x using K_a

Using the equilibrium expression, and neglecting x in comparison to 0.100 due to small dissociation, solve:K_a = \(10^{-3.62} = \frac{x^2}{0.100} \). Solve for x: x = \(\sqrt{10^{-3.62} \times 0.100}\).
05

Calculating [H鈧僌鈦篯 and pH

The concentration of H鈧僌鈦� is x. Calculate x using your answer from the previous step, then find pH = -log10(x).
06

Calculate Concentrations of Other Species

[C鈧塇鈧圢O鈧冣伝] equals the concentration of H鈧僌鈦� as x, and [HC鈧塇鈧圢O鈧僝 is 0.100 - x. Calculate [OH鈦籡 using: \([OH^-] = \frac{K_w}{[H_3O^+]}\), where K_w is 10^{-14}.
07

Calculate Percent Dissociation

For 0.0750 M hippuric acid, follow the ICE table method again, using \(x/0.0750 \times 100\) to find percent dissociation. Here, x is the concentration of dissociated ions at equilibrium.

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Key Concepts

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

Hippuric Acid
Hippuric acid, labeled chemically as HC鈧塇鈧圢O鈧�, is an interesting compound found naturally in the urine of horses. It's a weak acid, meaning it does not completely dissociate in water.
Understanding the dissociation of weak acids like hippuric acid is crucial to grasping acid-base equilibrium. This kind of acid partially breaks apart in water into its component ions.
  • Hippuric acid dissociates into hydronium ions (H鈧僌鈦�) and the conjugate base (C鈧塇鈧圢O鈧冣伝).
  • This process affects the acidity of the solution, which is measured by pH.
The understanding of hippuric acid's behavior in solution forms the basis for calculating related equilibria such as pH and concentration of involved species.
Percent Dissociation
Percent dissociation is a measure of how much of an acid ionizes in solution compared to how much remains in its original form. It's essentially a snapshot of the degree to which the acid breaks down into H鈦� ions in the solution.
To find the percent dissociation of hippuric acid:
  • Calculate the concentration of dissociated ions at equilibrium using an ICE table.
  • Divide the concentration of the dissociated form by the initial concentration of the acid, and multiply by 100 to get the percentage.
This value gives insight into how strong or weak the acid is, with lower values indicating weaker interactions.
pKa
The pKa value is a vital piece of information when dealing with acids. It's the negative logarithm of the acid dissociation constant (Ka) and signifies the strength of an acid.
In the case of hippuric acid, a pKa of 3.62 tells us about its affinity to donate protons.
  • Lower pKa values mean stronger acids that dissociate more in water, producing more H鈦� ions.
  • Because the pKa of hippuric acid is not extremely low, it indicates that it is a relatively weak acid.
Understanding pKa is essential for predicting how an acid-related reaction will proceed under various circumstances.
ICE Table Calculations
ICE tables are indispensable tools in chemistry used to track the concentrations of species in a chemical reaction at initial, change, and equilibrium periods.
In the context of hippuric acid, the ICE table helps us organize data and predict changes in concentrations as the dissociation reaction proceeds.
  • "I" stands for the initial concentrations of reactants and products before any reaction starts.
  • "C" represents the changes that occur as the system reaches equilibrium.
  • "E" stands for the equilibrium concentrations when the reaction has settled.
This method allows for systematic calculations and helps find unknowns like pH, concentrations of hydronium ions, and the acid itself at equilibrium.

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