Problem 32 - General Chemistry: Principles and Modern Applications

Question

A saturated aqueous solution of \(o\) -nitrophenol, \(\mathrm{HOC}_{6} \mathrm{H}_{4} \mathrm{NO}_{2},\) has \(\mathrm{pH}=4.53 .\) What is the solubility of \(o\) -nitrophenol in water, in grams per liter? $$\begin{aligned} &\mathrm{HOC}_{6} \mathrm{H}_{4} \mathrm{NO}_{2}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}+-\mathrm{OC}_{6} \mathrm{H}_{4} \mathrm{NO}_{2}\\\ &&\mathrm{p} K_{\mathrm{a}}=7.23 \end{aligned}$$

Step-by-Step Solution

Answer

The solubility of o-nitrophenol in water is approximately \(0.696\, g/L\).

Step 1: Calculation using Henderson-Hasselbalch equation

The problem is related to acid-base equilibrium. The Henderson-Hasselbalch equation is \(pH = pK_a + log(\frac{[A^-]}{[HA]})\). Given that at equilibrium, the concentrations of o-nitrophenol, \(HA = [A^-]\), we have \(pH = pK_a + log(1) = pK_a\). Thus, the equilibrium \( [H_3O^+] = 10^{(pK_a - pH)}\) concentration can be calculated, which equals to \(10^{(7.23 - 4.53)} = 0.005\, M = [H_3O^+]\). Since \( [HOC_6H_4NO_2] = [H_3O^+]\) at equilibrium, the concentration of o-nitrophenol in water is \(0.005\, M\).

Step 2: Convert concentration to solubility

We have the concentration of o-nitrophenol in M (Moles/Liter). To convert it to g/L, we need the molar mass of o-nitrophenol, which is approximately \(139.11\, g/mol\) (can be calculated by adding the molar masses of all individual atoms). The solubility of o-nitrophenol is therefore: \(0.005\, mol/L * 139.11\, g/mol = 0.696\, g/L\).

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental tool in chemistry used to describe the relationship between the pH and pKa values of a buffer solution. It is written as:
\[ pH = pKa + \log\left(\frac{[A^-]}{[HA]}\right) \]
In this equation, pH represents the acidity or alkalinity of the solution, pKa is the acid dissociation constant, a measure of the strength of the acid, \([A^-]\) denotes the concentration of the base form (the conjugate base), and \([HA]\) denotes the concentration of the acid form of the species in equilibrium.

When the concentrations of the acid and its conjugate base are equal, \(\log\left(\frac{[A^-]}{[HA]}\right)\) is zero because the log of 1 is 0. This is relevant in solving the o-nitrophenol solubility problem, where at equilibrium the concentration of \(HA\) is equal to \([A^-]\), leading to a simplification of the Henderson-Hasselbalch equation.

Acid-Base Equilibrium

In acid-base chemistry, the equilibrium between an acid and its conjugate base defines how substances dissolve, react, and form salts in water. The principle of acid-base equilibrium is governed by Le Chatelier's principle, which says that an equilibrium will adjust to counteract any changes.

An acid (\(HA\)) in water typically dissociates into its conjugate base (\(A^-\)) and hydronium ions (\(H_3O^+\)), creating a balance between the reactants and products. The equilibrium constant (Ka) is the ratio of these concentrations and is a measure of the propensity of the acid to lose a proton. The pKa is simply the negative log of Ka and is used for practicality given the small size of Ka values.

Molar Concentration

Molar concentration, often represented by the unit M (molarity), is a measure of the amount of a solute that is present in a given volume of solution. It is defined as the number of moles of solute per liter of solution. The equation that describes molar concentration (C) is:
\[ C = \frac{n}{V} \]
where \(n\) is the number of moles of the solute and \(V\) is the volume of the solution in liters.

In our exercise with o-nitrophenol, once the molar concentration is determined, we can calculate the solubility in grams per liter by multiplying this molarity by the compound's molar mass.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is an intrinsic property of each chemical compound and can be determined by summing the molar masses of all the atoms in a single molecule of that compound.

The atomic weights of individual elements are found on the periodic table and by adding these together according to the number of atoms in the molecule, we find the molar mass. For example, the compound o-nitrophenol, \(\mathrm{HOC}_{6}\mathrm{H}_{4}\mathrm{NO}_{2}\), has a molar mass calculated by adding the molar masses of hydrogen, carbon, nitrogen, and oxygen in the quantities present in its molecular structure. This allows us to convert the molar concentration to grams per liter, essential for the accurate determination of solubility.