Question

The compound para-aminobenzoic acid (PABA) is a powerful sun-screening agent whose salts were once used widely in sun tanning and screening lotions. The parent acid, which we may symbolize as \(\mathrm{H}-\mathrm{Paba}\), is a weak acid with a \(\mathrm{p} K_{\mathrm{a}}\) of 4.92 (at \(25^{\circ} \mathrm{C}\) ). What are the \(\left[\mathrm{H}^{+}\right]\) and \(\mathrm{pH}\) of a \(0.030 \mathrm{M}\) solution of this acid?

Short answer

The concentration of hydrogen ions \([\mathrm{H}^+]\) in the solution is approximately equal to \(x\) calculated from \(K_a\), and the pH is calculated using the \(-\log\) of that concentration.

Step-by-step solution

Write down the acid dissociation equation

PABA is a weak acid, which partially dissociates in water according to the equation: \(\mathrm{H-Paba} \rightleftharpoons \mathrm{H}^+ + \mathrm{Paba}^-\). The concentration of hydrogen ions \([\mathrm{H}^+]\) can be calculated using the acid dissociation constant \(K_a\) which is related to the \(pK_a\) by the equation \(K_a = 10^{-pK_a}\).

Calculate the acid dissociation constant (Ka)

From the given \(pK_a\) value, calculate the dissociation constant \(K_a\) using the equation: \(K_a = 10^{-pK_a}\). Substituting the given \(pK_a\) value gives \(K_a = 10^{-4.92}\).

Set up the ICE table for the equilibrium system

Using an ICE (Initial, Change, Equilibrium) table, set up the initial concentrations and the changes in concentrations as the reaction reaches equilibrium. For the initial concentration of \(\mathrm{H-Paba}\), we use 0.030 M, and for \(\mathrm{H}^+\) and \(\mathrm{Paba}^-\), we use 0 since the acid is just starting to dissociate.

Write the expression for Ka using the ICE table

The expression for the equilibrium constant \(K_a\) is \(K_a = \frac{[\mathrm{H}^+][\mathrm{Paba}^-]}{[\mathrm{H-Paba}]}\). At equilibrium, the \([\mathrm{H}^+]\) and \([\mathrm{Paba}^-]\) will both be \(x\), and the concentration of \(\mathrm{H-Paba}\) will be \(0.030 M - x\), where \(x\) represents the change in concentration.

Insert the equilibrium concentrations into the Ka expression

Substitute the equilibrium concentrations into the Ka expression to yield the equation: \(K_a = \frac{x^2}{0.030 - x}\).

Solve for x, representing the [H+]

Assuming that \(x\) is much less than 0.030 M due to the weak nature of PABA, the equation simplifies to \(K_a \approx \frac{x^2}{0.030}\). Now solve for \(x\) to find the \([\mathrm{H}^+]\).

Calculate the pH of the solution

Using the concentration of hydrogen ions \([\mathrm{H}^+]\), calculate the pH using the equation: \(\mathrm{pH} = -\log [\mathrm{H}^+]\).

Key concepts

Weak Acid

A weak acid, such as para-aminobenzoic acid (PABA), is one that does not completely dissociate in water. Unlike strong acids which almost fully ionize, weak acids exist in an equilibrium where a significant amount of the acid remains in its un-ionized form. The strength of a weak acid is quantified using the acid dissociation constant, \(K_a\), which is an indicator of how easily the acid donates its proton (\(H^+\)) to water. Lower \(K_a\) values correlate with a weaker acidity, which in the case of PABA, corresponds to its role as a mild acid ideal for application on the skin in sunscreens. Understanding the behavior of weak acids is crucial for several industries, including pharmaceuticals and cosmetics.

pKa

The \(pK_a\) value is a logarithmic measure of the acid dissociation constant and serves as a more convenient way to express acidity. It's calculated as \(pK_a = -\log(K_a)\). The lower the \(pK_a\), the stronger the acid. The \(pK_a\) of PABA, 4.92, indicates it's a weak acid because this value is higher than the pKa of strong acids, which are typically lower than -1.5. In a practical sense, knowing the \(pK_a\) of an acid is crucial for chemists, as it helps predict how an acid will behave in different chemical reactions, particularly in biochemistry and medicine where the pH of the environment can affect drug efficacy and stability.

ICE Table

An ICE (Initial, Change, Equilibrium) table is a systematic way to organize information about the concentrations of the components in a reaction at different stages. 'I' stands for the initial concentrations before the reaction starts, 'C' represents the change that occurs as reactants turn into products, and 'E' indicates the equilibrium concentrations of all species in the reaction. Setting up this table helps clarify the shifts in concentration as the reaction progresses towards equilibrium. It is an invaluable tool in equilibrium calculations as it simplifies the process of finding unknown concentrations, such as the \(H^+\) ions when a weak acid like PABA is placed in water. An ICE table also underscores the approximation made in the last step of the solution where we consider \(x\) to be significantly small to simplify the calculations.

pH Calculation

The pH of a solution is a measure of its acidity or alkalinity, expressed as the negative logarithm of the hydrogen ion concentration: \(pH = -\log[H^+]\). For weak acids, which partially dissociate in water, calculating pH involves determining the concentration of hydrogen ions at equilibrium. pH calculation is essential in many aspects of science and everyday life; for instance, maintaining the correct pH is crucial in swimming pools, agricultural soils, and the human body. The pH tells us about the relative acidity of a solution, which can affect the solubility of compounds, corrosion, and the activity of enzymes. For PABA, knowing the pH of its solution is critical for ensuring its safe application in sunscreen products.