Question

The \(\mathrm{pH}\) of an aqueous acid solution is 6.20 at \(25^{\circ} \mathrm{C}\). Calculate the \(K_{\mathrm{a}}\) for the acid. The initial acid concentration is \(0.010 \mathrm{M}\).

Short answer

The acid dissociation constant \(K_a\) is approximately \(3.98 \times 10^{-11}\).

Step-by-step solution

Calculate Hydronium Ion Concentration

The pH of the solution is given as 6.20. We can find the concentration of hydronium ions \([\mathrm{H}^+]\) using the formula \([\mathrm{H}^+]=10^{-\mathrm{pH}}\). Substitute the given pH into the formula: \([\mathrm{H}^+]=10^{-6.20}\approx 6.31 \times 10^{-7} \, \mathrm{M}\).

Set Up the Equilibrium Expression

For a monoprotic weak acid \(HA\) dissociating in water, the dissociation can be represented as \(HA \rightleftharpoons H^+ + A^-\). The expression for the acid dissociation constant \(K_a\) is \(K_a = \frac{[H^+][A^-]}{[HA]}\). Since the initial concentration of acid is much higher than \([H^+]\), we assume \([HA]\) at equilibrium is approximately equal to its initial concentration minus \([H^+]\), so \([HA] \approx 0.010 - 6.31 \times 10^{-7} \approx 0.010 \).

Solve for the Acid Dissociation Constant (\(K_a\))

At equilibrium, the concentration of \([H^+]\) and \([A^-]\) will be equal, both given by the calculated \([H^+]\). Substitute these values into the \(K_a\) expression: \(K_a = \frac{(6.31 \times 10^{-7})(6.31 \times 10^{-7})}{0.010}\). This simplifies to \(K_a = \frac{3.98 \times 10^{-13}}{0.010}\approx 3.98 \times 10^{-11}\).

Key concepts

pH calculation

Understanding pH calculation is fundamental to chemistry for predicting the acidity or basicity of solutions. pH is a scale used to specify the acidity or basicity of an aqueous solution. It is logarithmically based, meaning each whole pH value below 7 is ten times more acidic than the next higher value. The formula to calculate the pH of a solution is:- \( \text{pH} = -\log_{10} [H^+] \)Where \([H^+]\) is the concentration of hydrogen ions in moles per liter (M).
To find pH, you simply take the negative logarithm of the hydronium ion concentration. Remember that a lower pH means a higher concentration of hydronium ions, indicating a more acidic solution.
In our example, a pH of 6.20 indicates a relatively neutral solution since 7 is neutral on the pH scale.

Hydronium ion concentration

The hydronium ion concentration \([H^+]\) is directly related to a solution's acidity. It helps us understand how many hydrogen ions are present in the solution. You can calculate \([H^+]\) from pH using the formula:- \([H^+]=10^{-\text{pH}}\)In this exercise, we calculated \([H^+]\) to be approximately \(6.31 \times 10^{-7} \text{ M}\) from a pH of 6.20.
This concentration is a measure of how much the acid dissociates in water, which is crucial for calculating the dissociation constant.
A higher concentration usually means a lower pH and a stronger acid.

Equilibrium expression

An equilibrium expression is a mathematical representation of a chemical reaction at equilibrium. In the case of weak acids, the equilibrium involves the dissociation of the acid into its ions.
For the dissociation of a generic weak acid \(HA\), the reaction is:- \(HA \rightleftharpoons H^+ + A^-\)The equilibrium expression, which helps us determine \(K_a\), the acid dissociation constant, is written as follows:- \(K_a = \frac{[H^+][A^-]}{[HA^{\phantom{0}}]}\)At equilibrium, the concentrations of \([H^+]\) and \([A^-]\) are equal, and the concentration of \([HA]\) is slightly reduced.
Understanding how to set up and manipulate these expressions is vital for predicting the behavior of weak acids in solution.

Weak acid dissociation

Weak acid dissociation refers to the partial dissociation of an acid in solution. Unlike strong acids, weak acids do not fully ionize in water. This partial dissociation is important when calculating the acid dissociation constant. For a weak acid \(HA\), it will dissociate as:- \(HA \rightleftharpoons H^+ + A^-\)The equilibrium point achieved from this dissociation enables us to calculate \(K_a\), showing how readily a weak acid donates protons.
Because the initial concentration of the acid is much higher than the hydronium ion concentration, the change in the acid's concentration is negligible for the calculation.
This is why the initial concentration can often be approximated as the equilibrium concentration when calculating \(K_a\). In our steps, recognizing this allowed us to simplify our calculations and arrive at the correct \(K_a\) value for the acid.