Problem 27 - Chemistry and Chemical Reactivity

Question

A weak acid has a \(K_{\mathrm{a}}\) of \(6.5 \times 10^{-5} .\) What is the value of \(\mathrm{p} K_{\mathrm{a}}\) for the acid?

Step-by-Step Solution

Answer

The \(\mathrm{p}K_{\mathrm{a}}\) is approximately 4.19.

Step 1: Understand the Relationship between Ka and pKa

The relationship between the dissociation constant \(K_{\mathrm{a}}\) and \(\mathrm{p}K_{\mathrm{a}}\) is given by the formula: \(\mathrm{p}K_{\mathrm{a}} = -\log K_{\mathrm{a}}\). This means that to find \(\mathrm{p}K_{\mathrm{a}}\), we take the negative logarithm (base 10) of the \(K_{\mathrm{a}}\) value.

Step 2: Find the pKa Using the Known Ka

Given \(K_{\mathrm{a}} = 6.5 \times 10^{-5}\), we need to calculate \(\mathrm{p}K_{\mathrm{a}}\) by substituting this value into the formula. Compute \(\mathrm{p}K_{\mathrm{a}} = -\log(6.5 \times 10^{-5})\).

Step 3: Calculate and Interpret the pKa Result

Using a calculator, compute \(\mathrm{p}K_{\mathrm{a}} = -\log(6.5 \times 10^{-5})\). The result should be approximately \(4.19\). The \(\mathrm{p}K_{\mathrm{a}}\) value indicates the strength of the acid; the lower it is, the stronger the acid.

Key Concepts

Acid Dissociation Constant

In the study of acids, the acid dissociation constant, represented as \(K_a\), is a crucial concept. It reflects how easily an acid disassociates to produce hydrogen ions \((H^+)\) in a solution. This constant is specific to each acid and provides insight into its strength as an acid.
The simple equation that depicts the dissociation of a weak acid \(HA\) in water is:

• \(HA \leftrightarrow H^+ + A^-\)

At equilibrium, the acid dissociation constant \(K_a\) is expressed as:

• \(K_a = \frac{[H^+][A^-]}{[HA]}\)

Where \([H^+]\), \([A^-]\), and \([HA]\) represent the concentrations of the hydrogen ions, the conjugate base, and the undissociated acid respectively.A higher \(K_a\) value indicates more dissociation; hence, a stronger acid. However, note that many acids especially weak ones, have very small \(K_a\) values.

Weak Acid

Weak acids are those that only partially ionize in aqueous solutions. This means, in a solution, they exist in equilibrium with their ions. For weak acids, the acid dissociation constant \(K_a\) is small, indicating limited release of \(H^+\) ions.
Examples of weak acids include:

• Acetic acid \((CH_3COOH)\)
• Formic acid \((HCOOH)\)

Since weak acids do not fully disassociate, their ionization can be evaluated using the equilibrium constant \(K_a\). This small value signifies weak attraction to protons and thus maintaining most of its molecules intact in the solution.Understanding weak acids is essential in predicting the behavior and reactions of acids in various scientific and industrial applications.

Logarithmic Function

Logarithmic functions are a fundamental mathematical tool used to simplify multiplication and division into addition and subtraction. They are particularly useful in chemistry to handle numbers spanning vast ranges, making them easier to work with.
The relationship of interest in acid-base chemistry involving logarithms is:

• \(pK_a = -\log K_a\)

This equation tells us to take the negative base-10 logarithm of the \(K_a\) value to find \(pK_a\).

Logarithmic values help assess the strength of acids. For instance, as shown in the step-by-step solution, a \(K_a\) of \(6.5 \times 10^{-5}\) gives a \(pK_a\) of about 4.19. The negative logarithm makes charts and tables easier to navigate and interpret in scientific contexts, such as distinguishing between weak and strong acids based on their \(pK_a\) values.