Problem 28 - Chemistry and Chemical Reactivity

Question

If \(K_{\mathrm{a}}\) for a weak acid is \(2.4 \times 10^{-11},\) what is the value of \(\mathrm{p} K_{\mathrm{a}} ?\)

Step-by-Step Solution

Answer

pKₐ is approximately 10.62.

Step 1: Understanding pKₐ

The pKₐ of an acid is defined as the negative logarithm (base 10) of its Kₐ value: \[\text{p}K_a = -\log_{10}(K_a)\]We will apply this formula to find the pKₐ from the given Kₐ.

Step 2: Plugging in the given Kₐ value

Substitute the given Kₐ value into the pKₐ formula:\[\text{p}K_a = -\log_{10}(2.4 \times 10^{-11})\]Now we will calculate this expression.

Step 3: Calculating the logarithm

Use a calculator to find the common logarithm of the given Kₐ:\[\log_{10}(2.4 \times 10^{-11}) = \log_{10}(2.4) + \log_{10}(10^{-11})\]We know that \(\log_{10}(10^{-11}) = -11\). Calculate \(\log_{10}(2.4)\) using a calculator.

Step 4: Finding the final pKₐ value

The calculation yields:\[\log_{10}(2.4) \approx 0.3802\]Thus,\[\log_{10}(2.4 \times 10^{-11}) = 0.3802 - 11 = -10.6198\]Then calculate:\[\text{p}K_a = -(-10.6198) = 10.6198\]

Key Concepts

pKa calculation

Calculating the pKa of a weak acid is an important concept in chemistry that helps understand the acid's strength. The pKa is determined using the formula: \[\text{p}K_a = -\log_{10}(K_a)\]The "Kₐ" symbolizes the acid dissociation constant, offering insight into the degree of dissociation of an acid in water. A smaller Kₐ value indicates that the acid dissociates less in water, highlighting its weaker nature.
In the given exercise, the Kₐ of a weak acid is presented as \(2.4 \times 10^{-11}\), which implies a negligible dissociation. By substituting it into the pKa formula, you convert this complex number into a simple pKa value that's easier to compare.To compute the pKa:

• Calculate the logarithm base 10 of the Kₐ using a calculator. This involves taking the logarithm of both the coefficient (2.4) and the power of ten \(10^{-11}\).
• Add the results from these two calculations together to obtain the full logarithmic value of Kₐ.
• Finally, multiply by -1 to transition from the logarithmic Kₐ to the pKa.

This process translates the Kₐ into a user-friendly pKa value, helping chemists to easily assess acidity traits of various substances.

weak acid

Weak acids are characterized by their partial dissociation in aqueous solutions, as opposed to strong acids, which dissociate completely.
When dissolved in water, a weak acid like acetic acid (commonly found in vinegar) partially releases hydrogen ions (H⁺), contributing to a low concentration of these ions in the solution, which affects the solution’s acidity.
The acid dissociation constant (Kₐ) quantifies this dissociation balance. Typically:

• Smaller Kₐ values point towards weaker acids, indicating lesser dissociation and lower H⁺ ion concentration.
• Larger Kₐ values suggest stronger acids with higher dissociation levels.

For the weak acid addressed in the exercise, the exceptionally small Kₐ value of \(2.4 \times 10^{-11}\) confirms its minor dissociation properties. This means in a solution, most of the acid remains undissociated, preserving its molecular form.

pH and acidity

Understanding the pH scale is fundamental in grasping how acidic or basic a solution is. The scale ranges from 0 to 14:

• A pH less than 7 indicates an acidic solution.
• A pH of 7 is neutral, meaning the solution is neither acidic nor basic.
• A pH greater than 7 signifies a basic or alkaline solution.

Since pH is the negative logarithm of the hydrogen ion concentration (\([\text{H}^+]\)), weak acids with low levels of H⁺ ions often have higher pH values than strong acids. The pKa, being another logarithmic scale, helps predict the acid's behavior in varying pH environments.
In practical terms, a low pKa corresponds to a strong acid and a low pH, while a high pKa, like the 10.6198 calculated within the exercise, reflects a weaker acid which might contribute less to the acidity of the solution. Hence, it can have a larger pH comparing to stronger acids.