Problem 23 - 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

\(\mathrm{p} K_{\mathrm{a}} = 4.187\)

Step 1: Understanding the Problem

We are given the acid dissociation constant, \(K_{\mathrm{a}}\), of a weak acid, which is \(6.5 \times 10^{-5}\). We need to find \(\mathrm{p}K_{\mathrm{a}}\), which is the negative logarithm (base 10) of \(K_{\mathrm{a}}\).

Step 2: Applying the Formula for \(\mathrm{p}K_{\mathrm{a}}\)

The formula to calculate \(\mathrm{p}K_{\mathrm{a}}\) is \(\mathrm{p}K_{\mathrm{a}} = -\log_{10}(K_{\mathrm{a}})\). This means we need to take the base-10 logarithm of \(6.5 \times 10^{-5}\) and then multiply the result by -1.

Step 3: Calculating the Logarithm

Using a calculator, find \(\log_{10}(6.5 \times 10^{-5})\). This operation yields approximately \(-4.187\).

Step 4: Final Calculation

Now, apply \(-1\) to the logarithm result: \(-(-4.187) = 4.187\). Thus, \(\mathrm{p}K_{\mathrm{a}} = 4.187\).

Key Concepts

Weak Acid

In chemistry, a weak acid is one that only partially dissociates in water. This means that when a weak acid dissolves in water, not all of its molecules dissociate to release hydrogen ions or protons (H⁺).
Instead, there's a balance between the undissociated and dissociated forms of the acid.
A common example of a weak acid is acetic acid, the main component of vinegar. Characteristics of weak acids include:

• They do not fully ionize in water.
• The equilibrium favors the un-ionized form.
• They have higher pKa values than strong acids.

Understanding weak acids is important when calculating pKa, a measure of the acid's strength. The less a weak acid dissociates, the smaller the value of its acid dissociation constant (Ka), and the higher the pKa.

pKa Calculation

The pKa of an acid provides insight into its acid strength in solution. It's calculated using the acid dissociation constant (Ka).
The formula for pKa is: \[ \mathrm{p}K_{\mathrm{a}} = -\log_{10}(K_{\mathrm{a}}) \] In this operation, the negative sign indicates that as the dissociation constant decreases (as in weaker acids), the pKa value increases.For example, if an acid has a Ka of \(6.5 \times 10^{-5}\), we calculate its pKa as follows:

• First, find the base-10 logarithm of the Ka value: \(\log_{10}(6.5 \times 10^{-5})\).
• This results in approximately \(-4.187\).
• Multiply the result by \(-1\) to get \(\mathrm{p}K_{\mathrm{a}} = 4.187\).

This calculation helps chemists understand how strong or weak an acid is.

Logarithmic Calculations

Logarithms are mathematical operations that are the inverse of exponentiation.
In chemistry, they are often used to solve problems involving acid dissociation constants and pH calculations.
For logarithmic calculations in the context of pKa, the base-10 logarithm, also known as the common logarithm, is utilized.To calculate the pKa, you need:

• A scientific calculator or a tool that can compute logarithms.
• The ability to apply the logarithmic function to the Ka value, such as finding \(\log_{10}(6.5 \times 10^{-5})\).

Remember that when applying the logarithmic function:

• The result will often be a negative number, reflecting the small size of Ka for weak acids.
• You will need to turn this negative result into a positive one by multiplying by \(-1\).

This process allows easy comparison of differing acid strengths through their pKa values.

Chemical Equilibrium

In the context of acids and bases, chemical equilibrium refers to the balance between the forward and reverse reactions.
This state occurs when a weak acid partially dissociates in water, creating an equilibrium between the dissociated ions and the undissociated molecules.
The position of equilibrium is quantified by the acid dissociation constant (Ka). Key points about equilibrium with weak acids:

• The equilibrium constant, Ka, gives insight into how much of the acid dissociates.
• In an aqueous solution, the system reaches a point where the concentrations of ions and undissociated acid remain constant over time.
• The weaker the acid, the farther to the left the equilibrium lies, meaning more undissociated molecules are present than dissociated ions.

This equilibrium concept is crucial for understanding acid behavior, and it's closely tied to the pKa calculation since a higher pKa value indicates a weaker acid, with equilibration leaning towards the molecular form.