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

The value of \(pK_a\) is 4.1871.

Step 1: Understand the Relationship

Recall that the relationship between the acid dissociation constant \(K_a\) and its \( pK_a \) is given by the formula: \[ pK_a = -\log(K_a) \] This relationship shows how to convert the \(K_a\) value into the \(pK_a\) value by taking the negative logarithm.

Step 2: Apply the Formula

Using the formula from Step 1, we can substitute the given \(K_a\) value:\[ pK_a = -\log(6.5 \times 10^{-5}) \] This step involves calculating the logarithm of the \(K_a\) value.

Step 3: Calculate the Logarithm

Calculate \(-\log(6.5 \times 10^{-5})\). First, find \(\log(6.5)\) which is approximately \(0.8129\), and \(\log(10^{-5}) = -5\). So,\[ \log(6.5 \times 10^{-5}) = 0.8129 - 5 = -4.1871 \]

Step 4: Determine the pK_a

To find \( pK_a \), take the negative of the result from Step 3:\[ pK_a = -(-4.1871) = 4.1871 \] Thus, the \( pK_a \) value is the positive version of \(-4.1871\).

Key Concepts

Acid Dissociation Constant

The acid dissociation constant, denoted as \( K_a \), measures the strength of an acid in a solution. It represents the equilibrium constant for the dissociation of a weak acid into its ions. For a general weak acid, \( HA \), the dissociation can be represented as:\[ HA \rightleftharpoons H^+ + A^- \]The equilibrium expression for this reaction is:\[ K_a = \frac{[H^+][A^-]}{[HA]} \]Where:

• \([H^+]\) is the concentration of hydrogen ions.
• \([A^-]\) is the concentration of the anion.
• \([HA]\) is the concentration of the undissociated acid.

The higher the \( K_a \), the stronger the acid, meaning it dissociates more in solution. Conversely, a lower \( K_a \) indicates a weaker acid, which is less dissociated.

Logarithmic Conversion

Logarithmic conversion is essential in chemistry to simplify large ranges of numbers into a manageable form. When dealing with \( K_a \), which can vary greatly in magnitude, we use the \( pK_a \) scale. This conversion involves calculating the negative logarithm of the \( K_a \):\[ pK_a = -\log(K_a) \]This negative logarithm scale is useful because it turns very small numbers (like \( 10^{-5} \) for weak acids) into more straightforward numbers (like 5) for easy comparison.

• \( \log(6.5) \) gives you a value, for example, approximately 0.8129.
• \( \log(10^{-5}) = -5 \), turning a very small exponent into a simple negative integer.

After finding \( \log(K_a) \), we take its negative to obtain the \( pK_a \). This simplifies our understanding of an acid’s strength using these concise, comparable values.

Weak Acid

A weak acid is an acid that does not completely dissociate in a solution. It is characterized by a low \( K_a \) value, indicating partial ionization in water. For instance, imagine a weak acid like acetic acid:\[ CH_3COOH \rightleftharpoons H^+ + CH_3COO^- \]The equilibrium nature means that both the acid and its ions are present in significant amounts in the solution. Here are some key aspects of weak acids:

• They have high \( pK_a \) values due to their small \( K_a \) values. For instance, the example in the exercise shows a \( pK_a \) value of 4.1871, typical for weak acids.
• Because they don’t release many \( H^+ \) ions, their solutions have higher pH values compared to strong acids.
• In practical terms, this means weak acids are less aggressive and are often used as food preservatives or in buffer solutions due to their stability and mildness.

Understanding weak acids is crucial for studying equilibrium and reaction rates in chemistry.