Problem 234 - Introductory Chemistry Atoms First

Question

A quantity often used to characterize the acidity of an acid is its \(\mathrm{p} K_{\mathrm{a}}\). The \(\mathrm{pK}_{\mathrm{a}}\) of an acid is equal to \(-\log \left(K_{\mathrm{a}}\right)\). Knowing this, what can you say about acid #1 whose \(\mathrm{pK}_{\mathrm{a}}=2\), and acid \(\\# 2\) whose \(\mathrm{p} K_{\mathrm{a}}=6 ?\) Which is more acidic and by how much? (Hint: if \(\mathrm{pK}_{\mathrm{a}}=-\log \left(K_{\mathrm{a}}\right)\), then \(\left.K_{\mathrm{a}}=10^{-\mathrm{pKa}} .\right)\)

Step-by-Step Solution

Answer

Acid #1 is more acidic than Acid #2. Using the given pKa values and the formula to calculate Ka values, we find that Acid #1 has a Ka value of \(10^{-2}\) and Acid #2 has a Ka value of \(10^{-6}\). The higher Ka value indicates a more acidic nature, so Acid #1 is more acidic. To find out by how much, we compute the ratio of their Ka values: \(\frac{10^{-2}}{10^{-6}} = 10^{4}\). Thus, Acid #1 is 10,000 times more acidic than Acid #2.

Step 1: The formula for pKa and Ka are given as follows: \( pKa = -\log(Ka) \) and \( Ka = 10^{-pKa} \) #Step 2: Calculate Ka for both acids#

We are given the pKa values for Acid #1 (pKa = 2) and Acid #2 (pKa = 6). Using the formula for Ka, we can find out their Ka values. For Acid #1: \( Ka_{1} = 10^{-pKa_{1}} = 10^{-2} \) For Acid #2: \( Ka_{2} = 10^{-pKa_{2}} = 10^{-6} \) #Step 3: Compare Ka values to determine the acidic nature of the acids#

Step 2: The acid with the higher Ka value will be more acidic. Let's compare the calculated Ka values for Acid #1 and Acid #2. Since \( Ka_{1} = 10^{-2} \) and \( Ka_{2} = 10^{-6} \), we can clearly see that \( Ka_{1} > Ka_{2} \). This means that Acid #1 is more acidic than Acid #2. #Step 4: Determine the difference in acidity between the two acids#

To find out by how much Acid #1 is more acidic than Acid #2, we can calculate the ratio of their Ka values: Ratio of Ka values = \( \frac{Ka_{1}}{Ka_{2}} = \frac{10^{-2}}{10^{-6}} = 10^{4} \) This indicates that Acid #1 is 10^4 or 10,000 times more acidic than Acid #2.

Key Concepts

Acid Strength

The acid strength of a substance is a key characteristic in chemistry, indicating its ability to donate protons (H+) in an aqueous solution. A stronger acid readily donates its protons, leading to a higher concentration of hydrogen ions (H+) in a solution, which translates to a lower pH value. The strength of an acid is quantified by its dissociation constant, known as the Ka value. The larger the Ka, the stronger the acid, as it implies a greater degree of ionization.

Understanding the strength of an acid is crucial for predicting the behavior of chemical reactions, especially those involving acid-base equilibria. Stronger acids typically have a more pronounced impact on chemical equilibrium, which can be essential for processes ranging from industrial synthesis to biological metabolism.

pKa Calculation

Calculating the pKa of an acid is an essential process for understanding its acidity. The pKa is the negative base-10 logarithm of the acid dissociation constant (Ka). It can be represented mathematically as:

\[ pKa = -\log(Ka) \]
This calculation provides a more manageable number that inversely reflects the acid strength—the lower the pKa value, the stronger the acid and vice versa. In practice, chemists often use pKa values for practical reasons, such as the ease of comparing the relative strength of acids without dealing with very large or small numbers that are involved in Ka values.

Ka Value

The acid dissociation constant, represented as Ka, is an equilibrium constant that measures the strength of an acid in solution. It is defined by the extent to which an acid dissociates into its ions. The general expression for the Ka of an acid, HA, is given by:

\[ Ka = \frac{[H^+][A^-]}{[HA]} \]
In this equation, [H+] is the concentration of hydrogen ions, [A-] is the concentration of the conjugate base, and [HA] is the concentration of the undissociated acid. A higher Ka value indicates a stronger acid as it implies a greater concentration of hydrogen ions, suggesting that the acid ionizes more completely in solution. Ka values are typically very small for weak acids, necessitating the use of the pKa form for more convenient comparison.

Acidity Comparison

Comparing the acidity of different substances is fundamental in understanding their chemical behavior. To compare acidity, one can use either the Ka or pKa values. As shown in the earlier exercise, when comparing acids with different pKa values, it is clear that a lower pKa represents a stronger acid, and a higher pKa indicates a weaker acid.

For example, if an acid has a pKa of 2 and another has a pKa of 6, it means that the first acid is significantly more acidic than the second. Utilizing the relationship between Ka and pKa, one can calculate the relative strength. The difference in acidity can be expressed as a ratio of their Ka values, which, in this case, reveals the first acid to be 10,000 times more acidic than the second. This ratio is instrumental in predictive modeling for chemical reactions involving acids.