Problem 38 - Chemistry: Principles and Reactions

Question

Consider these acids $$\begin{array}{lllll}\hline \text { Acid } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} \\\K_{\mathrm{a}} & 1.6 \times 10^{-3} & 9 \times 10^{-4} & 2 \times 10^{-6} & 3 \times 10^{-4} \\\\\hline\end{array}$$ (a) Arrange the acids in order of increasing acid strength from weakest to strongest. (b) Which acid has the smallest \(\mathrm{pK}_{\mathrm{a}}\) value?

Step-by-Step Solution

Answer

Question: Arrange the given acids in order of increasing strength (weakest to strongest) and identify the acid with the smallest \(pK_a\) value. Given \(K_a\) values: Acid A: \(1.6 \times 10^{-3}\) Acid B: \(9 \times 10^{-4}\) Acid C: \(2 \times 10^{-6}\) Acid D: \(3 \times 10^{-4}\) Answer: The acids arranged in order of increasing strength are Acid C, Acid B, Acid D, and Acid A. Acid A has the smallest \(pK_a\) value.

Step 1: Compare \(K_a\) values

For each acid, examine its \(K_a\) value and compare it to the others. Remember, higher \(K_a\) values correspond to stronger acids. $$ \begin{array}{lllll}\hline \text { Acid } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} \\ K_{a} & 1.6 \times 10^{-3} & 9 \times 10^{-4} & 2 \times 10^{-6} & 3 \times 10^{-4}\\ \hline\end{array} $$

Step 2: Order acids from weakest to strongest

Based on the given \(K_a\) values, arrange the acids in order of increasing acid strength by their \(K_a\) values. The weakest acids will have the smallest \(K_a\) values, while the strongest acids will have the largest \(K_a\) values. Answer: (\(\mathrm{C}\), \(\mathrm{B}\), \(\mathrm{D}\), \(\mathrm{A}\)) #b) Finding the acid with the smallest \(pK_a\) value#

Step 1: Convert \(K_a\) to \(pK_a\) values

To answer this question, we need to find the \(pK_a\) values of each acid. This can be done using the formula: $$ pK_a = -\log_{10}K_a $$ Compute the \(pK_a\) values for each acid: Acid \(\mathrm{A}\): \(pK_a = -\log_{10}(1.6 \times 10^{-3})\) Acid \(\mathrm{B}\): \(pK_a = -\log_{10}(9 \times 10^{-4})\) Acid \(\mathrm{C}\): \(pK_a = -\log_{10}(2 \times 10^{-6})\) Acid \(\mathrm{D}\): \(pK_a = -\log_{10}(3 \times 10^{-4})\)

Step 2: Identify the acid with the smallest \(pK_a\) value

Compare the \(pK_a\) values calculated in the previous step and identify the acid with the smallest \(pK_a\) value. Remember, a smaller \(pK_a\) value means a stronger acid. Answer: Acid \(\mathrm{A}\) has the smallest \(pK_a\) value

Key Concepts

Ka values

The strength of an acid is an important concept in chemistry, and one way to measure this is by using the acid dissociation constant, known as \( K_a \). This constant reflects the degree to which an acid can dissociate in water. The equation for \( K_a \) is based on the expression for the chemical equilibrium of the acid \( \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \), detailing the dissociation of the acid. An acid with a large \( K_a \) value will dissociate more completely in water, thus releasing more \( \text{H}^+ \) ions, which indicates a stronger acid.

• The higher the \( K_a \), the stronger the acid, as it shows more significant dissociation into its ions in aqueous solutions.
• When comparing acids, it's crucial to look at these \( K_a \) values to determine their relative strengths.

Understanding \( K_a \) is essential not only for academic purposes but also for practical applications in predicting the behavior of chemical substances in various environments.

pKa values

The \( pK_a \) value is a more intuitive and handy representation of an acid’s strength. The \( pK_a \) is the negative base-10 logarithm of the \( K_a \) value:\( pK_a = -\log_{10}(K_a) \). This transformation turns the often unwieldy numbers of \( K_a \) into more manageable figures.

• A lower \( pK_a \) value means a stronger acid, correlating to a higher \( K_a \) value.
• Conversely, a higher \( pK_a \) indicates a weaker acid.

The use of \( pK_a \) provides a straightforward way to compare different acids. By examining \( pK_a \) values, chemists can easily identify which acids are stronger and propose their potential for being proton donors in reactions.

Acid Strength Comparison

When comparing the strength of acids, it's essential to utilize both \( K_a \) and \( pK_a \) values. The problem here originally requires comparing acids based on their \( K_a \) values.

• Acid strength directly relates to the \( K_a \) value: the larger the \( K_a \), the stronger the acid.
• When converting to \( pK_a \) values, remember that a smaller \( pK_a \) indicates a stronger acid.

For example, among acids A, B, C, and D, acid A with the highest \( K_a \) value of \( 1.6 \times 10^{-3} \) is the strongest. Converting these to \( pK_a \) values will affirm that acid A is strongest due to its smallest \( pK_a \), thereby making comparisons more accessible and clear, as acids are ranked from weak to strong based on their numerical \( pK_a \) order.

Logarithmic Calculations

Logarithmic calculations are fundamental in converting \( K_a \) values to \( pK_a \) values. The formula \( pK_a = -\log_{10}(K_a) \) is straightforward but requires understanding logarithms.

• Logarithms help compress a wide range of values into a manageable scale, which is a particularly useful tool in chemistry.
• For instance, since a \( K_a \) value might be incredibly small like \( 2 \times 10^{-6} \), its logarithmic counterpart \( pK_a \) simplifies comparison across a broad spectrum of acids.

To work through this in practice, calculate each \( pK_a \) from the given \( K_a \): for Acid C, with \( K_a = 2 \times 10^{-6} \), you compute \( pK_a = -\log_{10}(2 \times 10^{-6}) \). By utilizing logarithms, students and chemists can better understand and communicate the relative strengths of acids in a distilled and accessible format.