Problem 34 - Basic Principles of Organic Chemistry

Question

The ionization constants of 3- and 4-cyanobenzoic acids at \(30^{\circ}\) are \(2.51 \times 10^{-4}\) and \(2.82 \times 10^{-4}\), respectively. Benzoic acid has \(K_{a}\) of \(6.76 \times 10^{-5}\) at \(30^{\circ} .\) Calculate \(\sigma_{\text {meta }}\) and \(\sigma_{\text {para }}\) for the cyano substituent.

Step-by-Step Solution

Answer

\( \sigma_{\text{meta}} \approx 0.569 \) and \( \sigma_{\text{para}} \approx 0.620 \).

Step 1: Understand the Hammett Equation

The Hammett equation relates the acidity of a substituted benzoic acid to its substituent constant \( \sigma \). It is given by the formula:\[ \log \left(\frac{K_a}{K_0}\right) = \rho \cdot \sigma \]where \( K_a \) is the ionization constant of the substituted acid, \( K_0 \) is the ionization constant of unsubstituted benzoic acid, and \( \rho \) is the reaction constant which is typically 1 for meta and para positions when calculating substituent constants.

Step 2: Calculate \( \sigma_{\text{meta}} \)

For 3-cyanobenzoic acid (meta position), use the Hammett equation to find \( \sigma_{\text{meta}} \):\[ \log \left(\frac{K_{a, \text{meta}}}{K_{0}}\right) = \sigma_{\text{meta}} \]Substitute the values:\[ \log \left(\frac{2.51 \times 10^{-4}}{6.76 \times 10^{-5}}\right) = \sigma_{\text{meta}} \]Calculate the ratio within the logarithm:\[ \frac{2.51 \times 10^{-4}}{6.76 \times 10^{-5}} \approx 3.712 \]Then take the logarithm:\[ \sigma_{\text{meta}} = \log(3.712) \approx 0.569 \]

Step 3: Calculate \( \sigma_{\text{para}} \)

For 4-cyanobenzoic acid (para position), use the same formula:\[ \log \left(\frac{K_{a, \text{para}}}{K_{0}}\right) = \sigma_{\text{para}} \]Substitute the values:\[ \log \left(\frac{2.82 \times 10^{-4}}{6.76 \times 10^{-5}}\right) = \sigma_{\text{para}} \]Calculate the ratio:\[ \frac{2.82 \times 10^{-4}}{6.76 \times 10^{-5}} \approx 4.173 \]Then find the logarithm:\[ \sigma_{\text{para}} = \log(4.173) \approx 0.620 \]

Key Concepts

Substituent Constant

The substituent constant, symbolized as \( \sigma \), is a numerical value that tells us about the electronic effect of a substituent on an aromatic ring. It is an essential component of the Hammett equation, which relates chemical structure to reactivity. In the context of acidity, a higher substituent constant means a stronger electron-withdrawing effect, which tends to increase acidity by stabilizing the carboxylate ion formed after proton donation.

These constants are calculated using the known ionization constants of substituted and unsubstituted benzoic acids. Specifically, the substituent constant reflects how much the ionization constant of the acid changes when a particular group is attached to its aromatic ring. This change occurs due to a difference in electron distribution within the molecule.

By finding \( \sigma \) values for different positions, such as meta or para positions, you can predict and explain the behavior of various chemical reactions involving substituted benzoic acids.

Ionization Constant

The ionization constant (\( K_a \)) provides a measure of the acidity of a substance. It indicates the extent to which an acid can donate protons (\( H^+ \)) to water to form hydronium ions (\( H_3O^+ \)). The larger the \( K_a \), the stronger the acid, as it shows a greater ability to donate protons.

In the context of the Hammett equation, the ionization constants for substituted and unsubstituted benzoic acids are compared. For example, in this exercise, we calculate \( \sigma \) for cyano-substituted benzoic acids by comparing their \( K_a \) with that of plain benzoic acid.

The Benchmarks are:

• The ionization constant for benzoic acid: \( 6.76 \times 10^{-5} \)
• Meta-cyano-substituted: \( 2.51 \times 10^{-4} \)
• Para-cyano-substituted: \( 2.82 \times 10^{-4} \)

Calculations of \( \sigma_{\text{meta}} \) and \( \sigma_{\text{para}} \) involve taking the logarithm of the ratio of these ionization constants. This provides a quantitative measure of substituent effects on acidity when placed at different positions on the benzene ring.

Meta and Para Positions

In benzene ring chemistry, substituents can be attached to various positions, and these positions significantly influence the behavior of the molecule. The positions relevant to the Hammett equation are the meta and para positions, denoted as the 3- and 4-positions on the benzene ring.

When comparing substituents at these positions:

• Meta (3-position): This position is separated by two carbons from the primary reactive site (like a carboxyl group). Substituents here can influence the acidity moderately.
• Para (4-position): This position is directly opposite the reactive group on the benzene ring. It often has a more pronounced electronic effect due to resonance, affecting the acidity comparison significantly.

Depending on whether a substituent is electron-withdrawing or donating, these effects will either stabilize or destabilize the resultant ions. The para position often allows greater conjugation and resonance, leading to a stronger display of electronic effects, which is why \( \sigma_{\text{para}} \) might differ from \( \sigma_{\text{meta}} \). Understanding these differences is crucial in predicting and explaining chemical reactivity.