Question

Acid and base behavior can be observed in solvents other than water. One commonly used solvent is dimethyl sulfoxide (DMSO), which can be treated as a monoprotic acid \({ }^{\circ}\) HSol." Just as water can behave cither as an acid or a base, so HSol can behave either as a Bronsted-Lowry acid or base. (a) The equilibrium constant for self-dissociation of HSol (call it \(K_{\text {HSel }}\) ) is \(1 \times 10^{-35}\). Write the chemical equation for the self- dissociation reaction and the corresponding equilibrium equation. (Hint: The equilibrium equation is analogous to the equilibrium equation for \(K_{w}\) in the case of water.) (b) The weak acid HCN has an acid dissociation constant \(K_{a}=1.3 \times 10^{-13}\) in the solvent HSol. If \(0.010 \mathrm{~mol}\) of \(\mathrm{NaCN}\) is dissolved in \(1.00 \mathrm{~L}\) of \(\mathrm{HSol}\), what is the equilibrium concentration of \(\mathrm{H}_{2} \mathrm{Sol}^{+}\) ?

Short answer

The equilibrium concentration of \([\text{H}_2\text{Sol}^+]\) is \(1.3 \times 10^{-13}\, \text{M}.\)

Step-by-step solution

Write the self-dissociation reaction

The self-dissociation of dimethyl sulfoxide (HSol) is analogous to the autoionization of water. The reaction can be written as: \[\text{HSol} + \text{HSol} \rightleftharpoons \text{H}_2\text{Sol}^+ + \text{Sol}^-\]

Write the equilibrium expression for the self-dissociation

The equilibrium constant for the self-dissociation of HSol, \(K_{\text{HSol}}\), is defined by the concentrations of the ions in solution:\[K_{\text{HSol}} = [\text{H}_2\text{Sol}^+][\text{Sol}^-]\]

Setup the equilibrium expression for HCN in HSol

Given that the weak acid HCN has an acid dissociation constant in HSol, \(K_a = 1.3 \times 10^{-13}\), the dissociation reaction is:\[\text{HCN} + \text{Sol}^- \rightleftharpoons \text{CN}^- + \text{H}_2\text{Sol}^+\]The equilibrium expression is:\[K_a = \frac{[\text{CN}^-][\text{H}_2\text{Sol}^+]}{[\text{HCN}]}\]

Determine initial concentrations

Initially, when \(0.010\ \text{mol NaCN}\) is dissolved in \(1.00\ \text{L}\) of HSol, \(\text{CN}^-\) concentration is \(0.010\ \text{M}\), and \([\text{H}_2\text{Sol}^+]\) is assumed negligible.

Apply ICE table for the equilibrium

Setting up an ICE (Initial, Change, Equilibrium) table for the dissociation of HCN:- Initial concentrations: - \([\text{CN}^-] = 0.010\ \text{M}\) - \([\text{H}_2\text{Sol}^+] = 0\ \text{M}\) - \([\text{HCN}] = 0\ \text{M}\) since \(\text{NaCN}\) is a salt providing \(\text{CN}^-\).- Change: - \([\text{H}_2\text{Sol}^+] = +x\) - \([\text{CN}^-] = 0.010 - x\ \text{M}\) assuming \(x\) is small- Equilibrium: - Equilibrium \(\text{HCN}\) = 0 approximately

Solve for equilibrium concentration of \([\text{H}_2\text{Sol}^+]\)

Since \([\text{CN}^-] = 0.010\ - x\), assume \(0.010 - x \approx 0.010\):Substitute into \(K_a = \frac{x\times 0.010}{0.010}\)Thus, \(1.3 \times 10^{-13} = x\)Therefore, \([\text{H}_2\text{Sol}^+] = 1.3 \times 10^{-13}\ \text{M}\)

Key concepts

Dimethyl Sulfoxide (DMSO)

Dimethyl sulfoxide, often abbreviated as DMSO, is a versatile solvent that can dissolve both polar and nonpolar compounds. It is widely used in chemical reactions involving organic and inorganic substances. In the context of acid-base equilibrium, DMSO is frequently referred to as a monoprotic substance, indicated by the shorthand HSol. This signifies its potential to either donate or accept a proton, mimicking the behavior of water in acid-base reactions.

Characteristics of DMSO include:

• Strong solvation ability for a broad range of substances
• High boiling point making it suitable for reactions at elevated temperatures
• Low toxicity compared to other organic solvents, allowing for safer handling

When examining acids and bases in DMSO, it's important to consider its dual role, which leads us directly into the next concept, self-dissociation, just as water does.

Self-dissociation

Self-dissociation refers to the phenomenon where a solvent molecule reacts with itself to form ionic species. With water, this is commonly presented as the auto-ionization of water to form hydroxide (OH⁻) and hydronium (H₃O⁺) ions. Similarly, in DMSO, self-dissociation occurs as follows:

The chemical equation for the self-dissociation of DMSO is:\[\text{HSol} + \text{HSol} \rightleftharpoons \text{H}_2\text{Sol}^+ + \text{Sol}^-\]
This reaction illustrates how two DMSO molecules react to produce an ionic pair. Though the tendency for DMSO to self-dissociate is extremely low, as highlighted by its equilibrium constant, it still provides a path for acid-base chemistry in this solvent. Understanding this behavior is crucial when predicting the results of reactions in solvents other than water.

Equilibrium Constant

The equilibrium constant, denoted as \(K\), is a numerical value that represents the ratio of the concentrations of products to reactants at equilibrium. This is a key concept in understanding the behavior of substances in solution. For the case of DMSO self-dissociation, the equilibrium constant \(K_{\text{HSol}}\) can be expressed as:

\[K_{\text{HSol}} = [\text{H}_2\text{Sol}^+][\text{Sol}^-]\]
This equation dictates that at equilibrium, the product of the concentrations of the ions formed, \([\text{H}_2\text{Sol}^+]\) and \([\text{Sol}^-]\), remains constant. Given the extremely small value of \(1 \times 10^{-35}\), we can deduce that the concentration of these ions is negligible, reinforcing the neutrality of DMSO under normal conditions.

Recognizing how equilibrium constants work helps in predicting how reactions will proceed to achieve balance, whether in water, DMSO, or other solvents.

Bronsted-Lowry Acid and Base

The Bronsted-Lowry theory is a fundamental concept in chemistry which states that acids are proton (H⁺) donors, while bases are proton acceptors. In the context of DMSO, when it is referenced as HSol, it exemplifies a Bronsted-Lowry system where it can act as either an acid or a base, similar to water's dual nature.

Bronsted-Lowry behavior in DMSO:

• **As an acid:** HSol can donate a proton to form \(\text{Sol}^-\).
• **As a base:** HSol can accept a proton to form \(\text{H}_2\text{Sol}^+\).

This functionality allows for a wide range of acid-base reactions to occur in DMSO, encompassing both the donation and acceptance of protons. This concept is pivotal in preparing reactions for synthetic chemistry processes, offering alternative pathways when water-based mechanisms are not suitable.