Question

Given the two concentration of \(\mathrm{HCN}\left(\mathrm{K}_{a}=10^{-9}\right)\) are \(0.1 \mathrm{M}\) and \(0.001 \mathrm{M}\) respectively. What will be the ratio of degree of dissociation? (a) 1 (b) \(0.1\) (c) \(0.003\) (d) \(0.01\)

Short answer

The short answer is (b) 0.1.

Step-by-step solution

Write the equilibrium expression for HCN

The dissociation of hydrocyanic acid (HCN) can be represented as: HCN(aq) ⇌ H⁺(aq) + CN⁻(aq). The equilibrium constant expression (Ka) is given by Ka = [H⁺][CN⁻]/[HCN].

Express the degree of dissociation

Let the degree of dissociation for the 0.1 M solution be α1 and for the 0.001 M solution be α2. At equilibrium, [H⁺] = [CN⁻] = α[C] and [HCN] = C - α[C], where C is the initial concentration of HCN and α is the degree of dissociation.

Write the Ka expression for each concentration

For the 0.1 M HCN solution, Ka = α1²(0.1M) / (1 - α1)(0.1M). Simplify this to Ka = α1² / (1 - α1). For the 0.001 M HCN solution, similarly, Ka = α2² / (1 - α2). Since Ka is constant for a given acid at a particular temperature, and for weak acids α << 1, we can assume (1 - α) ≈ 1 thus, Ka = α²/C.

Find the ratio of the degrees of dissociation (α1/α2)

Using the approximation from Step 3 for both concentrations: Ka = α1² / 0.1 = α2² / 0.001. Rearrange to find the ratio α1/α2 = √(0.001/0.1) = √(1/100) = 1/10.

Key concepts

Acid Dissociation Constant

In chemistry, the acid dissociation constant, denoted as Ka, is a quantitative measure of the strength of an acid in solution. Specifically, it reflects the acid's tendency to donate a proton (hydrogen ion) to a base. The dissociation of weak acids, like hydrocyanic acid (HCN), doesn't go to completion, which means that not all of the acid molecules dissociate to form hydrogen ions and conjugate base ions in solution. Instead, these acids establish a chemical equilibrium.

For HCN, the dissociation can be represented by the reaction: HCN(aq) ⇌ H⁺(aq) + CN⁻(aq). The equilibrium condition is described mathematically by the equation: Ka = \( \dfrac{[H⁺][CN⁻]}{[HCN]} \).

The higher the value of Ka, the stronger the acid, because it means a greater concentration of H⁺ and CN⁻ ions relative to the undissociated HCN. In the case of hydrocyanic acid, Ka is extremely low (\(10^{-9}\)), indicating that it's a very weak acid and only dissociates to a small extent under typical conditions. Understanding Ka is crucial since it tells us about the propensity of an acid to lose a proton, which is a central aspect in many chemical reactions, particularly those in biological systems.

Chemical Equilibrium

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant over time. This does not imply that the reactants and products are in equal concentrations, but rather that their ratios do not change. In the context of acid dissociation, the equilibrium condition provides a snapshot of the dynamic balance between the undissociated acid molecules and the ions produced by their dissociation.

For our example with hydrocyanic acid, it's essential to note that at equilibrium, the concentrations used in the Ka expression are those at this balanced state. During this equilibrium, the forward reaction (dissociation of HCN to form H⁺ and CN⁻) is happening at the same rate as the reverse reaction (recombination of H⁺ and CN⁻ to form HCN), creating a constant ratio of products to reactants, described by the Ka value.

Dissociation of Weak Acids

Dissociation of weak acids is a partial process, where the acid partially ionizes in water. A weak acid has a low dissociation constant (Ka), meaning that it exists primarily in the form of undissociated molecules in solution. When a weak acid like HCN dissociates, it produces a relatively small amount of hydrogen ions and its conjugate base ions, establishing an equilibrium as mentioned earlier.

The degree of dissociation (α) signifies the fraction of the total acid that has dissociated. In our problem, HCN dissociates as: HCN(aq) ⇌ H⁺(aq) + CN⁻(aq). For solutions of different initial concentrations, the degree of dissociation varies, usually increasing as the concentration decreases due to the common ion effect and the shift in equilibrium dynamics.

In dilute solutions, the factor (1 - α), which represents the concentration of undissociated acid, is approximately 1 because α is very small. Therefore, the degree of dissociation is inversely proportional to the square root of the concentration, leading to a higher extent of dissociation in more dilute solutions, which in turn affects the pH and the buffer capacity of the solution.