The \(\mathrm{pH}\) of \(0.1 \mathrm{M}\) solution of cyanic acid (HCNO) is \(2.34\). Calculate the ionization constant of the acid and its degree of ionization in the solution.

Understanding the concept of pH is crucial when studying chemistry, particularly in the context of acids and bases. The pH is a measure of the acidity or alkalinity of a solution, expressed on a scale from 0 to 14, with 7 as neutral. Acids have pH values less than 7, and bases have values greater than 7. The pH of a solution is effectively the negative base-10 logarithm of the hydrogen ion concentration, denoted as \( [H^+] \). So mathematically, we can express this relationship with the formula \( \text{pH} = -\log([H^+]) \).

To calculate the pH from the [H+], you simply take the negative logarithm of the [H+] value. Conversely, to find the [H+] from the pH, you use the inverse of the logarithm function, which can be written as \([H^+] = 10^{-\text{pH}}\). This inverse log calculation is a foundational skill for many chemistry problems, including those involving ionization constants of weak acids.

Weak acids, such as cyanic acid (HCNO), do not fully dissociate in water, as strong acids do. Instead, they partially ionize, reaching an equilibrium between the undissociated acid and the ions produced. The ionization of a weak acid in water can be represented by a reversible chemical equation like \(\text{HA} \leftrightarrow H^+ + A^-\) where \(\text{HA}\) is the weak acid. At equilibrium, because the concentrations of \(H^+\) and \(A^-\) are equal, the dissociation can be described using an ionization constant expression, known as \(K_a\).

For instance, with cyanic acid in water, the reaction would be \(\text{HCNO} \leftrightarrow H^+ + \text{CNO}^-\). This equilibrium illustrates the essence of weak acid behavior—reluctance to release \(H^+\) ions—and is critical for understanding acid strength and the calculation of the ionization constant.

The degree of ionization is a term used to describe the extent to which an acid or base ionizes in solution. It's represented mathematically as the ratio of the concentration of ionized acid (or base) to the original concentration of the acid (or base) before ionization began. For example, if you have a weak acid like HCNO in aqueous solution, the degree of ionization (representing with the variable \(x\)) is the fraction of the original HCNO molecules that have given up their \(H^+\) ions to form \(CNO^-\).

The actual calculation of the degree of ionization involves determining \(x\), the concentration of \(H^+\) and \(CNO^-\) at equilibrium, and then dividing by the initial concentration of HCNO. Often, this calculation is simplified because the value of \(x\) is typically quite small for weak acids, thereby allowing us to make the approximation that the concentration of un-ionized acid remains roughly the same as the initial concentration, simplifying further computations.

The equilibrium constant expression for a weak acid's ionization, represented by \(K_a\), quantifies the acid's strength. It is determined by the concentrations of the reactants and products at equilibrium. Specifically, for the ionization of a weak acid \(\text{HA}\), the expression is \(K_a = \frac{[H^+][A^-]}{[HA]}\).

To calculate \(K_a\), you generally need the equilibrium concentrations of each species involved in the reaction. However, because the degree of ionization for weak acids is low, the equilibrium concentration of the un-ionized acid [HA] can be approximated as remaining constant, simplifying to \(K_a \approx \frac{x^2}{0.1}\), where \(x\) represents the concentrations of \(H^+\) and \(A^-\) produced by the acid's ionization. This allows us to calculate \(K_a\) directly from the pH of the solution and the initial concentration of the acid, streamlining the process for assessing the acid's strength.