Question

The \(\mathrm{pH}\) of a solution of salt of strong acid and weak base is given by a) \(\frac{1}{2}\left(\log K_{w^{\prime}}+\log K_{b}+\log C\right)\) (b) \(\frac{1}{2}\left(\log K_{w}-\log K_{b}-\log C\right)\) c) \(\frac{1}{2}\left(\log K_{w^{\prime}}-\log K_{b}+\log C\right)\) d) \(\frac{1}{2}\left(-\log K_{w^{\prime}}-\log K_{b}-\log C\right)\)

Short answer

The correct formula to calculate the pH of a salt solution of a strong acid and a weak base is \(\frac{1}{2}\left(\log K_{w}-\log K_{b}-\log C\right)\)

Step-by-step solution

Understanding the concept of pH

pH is a scale used to specify the acidity or basicity of an aqueous solution. It is the negative logarithm (base 10) of the concentration of hydronium ions (H+ or H3O+ ions). It is given by the formula \(-\log [H^+]\) or \(-\log [H_3O^+]\). Acids have a pH less than 7, bases have a pH greater than 7, and pH 7 indicates neutral solutions.

Understanding the terms in the given equations

In the given equations: \(K_w\)' is the ion product for water, \(K_b\) is the base ionization constant and \(C\) is the concentration of the salt.

Using logic and understanding of ph concept to choose the correct answer

According to the formula of pH (-log[H+]), the terms in the formula should not be added because the logarithmic properties state that the addition of log turns into multiplication, which makes no sense in the context of the problem. So the correct choice will not involve the sum of all logs. Thus, the correct answer is (b) \(\frac{1}{2}\left(\log K_{w}-\log K_{b}-\log C\right)\)

Key concepts

pH of Salt Solutions

The pH of a salt solution can be intriguing because it is not as straightforward as that of pure acids or bases. When salts are dissolved in water, they may react with the water in a process known as hydrolysis, influencing the pH. Salts formed from a strong acid and a weak base tend to make the solution acidic. This is because the anions of the weak base hydrolyze in water to produce OH^- ions and the cation of the strong acid does not react with water.
This results in an increase in the concentration of hydrogen ions (H⁺), which lowers the pH. To calculate the pH of such solutions, chemists derive equations that relate the ion product for water (K_w), the base ionization constant (K_b), and the salt concentration (C). Remember, pH is a logarithmic measure; therefore, the concentration terms in the pH calculation of salt solutions are expressed in terms of their logarithmic values, and the correct formula would involve subtracting these logarithmic terms, as indicated in the provided exercise solution.
When working with hydrolysis reactions, it's also essential to consider the stoichiometry and the equilibrium constants to determine the exact variation in pH.

Ion Product for Water (Kw)

The ion product for water (K_w) is a fundamental constant in acid-base chemistry. It represents the equilibrium concentration of hydrogen ions (H⁺) and hydroxide ions (OH^-) in pure water at a given temperature. At 25°C, K_w is always 1.0 x 10^-14. This value is crucial because it helps us understand and predict the behavior of aqueous solutions.
K_w is derived from the self-ionization of water: H₂O(l) → H⁺(aq) + OH^-(aq). Whenever you dissolve a salt in water, the equilibrium may shift, but the product of [H⁺] and [OH^-] at a given temperature will always be equal to K_w. In calculations, the logarithmic form of K_w is often used, specifically when working with pH, because pH is also defined logarithmically. Understanding the ion product for water is paramount for solving many problems in acid-base chemistry, including the calculation of pH in various types of solutions.

Base Ionization Constant (Kb)

The base ionization constant (K_b) is an equilibrium constant that measures the strength of a base in water. It reflects how well a base dissociates to produce hydroxide ions (OH^-) and the corresponding cation in an aqueous solution. The larger the value of K_b, the stronger the base. In essence, K_b is the ratio of the concentration of the ionized form of the base to the concentration of the non-ionized form.
For weak bases, which do not fully dissociate in water, K_b is especially important to determine the degree of ionization. K_b values also play a critical role in pH calculations for solutions containing weak bases, including salt solutions derived from such bases. When calculating pH, the K_b of the weak base from the salt must be factored in and is present in the logarithmic term in the pH formula referenced in the given exercise.

Acid-Base Chemistry

Acid-base chemistry revolves around the study of acids and bases and their reactions in an aqueous environment. The fundamental concept includes the transfer of protons (H⁺) from acids to bases. The pH scale is a measure of the concentration of H⁺ ions in a solution and thus provides insights into the solution’s acidity or basicity.
One key principle is the strength of acids and bases, defined by their tendency to donate or accept protons, respectively, and quantified by their dissociation constants (K_a for acids and K_b for bases). The nature of a salt's constituent acid and base determines the acidic, neutral, or basic character of the solution. For example, salts from strong acids and strong bases tend to be neutral.
Moreover, the principles of equilibrium apply when acids or bases are dissolved in water. Le Chatelier’s principle, for instance, explains how the equilibrium shifts in response to changes in concentration, temperature, or pressure. A comprehensive understanding of acid-base chemistry is paramount for various applications, including the calculation of pH in different chemical contexts.