A typical sample of vinegar has a pH of \(3.0\). Assuming that vinegar is only an aqueous solution of acetic acid \(\left(K_{\mathrm{a}}=1.8 \times\right.\) \(10^{-5}\) ), calculate the concentration of acetic acid in vinegar.

Understanding the pH of vinegar is crucial when we study its acidic properties. The pH scale is used to determine the acidity or basicity of an aqueous solution and ranges from 0 to 14. A pH of 7 is considered neutral, while a pH less than 7 indicates an acidic solution, and a pH greater than 7 indicates a basic solution.

Vinegar, which is primarily a dilute solution of acetic acid in water, typically has a pH around 3.0. This means vinegar is quite acidic. The pH value is used to calculate the hydrogen ion concentration \( [\mathrm{H^+}] \) in the solution, which is a direct measure of its acidity. The concentration of hydrogen ions is inversely related to the pH value and can be calculated using the equation \( [\mathrm{H^+}] = 10^{-\mathrm{pH}} \).

In the context of acetic acid in vinegar, knowing the pH allows us to deduce how much the acid has dissociated in the solution. It is this dissociation that gives vinegar its acidic properties and determines the concentration of acetic acid in the solution.

Acetic acid, with the chemical formula \(\mathrm{CH_3COOH}\), is a weak acid that partially dissociates in water to form hydrogen ions \(\mathrm{H^+}\) and acetate ions \(\mathrm{CH_3COO^-}\). The degree to which acetic acid dissociates is quantified by its acid dissociation constant, \(K_a\). The \(K_a\) expression is written as:

\[ K_a = \frac{[\mathrm{H^+}][\mathrm{CH_3COO^-}]}{[\mathrm{CH_3COOH}]} \]
This equilibrium expression relates the concentration of hydrogen ions and acetate ions to the concentration of undissociated acetic acid at equilibrium. It provides a numerical value that helps determine how strong or weak an acid is; the smaller the \(K_a\), the weaker the acid.

For acetic acid, the \(K_a\) value is \(1.8 \times 10^{-5}\), reflecting its weak acid nature. This information is essential when solving for the equilibrium concentration of acetic acid in a vinegar solution.

Determining the equilibrium concentration of acetic acid in vinegar involves using the \(K_a\) expression mentioned earlier. The equilibrium concentration tells us how much acetic acid remains undissociated at equilibrium.

To solve for the equilibrium concentration, we can assume an initial concentration of acetic acid, \(C\), and express the concentrations of hydrogen and acetate ions in terms of \(C\) and the known \(\mathrm{H^+}\) concentration from the pH of the solution. Given that the concentration of the acetate ion \(\mathrm{CH_3COO^-}\) is equal to the concentration of \(\mathrm{H^+}\) and the concentration of acetic acid \(\mathrm{CH_3COOH}\) is the initial concentration minus the \(\mathrm{H^+}\) concentration, you can set up the equation:

\[ K_a = \frac{(\mathrm{H^+})(\mathrm{H^+})}{(C - \mathrm{H^+})} \]
This equation will then be rearranged and solved for \(C\), giving the equilibrium concentration of acetic acid in the vinegar solution. This concentration is key to understanding the potency of vinegar and has practical implications in cooking, cleaning and other industrial processes where vinegar's acidic properties are utilized.