The pH of a \(0.02 \mathrm{M}\) solution of an acid was measured at 4.6. a. What is the \(\left[\mathrm{H}^{+}\right]\) in this solution? b. Calculate the acid dissociation constant \(K_{\mathrm{a}}\) and \(\mathrm{p} K_{\mathrm{a}}\) for this acid.

Understanding how to calculate the pH of a solution is essential for numerous scientific disciplines, particularly chemistry and biology. pH is a measure of the acidity or basicity of an aqueous solution, which can impact reactions, living organisms, and environmental factors. It is expressed on a logarithmic scale ranging from 0 to 14; a pH less than 7 indicates an acidic solution, a pH of 7 is neutral, and a pH greater than 7 indicates a basic solution.

To calculate pH, we use the formula:
\r\[\text{pH} = -\log\left(\left[\mathrm{H}^+\right]\right)\]
where \(\left[\mathrm{H}^+\right]\) represents the hydrogen ion concentration in moles per liter. In our exercise, we took the provided pH value of 4.6 and reversed the calculation to find the hydrogen ion concentration. This foundational step allows us to delve further into understanding the properties of the acidic solution in question.

The concentration of hydrogen ions \(\left[\mathrm{H}^+\right]\) in a solution is a critical parameter determining the solution's pH level. These hydrogen ions are what make a solution acidic. The higher the concentration of hydrogen ions, the lower the pH and the more acidic the solution is.

In our textbook example, we used the pH value to find the hydrogen ion concentration by transforming the pH equation to:\[\left[\mathrm{H}^+\right] = 10^{-\text{pH}}\]
By inserting the pH value of 4.6 into this formula, we determine the concentration of the hydrogen ions in the solution. This value is crucial as it is also used to compute the acid dissociation constant for the acid in the solution.

The calculation of the pKa value is a pivotal concept in acid-base chemistry, as it provides insights into the strength of an acid. It is defined as the negative logarithm of the acid dissociation constant (Ka):\[\text{pKa} = -\log(K_a)\]
This value is essential because it helps ascertain how much an acid dissociates in a solution. A low pKa value suggests a strong acid that readily gives off hydrogen ions, while a high pKa indicates a weaker acid.

In the provided exercise, we first had to determine the Ka using the calculated \(\left[\mathrm{H}^+\right]\) and the initial concentration of the acid. With Ka calculated, we could then move on to find the pKa by simply taking the negative logarithm of the Ka value, giving us an understanding of the acid's dissociation characteristics in solution.

The concept of acid-base equilibrium is related to the reversible reaction of acids donating hydrogen ions to bases. At equilibrium, the rate at which the acid dissociates to form hydrogen ions is equal to the rate at which these ions recombine to form the acid. The position of the equilibrium can be quantified by the Ka, which is the acid dissociation constant.

\rThe generic equilibrium for a weak acid HA in water is:
\r\[\mathrm{HA} \rightleftharpoons \mathrm{H}^+ + \mathrm{A}^-\]
The equilibrium expression for the dissociation of a weak acid is given by:
\r\[K_a = \frac{\left[\mathrm{H}^+\right]\times\left[\mathrm{A}^-\right]}{\left[\mathrm{HA}\right]}\]
In the problem provided, assuming the concentration of \(A^-\) is approximately equal to \(\left[\mathrm{H}^+\right]\) due to the 1:1 dissociation, we can then conclude that the Ka calculated reflects the extent of dissociation for this particular acid at the given concentration.