At \(28^{\circ} \mathrm{C}\) and 0.982 atm, gaseous compound HA has a density of \(1.16 \mathrm{~g} / \mathrm{L}\). A quantity of \(2.03 \mathrm{~g}\) of this compound is dissolved in water and diluted to exactly \(1 \mathrm{~L}\). If the \(\mathrm{pH}\) of the solution is 5.22 (due to the ionization of \(\mathrm{HA}\) ) at \(25^{\circ} \mathrm{C}\), calculate the \(K_{\mathrm{a}}\) of the acid.

Understanding the pH of a solution is fundamental to chemistry, particularly when diving into the properties of acids and bases. pH is a measure of the hydrogen ion concentration \( [H^+] \) in a solution, effectively indicating how acidic or basic that solution is. It follows a logarithmic scale, which is why a small change in pH translates to a significant change in \( [H^+] \) concentration.

To calculate pH, you use the formula \( pH = -\log[H^+] \). Conversely, if you know the pH and need to find the hydrogen ion concentration, you apply the inverse calculation: \( [H^+] = 10^{-pH} \). This calculation is pivotal because it links a physical measurement, the pH, with the chemical behaviors of the solution, like its reactivity, corrosiveness, or suitability for sustaining life.

Molar mass is a term that comes up quite often in chemistry and is pivotal when converting between the mass of a substance and the amount of substance in moles. It's defined as the mass of one mole of a given substance, typically expressed in units of grams per mole \( (g/mol) \). Calculating the molar mass is straightforward when dealing with individual elements as it's numerically equivalent to the atomic weight of the element from the periodic table.

For compounds, it involves summing up the atomic weights of all the atoms present within a single molecule of the compound. This calculation is instrumental in stoichiometry where you need to balance chemical equations and figure out how much of each reactant is required and how much of each product will be produced.

The ideal gas law is a cornerstone in the study of gases, providing a relationship between the pressure \( (P) \), volume \( (V) \), temperature \( (T) \), and the amount of gas in moles \( (n) \) through the equation \( PV = nRT \), where \( R \) is the ideal gas constant.

The Importance of the Ideal Gas Constant \( R \)

The ideal gas constant bridges the physical measurements of pressure, volume, and temperature with the chemical quantity of moles. It has a fixed value based on the units used for pressure, volume, and temperature, which in SI units is \( 8.314 \text{J/mol}\cdot\text{K} \).

Relevance to Molar Mass Determination

Moreover, with a known density of the gas, the ideal gas law allows us to rearrange the equation to solve for the molar mass, making it an invaluable equation for determining properties of gases without needing to engage with complex chemical reactions.

In chemical reactions, the equilibrium constant \( (K) \) is a quantification of the ratio of the concentrations of products and reactants at equilibrium. It's unique to each reaction and remains unchanged under constant temperature.

For the acid dissociation constant \( (K_a) \) specifically, it shows how well an acid dissociates in water, a crucial aspect when dealing with the acidity of a solution. It's calculated using the equilibrium concentrations of the acid \( (HA) \), the conjugate base \( (A^-) \), and the hydrogen ions \( (H^+) \) as follows: \( K_a = \frac{[H^+][A^-]}{[HA]} \). A higher \( K_a \) implies a stronger acid, which dissociates more in water to release \( H^+ \) ions. This concept is not only essential for predicting the direction and extent of chemical reactions but also for understanding buffer solutions and their ability to maintain a stable pH.