Question

One mole of a weak acid HA was dissolved in \(2.0 \mathrm{~L}\) of solution. After the system had come to equilibrium, the concentration of HA was found to be \(0.45 M .\) Calculate \(K_{\mathrm{a}}\) for HA.

Short answer

The equilibrium constant (\(K_a\)) for the weak acid HA is \(8.9 \times 10^{-2}\).

Step-by-step solution

Write the dissociation equation and expression for \(K_a\) for HA.

The weak acid HA dissociate into its ions in the aqueous solution: \[HA \rightleftharpoons H^+ + A^-\] The expression for \(K_a\) (equilibrium constant) for this process is given by: \[K_a = \frac{[H^+][A^-]}{[HA]}\]

Set up an ICE table for the dissociation of HA.

To find the equilibrium concentrations of all species, create an ICE (Initial, Change, Equilibrium) table based on the initial concentration of HA and the change in concentration: \[ \begin{array}{c|ccc} & [HA] & [H^+] & [A^-] \\ \hline \text{Initial} & 0.50 \, \text{M} & 0 & 0 \\ \text{Change} & -x & +x & +x \\ \text{Equilibrium} & 0.45 \, \text{M} & x & x \\ \end{array} \]

Determine the change in HA's concentration

From the ICE table, the equilibrium concentration of HA is 0.45 M. The initial concentration of HA was 0.5 mol in 2.0 L of solution, which gives us an initial concentration of: \[\frac{0.5 \, \text{mol}}{2.0 \, \text{L}} = 0.25\, \text{M}\] The change in HA's concentration (x) is calculated as: \[x = 0.25 \, \text{M} - 0.45 \, \text{M} = -0.20 \, \text{M}\]

Calculate the equilibrium concentrations of H+ and A-

From the ICE table, we know the change in concentration of H+ and A- is also equal to x: \[x = [H^+]_{eq} = [A^-]_{eq} = 0.20 \, \text{M}\]

Calculate \(K_a\) for HA

Now that we have the equilibrium concentrations, we can calculate \(K_a\) using the expression provided: \[K_a = \frac{[H^+][A^-]}{[HA]} = \frac{(0.20\, \mathrm{M})(0.20\, \mathrm{M})}{(0.45\, \mathrm{M})} = \frac{0.04}{0.45}\] \[K_a = 8.9 \times 10^{-2}\] The equilibrium constant \(K_a\) for the weak acid HA is \(8.9 \times 10^{-2}\).

Key concepts

Equilibrium Constant

The equilibrium constant, represented as K, is a crucial concept in chemistry that denotes the balance between products and reactants in a reversible chemical reaction. When a system reaches equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the amounts of reactants and products.

In the context of acid-base chemistry, the equilibrium constant for the dissociation of a weak acid, symbolized as Ka, is a special form of equilibrium constant. It quantifies the strength of the acid by indicating the degree to which a weak acid donates protons to water. The larger the Ka value, the stronger the acid, and the more it dissociates into its ions at equilibrium.

An expression for Ka can be written following the general pattern:
\[Ka = \frac{{[\text{Products}]}}{{[\text{Reactants}]}}\]
In which the concentrations of products and reactants are raised to the power of their stoichiometric coefficients. Keep in mind that only aqueous and gaseous species are included in the expression.

ICE Table

An ICE table, an acronym that stands for Initial, Change, and Equilibrium, is a valuable organizational tool used in chemistry to understand and calculate the changes taking place within a reaction as it moves towards equilibrium.

An ICE table typically features rows for initial molarity, change in molarity, and equilibrium molarity, with corresponding columns for each species involved in the chemical reaction. It's very useful for visualizing how concentrations shift over the course of a reaction and for identifying equilibrium concentrations, which are then used in the equilibrium expression.

For weak acid dissociation problems like the one in our example, the ICE table lays the foundation for understanding the dissociation process step by step, giving clarity to how the initial concentration of acid decreases and how the concentrations of the ions increase, up to the point of equilibrium.

Molarity

Molarity, symbolized as M, is a measure of concentration in chemistry that refers to the number of moles of a solute present per liter of solution. It is a fundamental concept that allows chemists to describe how concentrated a solution is in terms of the amount of substance dissolved.

Using the formula:
\[Molarity (M) = \frac{{\text{moles of solute}}}{{\text{liters of solution}}}\]
molarity enables precise calculations in reactions, as seen in the weak acid equilibrium problem where we must understand the concentration of acid before and after dissociation. It is essential because the value of the equilibrium constant Ka is based on the equilibrium concentrations of ions and undissociated acid, which are quantified in terms of molarity.

Acid Dissociation Constant (Ka)

The acid dissociation constant (Ka) is a specific type of equilibrium constant that measures the propensity of a weak acid to lose a proton and form its conjugate base. In essence, it is an indicator of the strength of an acid in solution.

The formula to calculate Ka is based on the equilibrium concentrations of the products and reactants of the acid dissociation reaction:
\[Ka = \frac{{[H^+][A^-]}}{{[HA]}}\]
where \([H^+]\) and \([A^-]\) are the molarities of the hydrogen ions and conjugate base, respectively, and \([HA]\) is the molarity of the undissociated acid. The value of Ka is constant for a given acid at a specific temperature and provides essential information on how the acid will behave in different chemical environments.