Question

(a) Calculate the percent ionization of \(0.125 \mathrm{M}\) lactic acid \(\left(K_{a}=1.4 \times 10^{-4}\right)\). (b) Calculate the percent ionization of \(0.125 \mathrm{M}\) lactic acid in a solution containing \(0.0075 \mathrm{M}\) sodium lactate.

Short answer

We can solve this quadratic equation for x, which represents the [\(H^+\)] at equilibrium. After solving, we get \(x = 0.00537\mathrm{M}\), which we can use to calculate the percent ionization for part (a) as: \[\text{Percent Ionization (a)} = \frac{0.00537}{0.125} \times 100\% \approx 4.30\%\] #tag_title# Step 3: Calculate equilibrium concentrations for part (b) #tag_content# In part (b), we have sodium lactate added to the solution. This will affect the equilibrium concentrations for the lactic acid ionization. The ICE table for this scenario can be written as: \[ \begin{array}{c|c|c|c} & [HA] & [H^+] & [A^-] \\ \hline \text{Initial} & 0.125 & 0 & 0.0075 \\ \text{Change } (-x) & -x & +x & +x \\ \text{Equilibrium} & 0.125-x & x & 0.0075+x \\ \end{array} \] We can rewrite the expression for Ka, using the updated equilibrium concentrations: \[K_a = \frac{x(0.0075+x)}{0.125 - x}\] Solving for x again, we get \(x = 0.00498\mathrm{M}\). We can then use this value to calculate the percent ionization for part (b) as: \[\text{Percent Ionization (b)} = \frac{0.00498}{0.125} \times 100\% \approx 3.98\%\] In conclusion, the percent ionization for 0.125 M lactic acid is approximately 4.30% without sodium lactate and 3.98% with sodium lactate present in the solution.

Step-by-step solution

Understand percent ionization

Percent ionization is the percentage of the initial acid concentration that has ionized or dissociated. For lactic acid, the ionization can be represented as \(HA \rightleftharpoons H^+ + A^-\), where \(HA\) represents lactic acid and \(A^-\) represents the lactate ion. Lactic acid has the Ka value given, which we can use to calculate the concentration of the ionized species in equilibrium.

Calculate equilibrium concentrations for part (a)

To find the equilibrium concentrations, we can set up an ICE table (Initial, Change, and Equilibrium) for the lactic acid reaction: \[ \begin{array}{c|c|c|c} & [HA] & [H^+] & [A^-] \\ \hline \text{Initial} & 0.125 & 0 & 0 \\ \text{Change } (-x) & -x & +x & +x \\ \text{Equilibrium} & 0.125-x & x & x \\ \end{array} \] Using the given \(K_a = 1.4 \times 10^{-4}\), we can write the expression for Ka as: \[K_a = \frac{x^2}{0.125 - x}\]

Key concepts

Acid Dissociation Constant (Ka)

Understanding the acid dissociation constant, or Ka, is essential for analyzing acid strength and calculating equilibrium concentrations in solutions. Ka reflects the extent to which an acid donates protons to water, forming its conjugate base and hydronium ions. In essence, the higher the Ka value, the stronger the acid.

For a general weak acid represented by HA, the dissociation in water can be shown as: \[\[\begin{align*}\inline \ HA \rightleftharpoons H^+ + A^- \end{align*}\]\]In this reaction, HA is the acid, H+ denotes the hydronium ion, and A- is the conjugate base. The Ka expression is a quotient calculated by dividing the product of the equilibrium concentrations of the products by the concentration of the reactant, minus any change as the reaction progresses.
\[\[\begin{align*}Ka = \frac{[H^+][A^-]}{[HA]}\end{align*}\]\]The brackets signify the molar concentrations at equilibrium. In the exercise provided, lactic acid (HA) has a Ka of \(1.4 \times 10^{-4}\), indicating it is a weak acid, as its dissociation in water is not complete and the value of Ka is relatively small.

ICE Table

The ICE table is a powerful tool used to organize data pertaining to initial concentrations, changes in concentrations, and equilibrium concentrations in a chemical reaction. ICE stands for Initial, Change, and Equilibrium, lining up vertically with the respective stages of the chemical process.

The table is set up with the reactants and products horizontally and the stages of the reaction vertically. Changes in concentrations are often represented with the variable 'x', where a decrease in reactant concentration (as the reaction proceeds to products) would be noted as '-x' and an increase in products as '+x'.

For an acid dissociation reaction, the ICE table allows us to visualize the initial concentration of the acid, the amount it changes as it ionizes, and the resulting concentrations of all species at equilibrium. By plugging these values into the Ka expression, we can solve for 'x', which represents the change in concentration of the ions. This will then be used to calculate the percent ionization of the acid.

Equilibrium Concentrations

Finding equilibrium concentrations is a fundamental step in solving acid-base equilibrium problems. These are the concentrations of the reactants and products once the system has reached a state of balance where the rate of the forward reaction equals the rate of the reverse reaction.

Equilibrium concentrations can be used alongside the acid dissociation constant (Ka) to determine the position of equilibrium and the extent of ionization in an aqueous solution of a weak acid. To calculate these concentrations, we typically start with the initial concentrations of reactants and products (often products start at zero), determine the changes that occur as the system reaches equilibrium, and then apply these changes to find the final concentrations.

When applying this approach to the exercise with lactic acid, we create an ICE table to delineate the starting concentration of lactic acid, the shift that occurs as some of it dissociates, and the equilibrium concentrations of lactic acid (HA), hydronium ions (H+), and lactate ions (A-). By solving the established equilibrium expression using the Ka, we ultimately obtain the value of 'x' that reveals the hydrogen ion concentration at equilibrium. This is a crucial step as it leads directly to calculating the percent ionization of the acid.