Question

Lactic acid \(\left(\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}\right)\) has one acidic hydrogen. A \(0.10 \mathrm{M}\) solution of lactic acid has a \(\mathrm{pH}\) of \(2.44\). Calculate \(K_{a}\).

Short answer

The \(K_{a}\) for lactic acid is \(1.38 \times 10^{-4}\).

Step-by-step solution

Convert pH to [H+] concentration

To find the concentration of \(H^+\) ions in the solution, we can use the formula: \[ [H^+] = 10^{-\mathrm{pH}}\] We are given the pH of the solution, so we can plug it into the formula: \[ [H^+] = 10^{-2.44} \] Calculate the concentration: \[ [H^+] = 3.63 \times 10^{-3} \mathrm{M} \]

Set up the ICE table

We will set up an ICE (Initial, Change, Equilibrium) table to help us keep track of the concentrations of all species involved in the dissociation. | | CH\(_3\)CH(OH)COOH | H\(^+\) | CH\(_3\)CH(OH)COO\(^-\) | |-------------------|-------------------|-------|--------------------| | Initial(moles/L) | 0.10 | 0 | 0 | | Change(moles/L) | -x | +x | +x | | Equilibrium | 0.10 - x | x | x |

Relate [H+] and [CH\(_3\)CH(OH)COO\(^-\)] concentrations

From the ice table, we can see that the equilibrium concentration of both \(H^+\) ions and CH\(_3\)CH(OH)COO\(^-\) ions are equal to x moles/L: \[ [\mathrm{H}^+] = [\mathrm{CH}_3\mathrm{CH}(\mathrm{OH})\mathrm{COO}^-] = x \] We calculated the concentration of \(H^+\) ions in Step 1: \[ x = 3.63 \times 10^{-3} \mathrm{M} \]

Calculate the equilibrium concentration of CH\(_3\)CH(OH)COOH

Now, we will find the equilibrium concentration of lactic acid (CH\(_3\)CH(OH)COOH). From the ICE table, the concentration at equilibrium is (0.10 - x) moles/L: \[ [\mathrm{CH}_3\mathrm{CH}(\mathrm{OH})\mathrm{COOH}]_{eq} = 0.10 - x \] Substitute the known value of x: \[ [\mathrm{CH}_3\mathrm{CH}(\mathrm{OH})\mathrm{COOH}]_{eq} = 0.10 - 3.63 \times 10^{-3} \] Calculate the equilibrium concentration of lactic acid: \[ [\mathrm{CH}_3\mathrm{CH}(\mathrm{OH})\mathrm{COOH}]_{eq} = 0.09637 \mathrm{M} \]

Calculate \(K_a\)

Now, we have equilibrium concentrations for all the species involved in the dissociation. We can use the formula for \(K_a\): \[ K_a = \frac{[\mathrm{H}^+][\mathrm{CH}_3\mathrm{CH}(\mathrm{OH})\mathrm{COO}^-]}{[\mathrm{CH}_3\mathrm{CH}(\mathrm{OH})\mathrm{COOH}]} \] Plug in the equilibrium concentrations: \[ K_a = \frac{(3.63 \times 10^{-3})(3.63 \times 10^{-3})}{0.09637} \] Calculate \(K_a\): \[ K_a = 1.38 \times 10^{-4} \] Thus, the \(K_{a}\) for lactic acid is \(1.38 \times 10^{-4}\).

Key concepts

Lactic Acid

Lactic acid is a fascinating organic compound that plays a key role in various biochemical processes. Its chemical formula is \( ext{CH}_{3} ext{CH}( ext{OH}) ext{COOH}\), and it possesses one acidic hydrogen, which means it can donate a proton in a chemical reaction. This ability to donate a proton classifies lactic acid as a weak acid. In many physiological scenarios, such as during intense exercise, lactic acid is produced in muscles, where it can cause temporary discomfort and fatigue.

In terms of its chemical behavior in solutions, lactic acid partially dissociates into hydrogen ions \( ext{H}^+\) and the lactate ion \( ext{CH}_3 ext{CH}( ext{OH}) ext{COO}^-\). This partial dissociation is quantitatively measured by the acid dissociation constant, \(K_a\), which reveals the strength of the acid in aqueous solutions. Understanding the behavior of lactic acid in solutions is crucial in settings ranging from biological to industrial.

pH Calculation

The pH of a solution is a measure of its acidity or basicity, often serving as a crucial parameter in chemical studies. It is calculated using the hydrogen ion concentration \([H^+]\). The formula to find pH is: \[ \text{pH} = -\log_{10}[H^+] \]In the given problem, the pH is used to find \([H^+]\) concentration for a 0.10 M solution of lactic acid with a pH of 2.44. This is done using the inverse logarithmic relationship:\[ [H^+] = 10^{-\text{pH}} \]Calculating this yields \([H^+] \approx 3.63 \times 10^{-3} \text{ M}\).

The pH gives insight into the concentration of hydrogen ions present in the solution, which is important for calculating the \(K_a\) of weak acids like lactic acid.

ICE Table

An ICE table is a helpful visual tool used in chemistry to calculate the changes in concentrations of substances during a chemical reaction. It stands for Initial, Change, and Equilibrium - three stages that a solution goes through in a reaction process.

Setting up an ICE table involves listing the initial concentrations of all reactants and products. If the initial concentration of the hydrogen ion \([H^+]\) is zero, this entry in the table would reflect that. The change line in the table accounts for the amount the species concentration increases or decreases, represented as \(x\) for a typical acid dissociation scenario. Finally, the equilibrium line in the table shows concentrations at complete equilibrium based on initial and change values.

For the dissociation of lactic acid, the ICE table helps calculate the equilibrium concentrations of \([H^+]\), \([ ext{CH}_3 ext{CH}( ext{OH}) ext{COO}^-]\), and \([ ext{CH}_3 ext{CH}( ext{OH}) ext{COOH}]\). Using these, you can easily find the acid dissociation constant \(K_a\) to analyze the acidity of the lactic acid solution in a more structured and simple way.