Question

(a) Calculate the percent ionization of 0.0075\(M\) butanoic acid \(\left(K_{a}=1.5 \times 10^{-5}\right) .\) (b) Calculate the percent ionization of 0.0075\(M\) butanoic acid in a solution containing 0.085\(M\) sodium butanoate.

Short answer

For part (a), the percent ionization of 0.0075 M butanoic acid is given by: \[ \% \text{ionization} = \frac{\sqrt{1.5 \times 10^{-5} \times 0.0075}}{0.0075} \times 100 \approx 2.87 \% \] For part (b), the percent ionization of 0.0075 M butanoic acid in a solution containing 0.085 M sodium butanoate is given by: \[ \% \text{ionization} = \frac{\frac{1.5 \times 10^{-5} \times 0.0075}{0.085}}{0.0075} \times 100 \approx 0.294 \% \]

Step-by-step solution

Write the chemical equilibrium equation

Since butanoic acid is a weak acid, it dissociates into the conjugate base butanoate ion and proton as follows: \[ \ce{CH3CH2CH2COOH <=> CH3CH2CH2COO- + H+} \]

Set up the ICE table for the initial problem

An ICE table represents the initial concentration, the change in concentration, and the equilibrium concentration for the reactants and products: Initial: \[ [\ce{CH3CH2CH2COOH}] = 0.0075\ M \] \[ [\ce{CH3CH2CH2COO^-}] = 0 \] \[ [\ce{H^+}] = 0 \] Change: \[ [\ce{CH3CH2CH2COOH}] = -x \] \[ [\ce{CH3CH2CH2COO^-}] = +x \] \[ [\ce{H^+}] = +x \] Equilibrium: \[ [\ce{CH3CH2CH2COOH}] = 0.0075 - x\ M \] \[ [\ce{CH3CH2CH2COO^-}] = x \] \[ [\ce{H^+}] = x \] In which \(x\) is the concentration of the ions at equilibrium.

Use the equilibrium expression to solve for x

The equilibrium expression is given by the acid dissociation constant, \(K_a\): \[ K_{a} = \frac{[\ce{CH3CH2CH2COO^-}][\ce{H^+}]}{[\ce{CH3CH2CH2COOH}]} \] Since \(K_a = 1.5 \times 10^{-5} \), \[ 1.5 \times 10^{-5} = \frac{x^2}{0.0075 - x} \] Considering the weak dissociation, we can assume that \(x \ll 0.0075\), making the equilibrium simplification of \[ 1.5 \times 10^{-5} \approx \frac{x^2}{0.0075} \] Now, we can solve for \(x\): \[ x = \sqrt{1.5 \times 10^{-5} \times 0.0075} \]

Calculate the percent ionization of butanoic acid

The percent ionization is given by the ratio of the concentration of dissociated protons and the initial concentration, multiplied by 100: \[ \% \text{ionization} = \frac{x}{0.0075} \times 100 \] Plug in the value of \(x\) to get the percent ionization of butanoic acid. For part (b), since we have 0.075 M butanoic acid and 0.085 M sodium butanoate: Initial: \[ [\ce{CH3CH2CH2COOH}] = 0.0075\ M \] \[ [\ce{CH3CH2CH2COO^-}] = 0.085 \] \[ [\ce{H^+}] = 0 \] Change: \[ [\ce{CH3CH2CH2COOH}] = -x \] \[ [\ce{CH3CH2CH2COO^-}] = +x \] \[ [\ce{H^+}] = +x \] Equilibrium: \[ [\ce{CH3CH2CH2COOH}] = 0.0075 - x\ M \] \[ [\ce{CH3CH2CH2COO^-}] = 0.085 + x \] \[ [\ce{H^+}] = x \]

Calculate the percent ionization of butanoic acid in the sodium butanoate solution

The new equilibrium expression with sodium butanoate present is as follows: \[ K_{a} = \frac{(x)(0.085 + x)}{(0.0075 - x)} \] Again, we can assume that \(x \ll 0.0075\), making the equilibrium simplification of \[ 1.5 \times 10^{-5} \approx \frac{x \times 0.085}{0.0075} \] Now, we can solve for \(x\): \[ x = \frac{1.5 \times 10^{-5} \times 0.0075}{0.085} \] Calculate the percent ionization as we did in Step 4: \[ \% \text{ionization} = \frac{x}{0.0075} \times 100 \] Plug in the value of \(x\) to get the percent ionization of butanoic acid in the sodium butanoate solution.

Key concepts

Weak Acids

Weak acids are a fascinating topic in chemistry because unlike strong acids, they do not completely dissociate in water. This means that when they are dissolved, only a small fraction of the acid molecules release hydrogen ions (\(H^+\)). This partial dissociation is what makes weak acids interesting, as we have to consider their equilibrium state.
For instance, butanoic acid is a typical example of a weak acid. In water, it reacts as follows:\[\ce{CH3CH2CH2COOH <=> CH3CH2CH2COO- + H+}\]In this reaction, butanoic acid (\(\ce{CH3CH2CH2COOH}\)) is partially converted into its conjugate base, known as the butanoate ion (\(\ce{CH3CH2CH2COO^-}\)), and a proton (\(H^+\)). The extent to which the weak acid dissociates is quantified by its acid dissociation constant (\(K_a\)).Understanding weak acids is crucial because their partial ionization affects properties like pH and conductivity, and plays a vital role in various chemical and biological processes.

Equilibrium Expressions

When dealing with weak acids, equilibrium expressions are an essential tool to understand how acids behave in solution. They are derived from the chemical equilibrium of the acid dissociating into its ions. The general equilibrium expression for a weak acid reaction is given by:\[K_a = \frac{[\ce{A^-}][\ce{H^+}]}{[\ce{HA}]}\]Here, \([\ce{A^-}]\) represents the concentration of the conjugate base, \([\ce{H^+}]\) is the concentration of the hydrogen ions, and \([\ce{HA}]\) is the concentration of the non-dissociated acid at equilibrium.
The acid dissociation constant, \(K_a\), indicates the strength of the acid: the larger the \(K_a\) value, the stronger the acid. In our case for butanoic acid, \(K_a = 1.5 \times 10^{-5}\), which tells us it is a weak acid since the \(K_a\) is relatively small.
To solve for concentrations at equilibrium, we can substitute known values into the equilibrium expression. By applying mathematical simplifications based on the assumption of weak dissociation, calculations become more manageable, helping to estimate the percent ionization accessible through these expressions.

ICE Table

An ICE table is a simple, systematic way to track changes in concentrations for equilibrium reactions like those of weak acids. ICE stands for Initial, Change, and Equilibrium, which helps students visualize and organize the concentrations of each species at different stages of the reaction.
Here's how it works:

• Initial: You start by filling in the initial concentrations of all reactants and products. For butanoic acid, initially, the conjugate base and protons have concentrations of zero because no dissociation has occurred yet.
• Change: You then consider changes in concentrations as the reaction proceeds. Typically, the acid's concentration decreases by \(-x\), while the concentrations of the conjugate base and protons increase by \(+x\).
• Equilibrium: Finally, you list the equilibrium concentrations by combining initial concentrations and the changes. These are used in the equilibrium expression to solve for \(x\), which represents the equilibrium concentration of the ions.

In summary, ICE tables are an invaluable tool for chemists and students to simplify complex equilibrium problems, providing a logical framework to determine how a weak acid like butanoic acid ionizes in solution.