Question

Sorbic acid $\left({\mathrm{C}}_5{\mathrm{H}}_7\mathrm{COOH}\right)$ is a weak monoprotic acid with $K_a=1.7\times {10}^{-5}$. Its salt (potassium sorbate) is added to cheese to inhibit the formation of mold. What is the $\mathrm{pH}$ of a solution containing $11.25\mathrm{g}$ of potassium sorbate in $1.75\mathrm{L}$ of solution?

Short answer

The short version of the answer is: To find the pH of a solution containing 11.25 g of potassium sorbate in 1.75 L of solution, first calculate the moles of potassium sorbate, then the concentration, and use the given $K_a$ value to find the concentration of H+ ions. Finally, calculate the pH using the formula pH = $-\log ([{\mathrm{H}}^{+}])$.

Step-by-step solution

Find the moles of potassium sorbate

We are given the mass of potassium sorbate (11.25 g) and need to convert it to moles. First, determine the molar mass of potassium sorbate (${\mathrm{KC}}_5{\mathrm{H}}_7\mathrm{COO}$). The molar mass is calculated using the following formula: Molar mass = (39.10 (K) + 5 * 12.01 (C) + 7 * 1.01 (H) + 2 * 16.00 (O) + 1 * 14.01 (N)) Now, we use the molar mass to convert the mass of potassium sorbate to moles: Moles = (mass of potassium sorbate) / (molar mass of potassium sorbate)

Find the concentration of potassium sorbate

We now need to find the concentration of potassium sorbate in the solution. It is given that the solution is 1.75 L, and we have already found the moles of potassium sorbate. To find the concentration, use the following formula: Concentration (M) = (moles of potassium sorbate) / (volume of solution in L)

Create an equilibrium expression using Ka value

As sorbic acid is a weak monoprotic acid, it will ionize partially in water: ${\mathrm{C}}_5{\mathrm{H}}_7\mathrm{COOH}\rightleftharpoons {\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}+{\mathrm{H}}^{+}$ Since potassium sorbate is a salt of sorbic acid, it will dissociate completely in water: ${\mathrm{KC}}_5{\mathrm{H}}_7\mathrm{COO}\to {\mathrm{K}}^{+}+{\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}$ From the reaction, we can see that the concentration of ${\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}$ is equal to the initial concentration of potassium sorbate. To create the equilibrium expression: $K_a=\frac{[{\mathrm{H}}^{+}][{\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}]}{[{\mathrm{C}}_5{\mathrm{H}}_7\mathrm{COOH}]}$

Solve for H+ ions concentration

Let the concentration of H+ ions and ${\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}$ be x and the concentration of ${\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}$ be C. Now, we can rewrite the expression as: $K_a=\frac{x\ast x}{C-x}$ Since $K_a$ is very small, we can assume that $x<<C$, so we can simplify the expression to: $K_a\approx \frac{x^2}{C}$ Now, solve for x (concentration of H+ ions): $x^2=K_a\ast C$ $x=\sqrt{K_a\ast C}$

Calculate pH

Now that we have the concentration of H+ ions (x), we can calculate the pH of the solution using the following formula: pH = -log($[{\mathrm{H}}^{+}]$) Plug in the value of x obtained in step 4 and calculate the pH of the solution.

Key concepts

Weak Acids

Weak acids only partially dissociate in water, meaning they don't release all their hydrogen ions into the solution. This partial ionization is what makes the acid weak. Unlike strong acids, which dissociate completely, the concentration of hydrogen ions from weak acids is much lower.

Understanding weak acids is essential because it influences how we calculate other properties like pH and equilibrium expressions. When you have a weak acid like sorbic acid, its dissociation in water is denoted by its dissociation constant, or acidity constant ($K_a$). The lower the $K_a$ value, the weaker the acid. Sorbic acid, with a $K_a$ of $1.7\times {10}^{-5}$, indicates it dissociates very little, meaning most sorbic acid molecules remain intact in solution.

pH Calculation

The pH of a solution is a measure of its acidity or basicity on a logarithmic scale. It represents the concentration of hydrogen ions ($[H^{+}]$) in a solution. For weak acids, calculating the pH involves determining how much the acid dissociates into these ions.

To find the pH, you first need the concentration of $[H^{+}]$. This is where equilibrium expressions come into play. After solving the expressions and finding the hydrogen ion concentration, you convert it into pH using the formula:

• $\text{pH}=-\log ([H^{+}])$

For instance, if the hydrogen ion concentration is $x$, plug it into the formula to find the pH. Remember, because the pH scale is logarithmic, even small changes in $[H^{+}]$ result in noticeable changes in pH.

Equilibrium Expressions

Equilibrium expressions help us understand the balance between reactants and products in weak acid solutions. These expressions show us the concentrations of all entities in a reaction at equilibrium.

For sorbic acid, the equilibrium is expressed by the dissociation:${\mathrm{C}}_5{\mathrm{H}}_7\mathrm{COOH}\rightleftharpoons {\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}+{\mathrm{H}}^{+}$Here, the equilibrium expression based on the $K_a$ value is$K_a=\frac{[{\mathrm{H}}^{+}][{\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}]}{[{\mathrm{C}}_5{\mathrm{H}}_7\mathrm{COOH}]}$

This equation shows the ratio of the products ($[{\mathrm{H}}^{+}][{\mathrm{C}}_5{\mathrm{H}}_7{\mathrm{COO}}^{-}]$) to the reactants ($[{\mathrm{C}}_5{\mathrm{H}}_7\mathrm{COOH}]$). An important step in these calculations involves assuming the concentration of hydrogen ions is small enough to not significantly change the initial concentration of the weak acid.

Molarity

Molarity is a way to express the concentration of a solution. It gives the number of moles of solute per liter of solution. The formula to find molarity (M) is:

• $M=\frac{\text{moles of solute}}{\text{volume of solution in liters}}$

When calculating molarity, especially for weak acids, it involves first finding the number of moles of the substance dissolved in the solution.

For example, if you dissolve potassium sorbate in water, calculate its molar mass first to convert grams into moles. Then, use this mole value to find its concentration in the solution. This molarity helps determine how the weak acid or its salt will interact in the solution, especially in equilibrium scenarios.