Question

Phosphoric acid is a common ingredient in traditional cola drinks. It is added to provide the drinks with a pleasantly tart taste. Although phosphoric acid is a triprotic acid, its protons are lost one at a time. Assuming that in cola drinks the concentration of phosphoric acid is \(0.007 M\), calculate the \(\mathrm{pH}\) in this solution.

Short answer

The pH of the 0.007 M phosphoric acid solution in cola drinks is approximately 3.19.

Step-by-step solution

Write the first ionization equation of phosphoric acid

The first ionization of phosphoric acid (H₃PO₄) can be written as: \[ H_{3}PO_{4} \rightleftharpoons H^{+} + H_{2}PO_{4}^{-} \]

Set up an ICE table for the ionization

We will use an ICE (Initial, Change, Equilibrium) table to set up our equilibrium concentrations: \[ \begin{array}{c|ccc} & [H_{3}PO_{4}] & [H^{+}] & [H_{2}PO_{4}^{-}]\\ \hline \text{Initial} & 0.007 & 0 & 0\\ \text{Change} & -x & +x & +x\\ \text{Equilibrium} & 0.007-x & x & x\\ \end{array} \] Where x is the change in concentration due to the ionization reaction.

Write the Kₐ expression for the first ionization of phosphoric acid

The Kₐ expression for the first ionization reaction is given by: \[ K_{a1} = \frac{[H^{+}][H_{2}PO_{4}^{-}]}{[H_{3}PO_{4}]} \] The first Kₐ value for phosphoric acid is \(K_{a1} = 7.25 \times 10^{-3}\).

Plug equilibrium concentrations into the Kₐ expression and solve for x

Now we will substitute the equilibrium concentrations from our ICE table into the Kₐ expression: \[ 7.25 \times 10^{-3} = \frac{x \times x}{(0.007 - x)} \] We assume that x is small compared to 0.007 because the acid is weak: \[ \text{K}_{a1} \approx \frac{x^2}{0.007} \] Now solve for x: \[ x^2 = 7.25 \times 10^{-3} \times 0.007 \\ x \approx 6.46 \times 10^{-4} \]

Calculate the pH of the solution

Now that we have found the concentration of hydrogen ions (H+), [H+] = x, we can calculate the pH value using the pH formula: \[ \mathrm{pH} = -\log[H^{+}] \] So our pH will be: \[ \mathrm{pH} = -\log(6.46 \times 10^{-4}) \approx 3.19 \] Therefore, the pH of the 0.007 M phosphoric acid solution in cola drinks is approximately 3.19.

Key concepts

Triprotic Acid

When we talk about acids, their strength and behavior in solution, the term 'triprotic acid' references an acid that has three protons, or hydrogen ions, it can donate. Phosphoric acid, with the chemical formula \( H_3PO_4 \), is a classic example of a triprotic acid. Each of these three protons can be released sequentially in reactions with water to form hydronium ions \( H_3O^+ \). This sequential release leads to three different dissociation equilibrium reactions, each with its distinct ionization constant \( K_a \). Understanding triprotic acids is important because it helps us grasp the concept that multiple equilibria can contribute to the acidity of a solution, with each ionization step progressively donating a proton and forming a new ion in solution.

For most calculations of pH in weak common acids, we primarily deal with the first ionization since it's the highest and usually the most important for creating \( H^+ \) ions in solution.

ICE Table

The ICE table is a critical tool for chemists. ICE stands for Initial, Change, Equilibrium, and it is used to keep track of the concentrations of all species in an equilibrium reaction. In the context of calculating the pH of phosphoric acid solution, the ICE table helps us organize our thoughts and calculations by systematically listing initial concentrations, changes in those concentrations once equilibrium is established, and the final equilibrium concentrations. The 'x' in the ICE table denotes the change in concentration as the reaction goes towards equilibrium. It represents the amount of \( H_3PO_4 \) that ionizes to form \( H^+ \) and \( H_2PO_4^- \).

The use of the ICE table simplifies the task of plugging equilibrium concentrations into the \( K_a \) expression to solve for the unknown ion concentrations, making it easier to understand the step-by-step progression of the reaction.

First Ionization

Ionization refers to the process by which an atom or molecule gains or loses an electron, forming ions. When we consider acids, like phosphoric acid, the first ionization is the initial step where the acid loses one proton to form its first conjugate base and a hydrogen ion. For phosphoric acid, the first ionization equation is \[ H_{3}PO_{4} \rightleftharpoons H^{+} + H_{2}PO_{4}^{-} \]. Understanding the first ionization is crucial because it generally has the highest \( K_a \) value and therefore contributes the most to the acidity of the solution.

Often, when calculating the pH of an acid solution, we focus on the first ionization because subsequent ionizations are much less significant, especially for weak acids like phosphoric acid, where the degree of ionization decreases with each step.

Kₐ Expression

The \( K_a \) expression is derived from the equilibrium constant for an acid's ionization reaction. It quantifies the strength of the acid in solution. For the first ionization of phosphoric acid, the \( K_a \) expression would describe the ratio of the product of the concentrations of the ions produced to the concentration of the intact acid, as illustrated by \[ K_{a1} = \frac{[H^{+}][H_{2}PO_{4}^{-}]}{[H_{3}PO_{4}]} \]. A larger value of \( K_a \) means a stronger acid, which dissociates more completely in solution.

Using the \( K_a \) expression and equilibrium concentrations calculated from the ICE table, chemists can determine the degree of ionization, which is fundamental in calculating the acidity (pH) of the solution.

pH Formula

The pH formula is a logarithmic measure of the hydrogen ion concentration in a solution and is expressed as \[ \mathrm{pH} = -\log[H^{+}] \]. Calculating the pH of a solution tells us how acidic or basic that solution is. In the case of the phosphoric acid solution, once the concentration of hydrogen ions \( [H^{+}] \) from the first ionization is determined using the ICE table and \( K_a \) expression, this value can be plugged into the pH formula to find the solution's pH. A pH lower than 7 indicates an acidic solution, which makes sense for typical cola drinks containing phosphoric acid that has a pleasantly tart taste.

The precise calculation of pH is essential in various fields, including food science, chemistry, and environmental science. Without the calculated pH value, it would be challenging to understand or predict the behavior of molecules in solution, their reactivity, or the biological systems they may interact with.