Question

The pigment cyanidin aglycone is one of the anthocyanin molecules that gives red cabbage (Brassica oleracea var. capitata \(\mathrm{f}\). rubra) its characteristic red coloration. Many young chemists have used this "red cabbage indicator" to study acid-base chemistry. Estimate the \(\mathrm{pH}\) range at which cyanidin aglycone shows a color change. $$ \text { Anth-H } \rightleftharpoons \text { Anth }^{-}+\mathrm{H}^{+} \quad K_{\mathrm{u}}=1.3 \times 10^{-7} $$

Short answer

The color change of cyanidin aglycone in red cabbage will occur within the pH range where there is a significant difference in the concentrations of Anth-H and Anth⁻. Although we cannot determine an exact pH range without the actual concentrations of Anth-H and Anth⁻, we can approximately suggest that the color change will take place around the pH values calculated in steps 2 and 3: For Anth-H dominant: \(\text{pH} \approx -\log (1.3 \times 10^{-7} [\text{Anth-H}])\) For Anth⁻ dominant: \(\text{pH} \approx -\log (1.3 \times 10^{-7} [\text{Anth}^-])\)

Step-by-step solution

Write the equilibrium expression for the reaction

Given the equilibrium reaction, we can write the equilibrium expression as: \[ K_{\mathrm{u}} = \frac{[\text{Anth}^-][\mathrm{H}^+]}{[\text{Anth-H}]} \] Now we need to consider the dominating species in the solution, depending on the pH of the solution.

Calculate the pH when Anth-H is the dominant species

Let's assume Anth-H is dominating, which means Anth-H concentration is much greater than Anth⁻ concentration. The equilibrium expression can be written as: \[ K_{\mathrm{u}} \approx \frac{[\mathrm{H}^+]}{[\text{Anth-H}]} \] Now, we know the pH value is defined as: pH = -log [\(\mathrm{H}^+\)]. We can write this as \([\mathrm{H}^+] \approx 10^{-\text{pH}}\) Using these definitions, we can write the equilibrium expression in terms of pH: \[ K_{\mathrm{u}} \approx \frac{10^{-\text{pH}}}{[\text{Anth-H}]} \] Approximately solving for pH, we have: \[ \text{pH} \approx -\log(K_{\mathrm{u}} [\text{Anth-H}]) \] Inserting the given values: \[ \text{pH} \approx -\log (1.3 \times 10^{-7} [\text{Anth-H}]) \]

Calculate the pH when Anth⁻ is the dominant species

Now let's assume Anth⁻ is dominating, which means Anth⁻ concentration is much greater than Anth-H concentration. The equilibrium expression can be written as: \[ K_{\mathrm{u}} \approx \frac{[\mathrm{H}^+]}{[\text{Anth}^-]} \] Using the definition of pH, we get: \[ K_{\mathrm{u}} \approx \frac{10^{-\text{pH}}}{[\text{Anth}^-]} \] Approximately solving for pH, we have: \[ \text{pH} \approx -\log(K_{\mathrm{u}} [\text{Anth}^-]) \] Inserting the given values: \[ \text{pH} \approx -\log (1.3 \times 10^{-7} [\text{Anth}^-]) \]

Determining the pH range of color change

We have shown the pH values when Anth-H is the dominant species and when Anth⁻ is the dominant species. The color change will occur when there is a significant change in the concentration of Anth-H and Anth⁻. Since the actual concentration of Anth-H and Anth⁻ are not given, we cannot determine an exact pH range for color change. However, we can predict that the color change will occur within the pH range where the concentrations of Anth-H and Anth⁻ are significantly different, which is approximately around the pH values we calculated in steps 2 and 3.

Key concepts

Acid-Base Chemistry

Acid-base chemistry is at the core of understanding chemical reactions in innumerable biological and environmental systems. It revolves around the exchange of proton(s) (H⁺ ions) between molecules, which leads to the formation of acids and bases. An acid is a substance that can donate a proton, while a base can accept it.

When an acid donates a proton to a base, an equilibrium is established between the original species (known as the conjugate acid-base pair) and the resulting products. This equilibrium plays a fundamental role in determining the properties of the acid and base, such as their strength, as well as the pH of the solution, which is the measure of the solution’s acidity or basicity.

Equilibrium Constant (Ku)

In the context of acid-base reactions, the equilibrium constant (Ku), also referred to as the dissociation constant when dealing with acids or bases, quantifies the extent to which a compound dissociates in a solution to give ions. It is expressed as a ratio of the concentration of the ionic products to the un-dissociated species at equilibrium. Formally for a weak acid HA dissociating into H⁺ and A^-, it is given as:\[\begin{equation}K_{a} = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\end{equation}\]When the concentration of H⁺ is increased, indicating increased acidity, the equilibrium shifts according to Le Chatelier's Principle, to favor the formation of more un-dissociated acid, HA.

pH Calculation

The pH of a solution is a logarithmic measure of the concentration of hydrogen ions (H⁺). It is determined by taking the negative logarithm of the H⁺ ion concentration:\[\begin{equation}pH = -\log [\mathrm{H}^+]\end{equation}\]To calculate the pH, it is often necessary to make simplifying assumptions, especially when dealing with weak acids or bases. For instance, if the concentration of the acid is much greater than its degree of dissociation, the pH can be approximated based on the initial concentration of the acid and its equilibrium constant. In doing so, we make the exercise more approachable while acknowledging that precision might slightly decrease. Understanding these approximations is crucial for young chemists studying acid-base reactions and pH estimations.

In Practice

For the red cabbage indicator, by plugging in the equilibrium constant and the assumption about the concentration of the dominant species, students can grasp an estimate of the pH at which a color change occurs.

Anthocyanin Molecules

Anthocyanins, such as the cyanidin aglycone from red cabbage, are naturally occurring pigments responsible for vivid colors in plants. These pigments belong to a class of compounds named flavonoids. They possess the ability to change color based on the pH of their environment due to structural changes in the molecules. This property makes them valuable as pH indicators.

Color Change and pH

At acidic pH levels, anthocyanins typically appear red, while in basic conditions, they shift towards blue or purple. The color transition occurs because of changes in the molecular structure, which affects the wavelengths of light absorbed. By monitoring these shifts, the pH range at which a color change is following the concentration of different species in the equilibrium can be estimated.