Question

A certain indicator HIn has a \(\mathrm{p} K_{\mathrm{a}}\) of 3.00 and a color change becomes visible when 7.00\(\%\) of the indicator has been converted to \(\mathrm{In}^{-}\) . At what pH is this color change visible?

Short answer

The color change of the indicator HIn becomes visible at a pH of 1.88. This is calculated using the Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]), with a ratio of 7:93 for the converted form (In-) to the unconverted form (HIn), which gives a decimal ratio of 0.0753.

Step-by-step solution

Determine the ratio between the concentrations of In- and HIn

We know that 7.00% of the indicator has been converted to In-. This means that the ratio of the converted form (In-) to the unconverted form (HIn) is 7:93.

Convert the ratio to decimals

To make it easier to work with, let's convert the ratio into decimals: \[ \frac{[In^-]}{[HIn]} = \frac{7}{93} = 0.0753 \]

Use the Henderson-Hasselbalch equation to find the pH

We can now substitute the values into the Henderson-Hasselbalch equation to find the pH. pH = pKa + log([In-]/[HIn]) pH = 3.00 + log(0.0753) pH ≈ 3.00 + (-1.12) = 1.88

Conclusion

The color change of the indicator HIn becomes visible at a pH of 1.88.

Key concepts

pH Calculation

Calculating pH is a fundamental aspect of understanding acids and bases. It's a measure of how acidic or basic a solution is. The pH scale ranges from 0 to 14, with lower values being more acidic, higher values more basic, and 7 being neutral.
To calculate pH, especially when dealing with a weak acid and its conjugate base, the Henderson-Hasselbalch equation is often used:

\[pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right)\]

• \(pK_a\) is the acid dissociation constant, which tells us the strength of the acid.
• \([A^-]\) is the concentration of the base form.
• \([HA]\) is the concentration of the acid form.

In practice, this means comparing the concentration of the dissociated (base) form to the undissociated (acid) form. A higher ratio of base to acid increases pH, making the solution more basic. Calculating pH this way is crucial for predicting how an indicator will behave in different environments.

Acid-Base Indicators

Acid-base indicators are substances that change color depending on the pH of the solution they're in. This color change occurs because the indicator itself exists in two forms: an acid form \([HIn]\) and a base form \([In^-]\). Each form has a different color.
These indicators are helpful for visually determining the acidity or basicity of a solution without sophisticated equipment.

Here's how an indicator works:

• The indicator's \(pK_a\) value is crucial, as it determines the pH range at which the color change is noticeable.
• A typical range is usually around \(\pm 1\) pH unit from its \(pK_a\).
• When a small percentage, such as 7% in this exercise, converts from \([HIn]\) to \([In^-]\), the color change can be visible. This indicates the pH value where the indicator begins to visually react.

The exact point where the color change occurs is dependent on the specific indicator and its chemical structure.

Chemical Equilibrium

Chemical equilibrium is a state in which the concentrations of reactants and products remain constant over time. For acid-base reactions, this implies a balance between the acid and its conjugate base.
The Henderson-Hasselbalch equation leverages this concept of equilibrium by relating it to pH calculation.

• At equilibrium, the reaction isn't stopped but occurs at rates allowing the concentrations to be stable.
• This balance is why specific ratios of \([In^-]\) and \([HIn]\) result in predictable pH values.
• The exercise illustrates this with a 7:93 ratio, indicating a slight shift towards the base form \([In^-]\).

Understanding chemical equilibrium helps in predicting how a system will behave when conditions like pH or concentration change. It's pivotal for mastering more complex chemical reactions.