Question

The \(\mathrm{p} K_{\mathrm{a}}\) of the indicator methyl orange is \(3.46 .\) Over what \(\mathrm{pH}\) range does this indicator change from 90 Dercent HIn to 90 percent In \(^{-}\) ?

Short answer

The pH range is approximately 2.51 to 4.41.

Step-by-step solution

Understanding the Problem

We need to find the pH range over which the methyl orange indicator changes from 90% protonated form (HIn) to 90% deprotonated form (In⁻). The question provides the \( ext{p}K_a\) value, which is 3.46.

Using the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation for this reaction is given by \( pH = pK_a + \log\left( \frac{[In^-]}{[HIn]} \right)\). We need to apply this equation for 90% HIn and 90% In⁻ conditions separately.

Calculating pH for 90% HIn

For 90% HIn, \([HIn] = 0.9\) and \([In^-] = 0.1\). Plug these into the Henderson-Hasselbalch equation: \(\begin{align*} pH &= pK_a + \log\left(\frac{[In^-]}{[HIn]}\right) &= 3.46 + \log\left(\frac{0.1}{0.9}\right) &= 3.46 + \log(0.111) &\approx 3.46 - 0.954 &\approx 2.51\end{align*}\)

Calculating pH for 90% In⁻

For 90% In⁻, \([In^-] = 0.9\) and \([HIn] = 0.1\). Plug these values in: \(\begin{align*} pH &= pK_a + \log\left(\frac{[In^-]}{[HIn]}\right) &= 3.46 + \log\left(\frac{0.9}{0.1}\right) &= 3.46 + \log(9) &\approx 3.46 + 0.954 &\approx 4.41\end{align*}\)

Conclusion

The pH range over which the indicator methyl orange changes from 90% HIn to 90% In⁻ is approximately 2.51 to 4.41.

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a mathematical relationship used to calculate the pH of a buffer solution. It is an essential tool in chemistry, particularly in understanding how indicators like methyl orange function. The equation is given by:
\[ pH = pK_a + \log\left( \frac{[A^-]}{[HA]} \right) \]
Here, \( pK_a \) is the acid dissociation constant, \([A^-]\) is the concentration of the base form, and \([HA]\) is the concentration of the acid form of the indicator. This equation helps you find the pH at which an acid (HA) is half-neutralized, resulting in equal concentrations of HA and A⁻.

• When the concentration of the base form \([A^-]\) increases, pH increases.
• When the concentration of the acid form \([HA]\) increases, pH decreases.

Using the Henderson-Hasselbalch equation allows you to understand at which pH the transition between acidic and basic forms of the indicator occurs, which is crucial in determining an indicator's effectiveness in signaling the endpoint of a titration.

pH range

The pH range is a critical concept in chemistry as it indicates the range from acidic to basic environments. In the context of methyl orange, the pH range refers to the specific range where its color changes occur.
For methyl orange, the transition from 90% protonated form (HIn) to 90% deprotonated form (In⁻) occurs between a pH of approximately 2.51 and 4.41. This specific range is determined using the Henderson-Hasselbalch equation by considering the ratio of the concentrations of the deprotonated and protonated forms of the indicator.

• A lower pH (2.51) indicates a greater concentration of the protonated form (HIn), meaning it remains in its acidic form.
• A higher pH (4.41) indicates a greater concentration of the deprotonated form (In⁻), meaning it turns to its basic form.

Understanding the pH range over which an indicator changes its color helps in selecting the right indicator for particular titrations, ensuring that the color change is sharp and distinct.

acid-base indicator

An acid-base indicator is a chemical compound that changes color depending on the pH of the solution it is placed in. Methyl orange is a common acid-base indicator used in titrations involving strong acids and weak bases.
Indicators work on the principle of indicating the endpoint of a titration, providing a visible signal of the completion of the reaction. Methyl orange is red in acidic solutions and changes to yellow as the solution becomes more basic. Unlike some other indicators, it is best used in a slightly acidic medium.

• In a strongly acidic solution, methyl orange appears red.
• As the solution moves to a slightly basic condition, it turns yellow.

Note that different indicators have different pH ranges for their color change. This property makes each indicator suitable for measuring particular types of chemical reactions. Methyl orange, with its sharp color change within its pH range, is especially useful in reactions where the endpoint lies within this range.

protonation state

The protonation state of a compound refers to whether it has gained or lost protons (H⁺ ions). This concept is crucial when discussing acid-base reactions and indicators. In the case of methyl orange, the protonation state determines whether it is in the protonated form (HIn) or the deprotonated form (In⁻).
When methyl orange is in its protonated state (HIn), it typically appears red. As the pH increases, the environment favors the loss of a proton, converting methyl orange to its deprotonated form (In⁻), where it takes on a yellow color.

• The protonation state at pH 2.51 shows 90% of methyl orange is in the HIn form.
• At pH 4.41, 90% of methyl orange is in the In⁻ form.

Understanding the protonation state of an indicator in various pH environments helps chemists predict and control the outcomes of chemical reactions.