Question

In the presence of acid, iodide is oxidized by hydrogen peroxide $$ 2 \mathrm{I}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}_{2}(a q)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q) \longrightarrow 4 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{I}_{2}(a q) $$ When \(\mathrm{I}^{-}\) and \(\mathrm{H}_{3} \mathrm{O}^{+}\) are present in excess, we can use the reaction's kinetics of the reaction, which is pseudo- first order in \(\mathrm{H}_{2} \mathrm{O}_{2},\) to determine the concentration of \(\mathrm{H}_{2} \mathrm{O}_{2}\) by following the production of \(\mathrm{I}_{2}\) with time. In one analysis the solution's absorbance at \(348 \mathrm{nm}\) was measured after \(240 \mathrm{~s}\). Analysis of a set of standard gives the results shown below. $$ \begin{array}{cc} {\left[\mathrm{H}_{2} \mathrm{O}_{2}\right](\mu \mathrm{M})} & \text { absorbance } \\ \hline 100.0 & 0.236 \\ 200.0 & 0.471 \\ 400.0 & 0.933 \\ 800.0 & 1.872 \end{array} $$ What is the concentration of \(\mathrm{H}_{2} \mathrm{O}_{2}\) in a sample if its absorbance is 0.669 after \(240 \mathrm{~s} ?\)

Short answer

The concentration of \(\mathrm{H}_2\mathrm{O}_2\) is approximately 284.62 µM.

Step-by-step solution

Understand the relationship between concentration and absorbance

To determine the concentration of \(\mathrm{H}_2\mathrm{O}_2\), we need to understand that absorbance is directly proportional to the concentration according to Beer-Lambert's law, \( A = \epsilon c \ell \), where \(A\) is absorbance, \(\epsilon\) is the molar absorptivity, \(c\) is the concentration, and \(\ell\) is the path length of the cell.

Establish a linear relationship from standard data

Using the provided standard data, we can establish a linear relationship between concentration \([\mathrm{H}_2\mathrm{O}_2]\) and absorbance. Given points are: (100.0, 0.236), (200.0, 0.471), (400.0, 0.933), and (800.0, 1.872). Such a linear relationship can be determined by calculating the slope and intercept of the line passing through these points.

Calculate the slope of the line

The slope \(m\) of the line \([\mathrm{H}_2\mathrm{O}_2]\) versus absorbance can be calculated by using any two points on the graph. For instance, slope \(m = \frac{1.872 - 0.236}{800.0 - 100.0} = \frac{1.636}{700} \approx 0.00234 \mathrm{ \mu M^{-1}}. \)

Derive the linear equation

Assuming a linear equation: \(\text{absorbance} = m \cdot [\mathrm{H}_2\mathrm{O}_2] + b\), where \(b\) is the y-intercept. Using one point, say (100.0, 0.236), we can solve for \(b\): \(0.236 = 0.00234 \times 100.0 + b\). This gives \(b \approx 0.002.\) Hence, the equation is \(\text{absorbance} = 0.00234 \cdot [\mathrm{H}_2\mathrm{O}_2] + 0.002.\)

Solve for unknown concentration

Given the absorbance of the sample is 0.669, substitute this value into the linear equation: \(0.669 = 0.00234 \cdot [\mathrm{H}_2\mathrm{O}_2] + 0.002\). Solving for \([\mathrm{H}_2\mathrm{O}_2]\) gives \([\mathrm{H}_2\mathrm{O}_2] = \frac{0.669 - 0.002}{0.00234} \approx 284.62 \mathrm{ \mu M}.\)

Key concepts

Absorbance

The concept of absorbance is a cornerstone in understanding the Beer-Lambert Law, which is crucial when studying reactions involving light absorption, like our iodide oxidation problem. Absorbance, denoted as \( A \), measures how much light a sample absorbs. Imagine a beam of light passing through a solution. Part of this light is absorbed by the molecules in the solution, reducing the light's intensity. Absorbance quantifies this reduction.In the Beer-Lambert equation, \( A = \epsilon c \ell \), absorbance depends on three factors:

• \( \epsilon \) (molar absorptivity)
• \( c \) (concentration of the absorbing species)
• \( \ell \) (the path length the light travels through the solution)

Molar absorptivity is specific to each substance and indicates how strongly the molecules absorb light at a particular wavelength. Hence, by measuring absorbance, we can deduce important information about the concentration of substances, such as \( \mathrm{H}_2\mathrm{O}_2 \) in this problem, using a known molar absorptivity and light path length.

Molar Absorptivity

Molar absorptivity, denoted as \( \epsilon \), plays a significant role in the Beer-Lambert Law equation \( A = \epsilon c \ell \). This constant tells us how well a certain substance absorbs light at a specific wavelength. Think of it as the substance's fingerprint in how it interacts with light.A high molar absorptivity means the substance is very effective at absorbing light, resulting in higher absorbance for a given concentration. This is crucial in determining concentration levels from absorbance measurements. For example, if a solution has a high molar absorptivity, even a small concentration will have a noticeable absorbance, making it easier to detect.In laboratory settings, molar absorptivity has practical applications. It allows chemists to create calibration curves with known concentrations and absorbance measures, just like in our exercise. Through this understanding, they can predict unknown concentrations of \( \mathrm{H}_2\mathrm{O}_2 \) by comparing their measured absorbance against this standard reference.

Pseudo-first Order Kinetics

When studying reactions, it's crucial to understand the order of reactions, as it dictates how concentration changes over time. In our problem, we're dealing with pseudo-first order kinetics. But what does this mean?In "normal" first order reactions, the rate of the reaction depends on the concentration of one specific reactant. However, in pseudo-first order reactions, one reactant is present in such excess that its concentration effectively doesn't change during the reaction. This simplifies the reaction order determination: it can be treated as first order with respect to the limiting reactant.In this iodide oxidation reaction, iodide and hydronium ions are in excess. Thus, the reaction rate primarily depends on the hydrogen peroxide concentration. This simplification is practical because it allows us to track the reaction kinetics as if it were only dependent on \( \mathrm{H}_2\mathrm{O}_2 \), making calculations, like determining concentration changes from absorbance readings, much less complex.