The decomposition of hydrogen peroxide (H₂O₂) into water and oxygen is a classic example of first-order kinetics:

\[\ce{2H2O2 -> 2H2O + O2}\]

The rate of decomposition follows first-order kinetics, which means:

\[\frac{d[H_2O_2]}{dt} = -k[H_2O_2]\]

where: - \([H_2O_2]\) is the concentration of hydrogen peroxide in \(\text{mol}\cdot\text{L}^{-1}\) - \(k\) is the rate constant in \(\text{s}^{-1}\) - \(t\) is time in seconds

The integrated rate law gives us:

\[[H_2O_2]_t = [H_2O_2]_0 e^{-kt}\]

When we plot \(\ln([H_2O_2]_t/[H_2O_2]_0)\) versus time, we get a straight line with slope \(-k\):

\[\ln\left(\frac{[H_2O_2]_t}{[H_2O_2]_0}\right) = -kt\]

At room temperature (\(25^\circ\text{C}\)), the half-life (\(t_{1/2}\)) of this reaction is approximately 24 hours. We can calculate the rate constant using:

\[k = \frac{\ln(2)}{t_{1/2}} = \frac{0.693}{86400\text{ s}} = 8.02 \times 10^{-6}\text{ s}^{-1}\]

This means that in any given second, about 0.0008% of the remaining H₂O₂ molecules decompose.

The activation energy (\(E_a\)) for this reaction is approximately \(75\text{ kJ}\cdot\text{mol}^{-1}\), which we can plug into the Arrhenius equation:

\[k = A e^{-\frac{E_a}{RT}}\]

where: - \(A\) is the pre-exponential factor - \(R\) is the gas constant (\(8.314\text{ J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\)) - \(T\) is temperature in Kelvin

Experimental Data

Here’s some sample data from our H₂O₂ decomposition experiment at 25°C:

We can visualize this data with a plot:

As we can see from both the data and the plot, the relationship between ln([H₂O₂]/[H₂O₂]₀) and time is linear, confirming that this is indeed a first-order reaction.

The slope of this line gives us our rate constant \(k\):

\[k = -\text{slope} = -\left(\frac{-1.033}{30\text{ hr}}\right) = 0.0344\text{ hr}^{-1}\]

Converting to SI units: \[k = 0.0344\text{ hr}^{-1} \times \frac{1\text{ hr}}{3600\text{ s}} = 9.56 \times 10^{-6}\text{ s}^{-1}\]

This experimental value is close to our theoretical calculation from earlier!

This example shows how mathematics helps us understand and predict chemical reactions. The beauty of first-order kinetics lies in its simplicity and wide applicability across different chemical systems.

A Note on Measurement Units

In chemistry, we often deal with various units. Here are some common ones used in kinetics:

• Concentration: \(\text{mol}\cdot\text{L}^{-1}\) or \(\text{M}\)
• Rate constants: \(\text{s}^{-1}\) (first order)
• Temperature: \(\text{K}\) or \(^\circ\text{C}\)
• Energy: \(\text{kJ}\cdot\text{mol}^{-1}\)

Remember that proper unit analysis is crucial in chemical calculations!