Question

At \(25^{\circ} \mathrm{C}\), the rate constant for the ozone-depleting reaction: $$ \mathrm{O}(g)+\mathrm{O}_{3}(g) \longrightarrow 2 \mathrm{O}_{2}(g) $$ is \(7.9 \times 10^{-15} \mathrm{~cm}^{3} / \mathrm{molecule} \cdot \mathrm{s}\). Express the rate constant in units of \(1 / M \cdot \mathrm{s}\).

Short answer

The rate constant is \(4.756 \times 10^{12} \; 1/M \cdot \mathrm{s}\).

Step-by-step solution

Understand the Units

The given rate constant is in the units of \(\mathrm{cm}^{3} / \mathrm{molecule} \cdot \mathrm{s}\). We need to convert it to \(1 / M \cdot \mathrm{s}\), where \(M\) stands for molarity (moles per liter).To achieve this, we require the conversion between molecules and moles using Avogadro's number, \(N_A = 6.022 \times 10^{23} \; \mathrm{molecules/mol}\).We also need to convert cubic centimeters to liters (where \(1 \; \mathrm{L} = 10^3 \; \mathrm{cm}^3\)).

Conversion from Cubic Centimeters to Liters

Convert \(\mathrm{cm}^3\) to liters for the volume component of the rate constant.Since \(1 \; \mathrm{L} = 10^3 \; \mathrm{cm}^3\),do the following transformation:\[7.9 \times 10^{-15} \; \mathrm{cm}^{3} / \mathrm{molecule} \cdot \mathrm{s} = 7.9 \times 10^{-15} \times 10^3 \; \mathrm{L} / \mathrm{molecule} \cdot \mathrm{s}.\]This results in:\(7.9 \times 10^{-12} \; \mathrm{L} / \mathrm{molecule} \cdot \mathrm{s}\).

Use Avogadro's Number to Convert Molecules to Moles

Next, convert molecules to moles:The mole conversion involves substituting \(\mathrm{molecule}^{-1}\) with \(\mathrm{mol}^{-1}\) using Avogadro's number\(N_A = 6.022 \times 10^{23} \; \mathrm{molecules/mol}\).Use the transformation:\[(7.9 \times 10^{-12} \; \mathrm{L} / \mathrm{molecule} \cdot \mathrm{s}) \times 6.022 \times 10^{23} \; \mathrm{molecules/mol}.\]This yields:\[7.9 \times 10^{-12} \times 6.022 \times 10^{23} \; \mathrm{L} / \mathrm{mol} \cdot \mathrm{s} = 4.756 \times 10^{12} \; \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}.\]

Express in Molarity Units

The result from Step 3,\(4.756 \times 10^{12} \; \mathrm{L/mol} \cdot \mathrm{s}\),is equivalent to\( 4.756 \times 10^{12} \; 1/M \cdot \mathrm{s}\),since 1 M = 1 mole/liter.Thus, the final value in desired units is:\(\boxed{4.756 \times 10^{12} \; 1/M \cdot \mathrm{s}}.\)

Key concepts

Ozone-Depleting Reaction

Ozone-depleting reactions are chemical processes that reduce the concentration of ozone in the Earth's stratosphere. One important reaction involves atomic oxygen (\(\text{O}\)) reacting with ozone (\(\text{O}_3\)) to produce oxygen gas (\(\text{O}_2\)). This reaction contributes to the thinning of the ozone layer, which protects the planet from harmful ultraviolet radiation. Understanding the rate constant of such reactions is essential to assess their impact on atmospheric chemistry. By converting the units of the rate constant, scientists can better relate these values to conditions useful in environmental studies.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, defined as the number of atoms, ions, or molecules in one mole of a substance. Its value is approximately \(6.022 \times 10^{23}\). This large number allows chemists to count particles at the microscopic level by relating them to a quantity, the mole, which is easier to work with in the lab and calculations.
When converting units from molecules to moles, Avogadro's number is crucial. In this problem, it transforms the rate constant from per molecule to per mole, allowing the rate to be expressed in terms of molarity, which is standard in chemical kinetics.

Molarity

Molarity represents the concentration of a solute in a solution. It's expressed in moles of solute per liter of solution \((M =\text{mol/L})\).
Molarity is an important concept because it provides a simple way to express concentration that is widely applicable in chemical reactions. In this exercise, converting the rate constant to units involving molarity makes it more accessible for use in lab experiments and calculations. This unit conversion allows scientists to predict how fast reactions occur under different conditions in a way that is relatable to experimental setups.

Unit Conversion

Unit conversion involves changing a measurement from one set of units to another. It is essential in science to ensure that values can be easily compared and used in different contexts.
For this problem, the conversion from \(\text{cm}^3/\text{molecule}\cdot\text{s} \) to \(1/M\cdot\text{s}\) requires two main steps:

• Converting cubic centimeters to liters using the relation \(1 \text{ L} = 1000 \text{ cm}^3\).
• Using Avogadro's number to transition from molecules to moles.

By doing these conversions, we effectively translate the rate constant into a format that aligns with standard chemical practices, enhancing its interpretability and utility in experiments.