Problem 45 - General Chemistry

Question

Chlorine dioxide, \(\mathrm{ClO}_{2}\), is a reddish yellow gas used in bleaching paper pulp. The average speed of a \(\mathrm{ClO}_{2}\) molecule at \(25^{\circ} \mathrm{C}\) is \(306 \mathrm{~m} / \mathrm{s}\). What is the kinetic energy (in joules) of a \(\mathrm{ClO}_{2}\) molecule moving at this speed?

Step-by-Step Solution

Answer

The kinetic energy is approximately \( 5.24 \times 10^{-21} \text{ J} \).

Step 1: Understanding Kinetic Energy Formula

The kinetic energy (KE) of a moving object can be found using the formula: \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. In this problem, \( v \) is given as \(306 \text{ m/s}\).

Step 2: Determining the Mass of a ClO2 Molecule

The molar mass of \( ClO_2 \) is approximately \( 67.45 \text{ g/mol} \). To find the mass of a single \( ClO_2 \) molecule, convert this to kilograms (since 1 g = 0.001 kg) and divide by Avogadro's number (\( 6.022 \times 10^{23} \text{ molecules/mol} \)). Thus, \( m = \frac{67.45 \times 10^{-3} \text{ kg/mol}}{6.022 \times 10^{23} \text{ molecules/mol}} \approx 1.12 \times 10^{-25} \text{ kg/molecule} \).

Step 3: Calculating the Kinetic Energy of ClO2

Now plug the mass \( m = 1.12 \times 10^{-25} \text{ kg} \) and velocity \( v = 306 \text{ m/s} \) into the kinetic energy formula \( KE = \frac{1}{2}mv^2 \): \( KE = \frac{1}{2} \times 1.12 \times 10^{-25} \times (306)^2 \). \ Calculating gives: \( KE = 5.24 \times 10^{-21} \text{ J} \).

Step 4: Conclusion on Kinetic Energy

The kinetic energy of a \( ClO_2 \) molecule moving at 306 m/s is approximately \( 5.24 \times 10^{-21} \text{ J} \).

Key Concepts

Molecular Speed

Molecular speed refers to the velocity at which a molecule moves, often influenced by the temperature of the environment. The average molecular speed provides insight into the energy and dynamics of molecules in a gas. Since molecules are constantly in motion, understanding this speed is crucial for calculating other properties like kinetic energy.

• The equation most commonly used is the root-mean-square speed, which is derived from the kinetic theory of gases.
• This speed typically increases with temperature, as molecules absorb more energy and move faster.

In the case of chlorine dioxide (ClO₂) at room temperature (25°C), its average speed is 306 m/s.

Chlorine Dioxide

Chlorine dioxide (ClO₂) is a reddish-yellow gas widely used for bleaching and disinfecting, especially in the paper and pulp industry. It is different from chlorine gas (Cl₂), primarily due to its unique chemical structure and properties that make it a more effective bleaching agent.

• Unlike chlorine, it's a neutral molecule, combining chlorine and oxygen, which makes it less corrosive.
• It decomposes rapidly under sunlight, thus is often stored and used under controlled conditions.

ClO₂'s chemical properties and applications benefit various industrial processes, ensuring materials are treated safely and effectively.

Kinetic Energy Calculation

Kinetic energy is the energy that a molecule possesses due to its motion. It's calculated using the formula: \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the molecule. This formula helps determine how much energy a molecule like ClO₂ carries as it travels at a given speed.

• In the example, the speed of ClO₂ is 306 m/s.
• After determining the mass of a single ClO₂ molecule, one can substitute these values into the formula to find its kinetic energy.

Hence, understanding these variables and their relation is key to calculating kinetic energy in molecular physics.

Molecular Mass

Molecular mass is the sum of the masses of all atoms in a molecule. It's an essential aspect of molecular science, providing the basis for converting between mass and number of molecules. Molecular mass is typically measured in grams per mole (g/mol), which assists in calculations involving Avogadro's number for individual molecule mass.

• The molecular mass of ClO₂ is approximately 67.45 g/mol.
• To find the mass of a single ClO₂ molecule, divide the molar mass by Avogadro's number.
• This converts the mass into kilograms, fitting kinetic energy calculations.

Accurately determining molecular mass is crucial for reliable scientific and industrial applications.

Temperature Effect on Molecules

Temperature is a measure of the average energy of motion of molecules in a substance. As temperature increases, molecules move faster—that's why temperature is a pivotal factor in molecular kinetics.

• Higher temperatures provide molecules with more kinetic energy, increasing their speed.
• This change in speed with temperature is crucial for reactions, stability, and phase changes.

For molecules like ClO₂, an increase in temperature translates to a higher average speed, affecting how they interact and react in their environment.