Question

The diffusion coefficient for \(\mathrm{CO}_{2}\) at \(273 \mathrm{K}\) and 1 atm is \(1.00 \times 10^{-5} \mathrm{m}^{2} \mathrm{s}^{-1}\). Estimate the collisional cross section of \(\mathrm{CO}_{2}\) given this diffusion coefficient.

Short answer

The collisional cross section (d) of CO2 can be estimated using the given diffusion coefficient (D) and the formula \( d = \sqrt{\frac{kt}{3 \pi D N_{A}\sqrt{2}}} \). By substituting the provided values and constants into the equation, we find that the estimated collisional cross section of CO2 is approximately \(3.56 \times 10^{-10}\) m.

Step-by-step solution

Write down the provided data

The problem gives us the following values: - Diffusion coefficient (D) = \( 1.00 \times 10^{-5} \mathrm{m}^{2} \mathrm{s}^{-1} \) - Temperature (t) = \( 273 \mathrm{K} \)

Define the constants

We'll also need the following constants: - Boltzmann constant (k) = \(1.381 \times 10^{-23} J \mathrm{K}^{-1}\) - Avogadro's number (N_A) = \(6.022 \times 10^{23} \mathrm{mol}^{-1}\)

Rearrange the equation to solve for the collisional cross section (d)

We want to find the value of d, so let's rearrange the formula to isolate d: \( D = \frac{1}{3} \frac{kt}{\pi d^{2}N_{A}\sqrt{2}} \) \( d = \sqrt{\frac{kt}{3 \pi D N_{A}\sqrt{2}}} \)

Substitute the given values and constants into the equation

Now, let's substitute the given values and constants into the equation: \( d = \sqrt{\frac{(1.381 \times 10^{-23} \mathrm{J} \mathrm{K}^{-1})(273 \mathrm{K})}{3 \pi (1.00 \times 10^{-5} \mathrm{m}^{2} \mathrm{s}^{-1}) (6.022 \times 10^{23} \mathrm{mol}^{-1})\sqrt{2}}}\)

Solve for the collisional cross section (d)

Finally, we can solve for the collisional cross section: \( d \approx 3.56 \times 10^{-10} \mathrm{m} \) So, the estimated collisional cross section of CO2 is approximately \(3.56 \times 10^{-10}\) m.

Key concepts

Diffusion Coefficient

The diffusion coefficient, often represented by the symbol 'D', plays a critical role in estimating how rapidly particles spread out in a given medium. It can be thought of as the measure of the particle's mobility: the higher the diffusion coefficient, the faster the molecules of a substance will spread throughout their environment.

The diffusion coefficient for a gas is influenced by factors such as the temperature of the system and the mass and size of the diffusing particles. Understanding this concept is essential when analyzing phenomena such as the mixing of gases or the rate at which pollutants disperse in the atmosphere.

In the provided exercise, the diffusion coefficient for carbon dioxide (\( \text{CO}_2 \)) helps us estimate the collisional cross section—a measure of the likelihood for two particles (like molecules of \( \text{CO}_2 \) in this case) to collide with each other within a given system.

Boltzmann Constant

The Boltzmann constant (k), named after the Austrian physicist Ludwig Boltzmann, is a fundamental physical constant that relates the average kinetic energy in a particle with the temperature of the system. Its value is approximately \(1.381 \times 10^{-23} J \text{K}^{-1}\) and is a key factor in various equations across thermodynamics and statistical mechanics.

This constant provides a bridge between macroscopic variables, like temperature, and microscopic entities such as individual atoms or molecules. For instance, the kinetic energy of gas particles in thermal equilibrium is directly proportional to the temperature and Boltzmann constant. Therefore, the Boltzmann constant essentially helps translate temperature into an equivalent measure of energy at the atomic and molecular scale, making it indispensable in calculating properties like the collisional cross section in the exercise.

Avogadro's Number

Avogadro's number, denoted as \( N_A \), represents the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. Its value is \(6.022 \times 10^{23} \text{mol}^{-1}\) and is named after the Italian scientist Amedeo Avogadro.

This exceptionally large number is fundamental in chemistry because it provides a link between microscopic entities and macroscopic quantities that can be easily measured in a laboratory. For example, Avogadro's number allows chemists to determine how many molecules are in a given mass of a substance and vice versa. In the context of our exercise, it is used within the equation that estimates the collisional cross section, connecting the molecular dimension to macroscopic measurements.