Problem 44 - Chemistry and Chemical Reactivity

Question

Calculate the rms speed for CO molecules at \(25^{\circ} \mathrm{C} .\) What is the ratio of this speed to that of \(\mathrm{Ar}\) atoms at the same temperature?

Step-by-Step Solution

Answer

The ratio of rms speeds of CO to Ar at 25°C is \(\sqrt{28.01/39.95}\).

Step 1: Convert Temperature to Kelvin

To calculate the root mean square (rms) speed, we need the temperature in Kelvin. The temperature in Celsius is given as 25°C. Convert this to Kelvin by adding 273.15: \[T = 25 + 273.15 = 298.15 \text{ K}\]

Step 2: Use the RMS Speed Formula

The formula for the root mean square speed is given by \[v_{rms} = \sqrt{\frac{3kT}{m}}\]where:- \(k\) is the Boltzmann constant \((1.38 \times 10^{-23} \text{ J/K})\)- \(T\) is the temperature in Kelvin- \(m\) is the mass of a single molecule in kilograms. For CO, the molar mass is 28.01 g/mol, so the mass of one molecule is 28.01 g/mol divided by Avogadro's number \((6.022 \times 10^{23} \text{ mol}^{-1})\) converted to kg.\[m_{CO} = \frac{28.01 \times 10^{-3}}{6.022 \times 10^{23}} \text{ kg}\]

Step 3: Calculate RMS Speed for CO

Substitute the values into the RMS speed formula:\[v_{rms, CO} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298.15}{\frac{28.01 \times 10^{-3}}{6.022 \times 10^{23}}}}\]Calculate \(v_{rms, CO}\).

Step 4: Calculate RMS Speed for Ar

Repeating the same process for Argon (Ar), which has a molar mass of 39.95 g/mol:\[m_{Ar} = \frac{39.95 \times 10^{-3}}{6.022 \times 10^{23}} \text{ kg}\]Use the RMS speed formula:\[v_{rms, Ar} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298.15}{\frac{39.95 \times 10^{-3}}{6.022 \times 10^{23}}}}\]

Step 5: Calculate the Ratio of RMS Speeds

Finally, calculate the ratio of the rms speeds of CO to Ar:\[\text{Ratio} = \frac{v_{rms, CO}}{v_{rms, Ar}}\]Substitute values from the previous calculations to find the ratio.

Key Concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory is a fundamental concept in physics and chemistry that explains the behavior of gases. It provides a detailed insight into how gas molecules move and interact. The theory suggests that gases are made up of a huge number of tiny particles, which are in constant, random motion. This motion is responsible for the pressure exerted by gases in a container.

• Molecules move in straight lines until they collide with each other or the walls of a container.
• Collisions are elastic, which means that the total kinetic energy is conserved.
• The average kinetic energy of gas molecules is proportional to the temperature in Kelvins.

The root mean square (rms) speed is an application of this theory. It specifically relates to the average speed of gas particles in a sample and is a measure of how fast the molecules are moving on average. In essence, the higher the temperature, the faster the molecules move, which is consistent with an increase in the rms speed.

Molecular Mass

Molecular mass, often referred to as molar mass, is the mass of a given molecule. It is calculated as the sum of the atomic masses of its constituent atoms. Molecular mass is crucial when determining the rms speed, as it affects how fast or slow a molecule can go.

• It is usually expressed in grams per mole (g/mol).
• To find the mass of a single molecule, the molecular mass is divided by Avogadro's number (approximately \(6.022 \times 10^{23}\) molecules per mole).

Take carbon monoxide (CO) as an example. Its molecular mass is 28.01 g/mol. Converting it to kg per molecule involves dividing this number by Avogadro's number and converting grams to kilograms. Hence, a molecule's molecular mass plays a key role in computing its rms speed since heavier molecules will generally move more slowly than lighter ones at the same temperature.

Boltzmann Constant

The Boltzmann constant (\(k\)) is a key player in physical chemistry and thermodynamics, especially within the realm of gases. It relates the average kinetic energy of particles in a gas with the temperature of the gas.

• Its value is approximately \(1.38 \times 10^{-23} \text{ J/K}\).
• It acts as a bridge, connecting the microscopic world, like atoms and molecules, with macroscopic properties such as temperature.

In the formula for rms speed, the Boltzmann constant helps relate the temperature of a system to the average kinetic energy and hence the speed of its molecules. By understanding this constant, one can appreciate how molecular motion correlates to temperature, providing a deeper understanding of the behavior of gases through the lens of the Kinetic Molecular Theory.