Question

Determine the root-mean-square speed of \(\mathrm{CO}_{2}\) molecules that have an average kinetic energy of \(3.2 \times 10^{-21} \mathrm{J}\) per molecule.

Short answer

Answer: The root-mean-square speed of CO2 molecules with an average kinetic energy of \(3.2 \times 10^{-21} \mathrm{J}\) per molecule is approximately 348 m/s.

Step-by-step solution

Determine the formula for the root-mean-square speed

For an ideal gas, the root-mean-square speed (\(v_\text{rms}\)) is related to the average kinetic energy (\(KE\)) per molecule using the formula: \(KE = \frac{1}{2} m v_\text{rms}^2\) Here, \(m\) represents the mass of one CO2 molecule.

Calculate the mass of one CO2 molecule

First, we need to determine the mass of one CO2 molecule. CO2 has one carbon atom (mass = 12 amu) and two oxygen atoms (mass = 16 amu each). The molecular mass of CO2 is therefore: \(M_\text{CO2} = 12 + (2\times 16) = 44 \thinspace \text{amu}\) Next, we need to convert this mass from atomic mass units (amu) to kilograms, using the conversion factor 1 amu = \(1.66 \times 10^{-27} \mathrm{kg}\). \(m = M_\text{CO2} \times (1.66 \times 10^{-27} \mathrm{kg/amu})\) \(m = 44 \times (1.66 \times 10^{-27} \mathrm{kg}) = 7.3 \times 10^{-26} \mathrm{kg}\)

Rearrange the formula to solve for \(v_\text{rms}\)

Now, we will rearrange the formula relating average kinetic energy and root-mean-square speed to solve for \(v_\text{rms}\): \(v_\text{rms} = \sqrt{\frac{2\times KE}{m}}\)

Calculate the root-mean-square speed

Input the values for \(KE\) and \(m\) into the equation to find the root-mean-square speed of CO2 molecules: \(v_\text{rms} = \sqrt{\frac{2\times (3.2 \times 10^{-21} \mathrm{J})}{(7.3 \times 10^{-26} \mathrm{kg})}}\) \(v_\text{rms} = \sqrt{\frac{6.4 \times 10^{-21} \mathrm{J}}{7.3 \times 10^{-26} \mathrm{kg}}}\) \(v_\text{rms} \approx 348 \thinspace\mathrm{m/s}\) The root-mean-square speed of CO2 molecules with an average kinetic energy of \(3.2 \times 10^{-21} \mathrm{J}\) per molecule is approximately 348 m/s.

Key concepts

Kinetic energy

Kinetic energy is the energy possessed by an object due to its motion. For gas molecules, kinetic energy relates to how quickly they move. The formula often used to represent kinetic energy (\(KE\)) of a single molecule is:

• \(KE = \frac{1}{2} m v^2\)

Here, \(m\) stands for the mass of the molecule, and \(v\) is its velocity.
In the context of gases, especially ideal gases, we think about many molecules moving at various speeds.
Since calculating the speed of each molecule isn't feasible, we use measures like the average kinetic energy, which provides an average value of energy across all molecules in the gas.
This average energy reflects in the temperature of the gas, as higher temperatures indicate greater kinetic energies. By knowing the kinetic energy of molecules, like in our example of \(\mathrm{CO}_2\), we can further explore concepts like root-mean-square speed which offers an average speed accounting for all molecules' kinetic energies.

Ideal gas law

The Ideal Gas Law is a fundamental equation in thermodynamics that describes relationships between pressure, volume, temperature, and the number of moles of a gas. The equation is:

• \(PV = nRT\)

where:

• \(P\) represents pressure,
• \(V\) is volume,
• \(n\) stands for the number of moles,
• \(R\) is the ideal gas constant,
• and \(T\) is temperature.

This law assumes all gas particles are in constant random motion and experience elastic collisions—where kinetic energy is conserved.
While the ideal gas law itself doesn't directly calculate properties like root-mean-square speed or kinetic energy, it frames the behaviors of gases and underpins other scientific principles.
For example, with known pressures, volumes, and temperatures, we can infer information about the density and kinetic behavior of a gas. It also helps modify other kinetic equations to include more variables, showing ideal gases' predictability and reliability under controlled conditions.

Molecular mass calculation

Molecular mass is crucial for understanding the characteristics of substances, particularly in determining the behavior of gases. In molecular mass calculations, we sum the atomic masses of all atoms in a molecule.

• For a \(\mathrm{CO}_2\) molecule, this involves adding the mass of one carbon atom (12 amu) to the mass of two oxygen atoms (each 16 amu), resulting in a molecular mass of 44 amu.

To work with gases at a macroscopic level, we convert this molecular mass to kilograms because the SI unit of mass is the kilogram.
The conversion uses: 1 amu = \(1.66 \times 10^{-27}\) kg.
By calculating the mass of a single molecule in kilograms, we establish its relevance in kinetic equations determining the speed or energy of gas molecules.Understanding molecular mass underlies tasks like deducing a molecule's speed in an ideal gas.
It serves as a bridge between atomic-level measurements and practical applications in calculations around energy and speed, especially when exploring concepts like root-mean-square speed.