Question

Calculate the root mean square velocities of \(\mathrm{CH}_{4}(g)\) and \(\mathrm{N}_{2}(g)\) molecules at \(273 \mathrm{K}\) and \(546 \mathrm{K}\).

Short answer

The root mean square velocities of CH4 and N2 at 273 K and 546 K are as follows: CH4 at 273 K: \(606.62 \mathrm{m/s}\), N2 at 273 K: \(455.57 \mathrm{m/s}\), CH4 at 546 K: \(858.80 \mathrm{m/s}\), and N2 at 546 K: \(644.60 \mathrm{m/s}\).

Step-by-step solution

Calculate molecular masses of CH4 and N2

To calculate the molecular masses, we need to use the atomic masses of carbon (C), hydrogen (H), and nitrogen (N). Atomic masses (in atomic mass units - amu): - Carbon (C): 12.01 amu - Hydrogen (H): 1.01 amu - Nitrogen (N): 14.01 amu Molecular mass of CH4: 1 C atom + 4 H atoms = (1 x 12.01) + (4 x 1.01) = 12.01 + 4.04 = 16.05 amu Molecular mass of N2: 2 N atoms = 2 x 14.01 = 28.02 amu To convert these molecular masses to kilograms, we need to multiply by the conversion factor: 1 amu = \(1.66 \times 10^{-27} \mathrm{kg}\) Mass of CH4: 16.05 amu x \(1.66 \times 10^{-27} \mathrm{kg/amu}\) = \(2.66 \times 10^{-26} \mathrm{kg}\) Mass of N2: 28.02 amu x \(1.66 \times 10^{-27} \mathrm{kg/amu}\) = \(4.65 \times 10^{-26} \mathrm{kg}\)

Calculate root mean square velocities at 273 K

Now we will use the root mean square velocity formula to calculate the velocities at 273 K: CH4 at 273 K: \(v_{rms} = \sqrt{\frac{3(1.38 \times 10^{-23})(273)}{2.66 \times 10^{-26}}}\) = \(606.62 \mathrm{m/s}\) N2 at 273 K: \(v_{rms} = \sqrt{\frac{3(1.38 \times 10^{-23})(273)}{4.65 \times 10^{-26}}}\) = \(455.57 \mathrm{m/s}\)

Calculate root mean square velocities at 546 K

Now we will calculate the velocities at 546 K: CH4 at 546 K: \(v_{rms} = \sqrt{\frac{3(1.38 \times 10^{-23})(546)}{2.66 \times 10^{-26}}}\) = \(858.80 \mathrm{m/s}\) N2 at 546 K: \(v_{rms} = \sqrt{\frac{3(1.38 \times 10^{-23})(546)}{4.65 \times 10^{-26}}}\) = \(644.60 \mathrm{m/s}\)

Present the results

The root mean square velocities of CH4 and N2 are: - CH4 at 273 K: \(606.62 \mathrm{m/s}\) - N2 at 273 K: \(455.57 \mathrm{m/s}\) - CH4 at 546 K: \(858.80 \mathrm{m/s}\) - N2 at 546 K: \(644.60 \mathrm{m/s}\)

Key concepts

Molecular Mass

Understanding molecular mass is the first step in calculating the root mean square velocities of gas molecules like methane (CH _4 ) and nitrogen ( N _2 ). Molecular mass is the sum of the atomic masses of all atoms in a molecule. It's usually measured in atomic mass units (amu), but sometimes you need to convert it to kilograms for various calculations.
Here's how it works:

• For CH _4 , the molecular mass is calculated by adding the mass of one carbon atom (12.01 amu) to the mass of four hydrogen atoms (4 x 1.01 amu), totaling 16.05 amu.
• For N _2 , it's simply twice the mass of a nitrogen atom (2 x 14.01 amu), resulting in 28.02 amu.

Understanding these calculations ensures accurate velocity predictions for different gases.

Kinetic Theory of Gases

The kinetic theory of gases forms the basis for understanding gas behavior at a molecular level. This theory assumes that gas molecules are in constant, random motion, colliding with each other and the walls of their container.
This movement explains macroscopic properties like pressure and temperature. Particularly, the root mean square velocity (v_rms) is used to describe the average speed of gas molecules.

• The formula is v_rms = \( \sqrt{\frac{3kT}{m}} \), where k is the Boltzmann constant, T the temperature in Kelvin, and m the molecular mass in kilograms.

This helps predict how changes in conditions like temperature affect gas velocity.

Temperature Effect on Gases

Temperature plays a crucial role in determining the behavior of gases. As temperature increases, gas molecules move faster. This can be quantified using the root mean square velocity.
Consider the example:

• At 273 K, the v _ rms of CH _4 is 606.62 m/s, and for N _2 it's 455.57 m/s.
• Increasing the temperature to 546 K raises the velocity to 858.80 m/s for CH _4 and 644.60 m/s for N _2 .

Thus, higher temperatures result in greater molecular speeds, influencing how gases expand and how pressure builds in a confined space.

Gas Molecules

Gas molecules like CH _4 and N _2 exhibit specific behaviors under different conditions. These molecules constantly collide, and these collisions depend on the molecular size and velocity.
Gas molecules' movement and speed can be understood through the root mean square velocity, which indicates how fast molecules move on average.
When dealing with energy and reaction rates, knowing the velocity helps predict how molecules interact. This underlines why temperature, pressure, and volume need careful control in chemical reactions and industrial processes involving gases.

Atomic Mass Unit

The atomic mass unit (amu) is a standard unit for measuring the mass of atoms and molecules. Understanding this unit is essential for calculating other properties like molecular mass.
One amu is defined as \(1.66 \times 10^{-27} \mathrm{kg}\), a conversion crucial for finding molecular masses in kilograms.
For example, using this conversion, the molecular mass of CH_4 becomes \(2.66 \times 10^{-26} \mathrm{kg}\), and N_2 becomes \(4.65 \times 10^{-26} \mathrm{kg}\).
This step is vital in computing root mean square velocities, which rely on accurate mass measurements to predict molecular behavior accurately.