Question

Which one of the following gases has the maximum value of root mean square velocity? (a) \(\mathrm{CH}_{4}\) (b) \(\mathrm{CO}_{2}\) (c) \(\mathrm{H}_{2}\) (d) \(\mathrm{CO}\)

Short answer

The gas with the maximum root mean square velocity is \(\mathrm{H}_2\) as it has the smallest molar mass amongst the given gases.

Step-by-step solution

Determine the Molar Masses

We need to find the molar masses of the gases \(\mathrm{CH}_{4}\), \(\mathrm{CO}_{2}\), \(\mathrm{H}_{2}\), and \(\mathrm{CO}\). Molar mass can be calculated by adding up the atomic masses of each of the elements in the molecular formula.

Compare the Molar Masses

After calculating the molar masses, we then compare them. Remember, the gas with the lowest molar mass will have the highest root mean square velocity.

Identify the Gas with the Maximum Root Mean Square Velocity

Once the comparisons are done, we should be able to identify which one of the gases \(\mathrm{CH}_{4}\), \(\mathrm{CO}_{2}\), \(\mathrm{H}_{2}\), and \(\mathrm{CO}\) has the maximum root mean square velocity based on the inverse relationship between root mean square velocity and molar mass.

Key concepts

Molar Mass

Molar mass is a fundamental chemical property that represents the mass of one mole of a given substance. It is usually expressed in grams per mole (g/mol). To calculate the molar mass of a compound, you simply add up the atomic masses of all the atoms in the molecule.
For example:

• Methane (\(\text{CH}_4\)): Carbon has an atomic mass of about 12 amu, and hydrogen is approximately 1 amu. Hence, the molar mass of methane is \(12 + 4 \times 1 = 16\) g/mol.
• Carbon Dioxide (\(\text{CO}_2\)): Carbon is 12 amu, and oxygen is about 16 amu. Therefore, the molar mass of carbon dioxide is \(12 + 2 \times 16 = 44\) g/mol.
• Hydrogen (\(\text{H}_2\)): With each hydrogen atom about 1 amu, hydrogen gas has a molar mass of \(2 \times 1 = 2\) g/mol.
• Carbon Monoxide (\(\text{CO}\)): With carbon at 12 amu and oxygen at 16 amu, the molar mass is \(12 + 16 = 28\) g/mol.

This property is vital because it helps in understanding how different gases will behave, especially when considering their root mean square velocities.

Kinetic Theory of Gases

The kinetic theory of gases provides a model that helps explain the behavior and movement of gas particles. This theory is based on several interesting assumptions that simplify the study of gases. Here's a quick breakdown:

• Rapid and Random Motion: Gas particles move quickly and randomly. They collide with each other and with the walls of their container, which results in pressure exerted by the gas.
• Negligible Volume: The actual volume of gas particles is very small compared to the total volume that the gas occupies.
• No Intermolecular Forces: The forces between gas particles are very weak or negligible, which allows them to move freely.
• Elastic Collisions: When gas particles collide, they do not lose energy, as these collisions are perfectly elastic.

From these assumptions, we can derive that the temperature of a gas is a measure of the average kinetic energy of the particles. Importantly, the root mean square velocity (\(v_\text{rms}\)) can be derived using the formula \(v_\text{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is Boltzmann's constant, \(T\) is temperature (in Kelvin), and \(m\) is the mass of a gas particle. Therefore, lighter gases, having lower molar masses, will have higher root mean square velocities at the same temperature.

Atomic Mass

Atomic mass refers to the mass of a single atom, usually expressed in atomic mass units (amu). It is a crucial concept when calculating molar masses and helps in understanding gas behavior. Each element in the periodic table has a specific atomic mass, which is an average reflecting the presence of isotopes.
This atomic mass is determined by counting the protons and neutrons in an atom, as these are the subatomic particles that contribute most to an atom's total mass. Electrons are much lighter and don't significantly affect the atomic mass.
Let's illustrate this:

• Carbon (\(\text{C}\)): Carbon has an atomic mass of approximately 12 amu because it typically has 6 protons and 6 neutrons.
• Oxygen (\(\text{O}\)): Its atomic mass is about 16 amu, correlating with its 8 protons and 8 neutrons.

Understanding atomic mass allows chemists to calculate the molar mass of compounds, which in turn is essential for applications like determining root mean square velocity in gases. The inversely proportional relationship between molar mass and root mean square velocity essentially stems from atomic mass.