Question

Natural gas contains methane, \(\mathrm{CH}_{4}\), ethane, \(\mathrm{C}_{2} \mathrm{H}_{6},\) and hydrogen sulfide, \(\mathrm{H}_{2} \mathrm{~S}\). Which molecules have the fastest velocity?

Short answer

Methane \(\mathrm{CH}_{4}\) has the fastest velocity due to the lowest molar mass.

Step-by-step solution

Understand the Contents

Identify the components in natural gas: methane \(\mathrm{CH}_{4}\), ethane \(\mathrm{C}_{2}\mathrm{H}_{6}\), and hydrogen sulfide \(\mathrm{H}_{2}\mathrm{~S}\).

Recall the Formula for Molecular Velocity

The velocity of a gas molecule is inversely proportional to the square root of its molar mass. The formula to determine this is derived from the Kinetic Molecular Theory: \( v \propto \frac{1}{\sqrt{M}} \), where \( v \) is the velocity and \( M \) is the molar mass of the molecule.

Calculate Molar Masses

Calculate the molar masses of the gases: \( \mathrm{CH}_{4} \) has a molar mass of \( 12 + 4 \times 1 = 16 \) g/mol, \( \mathrm{C}_{2}\mathrm{H}_{6} \) has a molar mass of \( 2 \times 12 + 6 \times 1 = 30 \) g/mol, and \( \mathrm{H}_{2}\mathrm{~S} \) has a molar mass of \( 2 \times 1 + 32 = 34 \) g/mol.

Relate Molar Mass to Velocity

Since velocity is inversely proportional to the square root of molar mass, the molecule with the smallest molar mass has the highest velocity.

Determine the Fastest Molecule

\( \mathrm{CH}_{4} \) has the smallest molar mass among \( \mathrm{CH}_{4}\), \( \mathrm{C}_{2}\mathrm{H}_{6}\), and \( \mathrm{H}_{2}\mathrm{~S} \), so it will have the fastest velocity.

Key concepts

Molecular Velocity

The concept of molecular velocity is an intriguing aspect of the Kinetic Molecular Theory, which relates the speed of molecules to their physical properties. According to this theory, the velocity of a gas molecule is not constant because molecules are always in motion, colliding with one another and the walls of their container. However, there's a way to approximate the average speed using the molar mass of the molecule.
One important formula related to this is the relationship between velocity and molar mass:\[ v \propto \frac{1}{\sqrt{M}} \]where \( v \) represents the molecular velocity and \( M \) is the molar mass of the gas. This implies that as the molar mass increases, the speed decreases, and vice versa. This inverse relationship is crucial for predicting molecular behaviors in various conditions.
When applying this to natural gases like methane (\( \mathrm{CH}_4 \)), ethane (\( \mathrm{C}_2\mathrm{H}_6 \)), and hydrogen sulfide (\( \mathrm{H}_2\mathrm{S} \)), it's clear that methane, with its lower molar mass, will generally move faster than the other heavier molecules.

Molar Mass

Molar mass is a fundamental property of substances that plays an essential role in determining molecular velocity, as discussed above. It is defined as the mass of one mole of a given substance, typically expressed in grams per mole (g/mol).
The molar mass can be calculated by summing the atomic masses of all the atoms in a molecule. For instance:

• Methane (\(\mathrm{CH}_4\)): The carbon atom has an atomic mass of 12 g/mol, and each hydrogen atom has an atomic mass of 1 g/mol, resulting in a total molar mass of 16 g/mol.
• Ethane (\(\mathrm{C}_2\mathrm{H}_6\)): With two carbon atoms and six hydrogen atoms, its total molar mass is 30 g/mol.
• Hydrogen sulfide (\(\mathrm{H}_2\mathrm{S}\)): Two hydrogen atoms and one sulfur atom sum up to a molar mass of 34 g/mol.

Understanding molar mass allows scientists and students to predict and compare various molecular properties, like determining which molecules in a gas mixture will move the fastest, as each molecule's speed is reliant on its molar mass.

Natural Gas Composition

Natural gas is primarily a mix of methane, ethane, and sometimes hydrogen sulfide, among other minor components. Each of these gases has different properties that make them suitable for various applications, mostly in energy production.
Methane (\(\mathrm{CH}_4\)) is the primary constituent of natural gas and contributes significantly to its energy-generating capabilities. Due to its small size and light molar mass, methane molecules typically travel faster than heavier molecules like ethane or hydrogen sulfide.
Ethane (\(\mathrm{C}_2\mathrm{H}_6\)) is another prominent component often separated for use as a petrochemical feedstock. It has higher energy content per mole than methane but moves slower due to its greater molar mass.
In smaller quantities, hydrogen sulfide (\(\mathrm{H}_2\mathrm{S}\)) is found in natural gas, known for being toxic and having a rotten egg smell. It is heavier than both methane and ethane, so it contributes less significantly to the molecular velocity average of the gas mixture. Understanding these components provides insights into the behavior and uses of natural gas in various industrial applications.