Problem 19 - An Introduction to Thermal Physics

Question

Suppose you have a gas containing hydrogen molecules and oxygen molecules, in thermal equilibrium. Which molecules are moving faster, on average? By what factor?

Step-by-Step Solution

Answer

Hydrogen molecules move faster; they are 4 times faster than oxygen molecules.

Step 1: Understanding Molecular Speed

The average speed of a gas molecule is related to its mass and the temperature of the gas. According to the kinetic theory of gases, lighter molecules move faster than heavier molecules at the same temperature because they have more kinetic energy.

Step 2: Using the Kinetic Energy Formula

The average kinetic energy of a gas molecule is given by the formula: \[ \frac{1}{2} m v^2 = \frac{3}{2} k T \] where \( m \) is the mass of the molecule, \( v \) is the speed, \( k \) is the Boltzmann constant, and \( T \) is the absolute temperature. For molecules at the same temperature, the kinetic energy is equal.

Step 3: Expressing Speed in Terms of Mass

Rearranging the kinetic energy formula to solve for speed gives:\[ v = \sqrt{\frac{3kT}{m}} \]Since the temperature \( T \) and the Boltzmann constant \( k \) are constants for both gases, \( v \) varies inversely with the square root of the mass \( m \).

Step 4: Calculating the Speed Ratio

The molecular mass of hydrogen (\(H_2\)) is approximately 2 amu, and for oxygen (\(O_2\)), it is approximately 32 amu. Thus, the ratio of speeds is determined by the inverse square root of their masses:\[ \frac{v_{H_2}}{v_{O_2}} = \sqrt{\frac{m_{O_2}}{m_{H_2}}} = \sqrt{\frac{32}{2}} = 4 \]

Step 5: Conclusion

Hydrogen molecules move faster on average than oxygen molecules in thermal equilibrium. The speed of hydrogen molecules is \(4\) times the speed of oxygen molecules.

Key Concepts

Molecular Speed

In the realm of gases, molecular speed refers to how quickly individual molecules zip around their container. The kinetic theory of gases gives us a deeper understanding of this concept. It tells us that molecules in a gas are in constant, random motion. The speed at which they move is influenced by two key factors:

• The mass of the molecules
• The temperature of the gas

Lighter molecules tend to move faster than heavier ones when they are at the same temperature. This is because lighter molecules need to travel faster to have the same kinetic energy as their heavier counterparts.
In our given example, hydrogen molecules, which are lighter than oxygen molecules, move faster in thermal equilibrium. By analyzing the mass of hydrogen ( 2 atomic mass units, or amu) and oxygen (32 amu), you see why hydrogen zips by more quickly.

Kinetic Energy

Kinetic energy is the energy of motion, and for gas molecules, it measures how much energy is in their movements. According to the formula:\[\frac{1}{2} m v^2 = \frac{3}{2} k T\]This states that the average kinetic energy of a gas molecule depends on its mass (\(m\)) and speed (\(v\)), alongside the temperature (\(T\)) and the Boltzmann constant (\(k\)).

When you're exploring gases, the fascinating aspect is that all molecules at the same temperature will have the same average kinetic energy. It's about balance—faster speeds for lighter molecules and slower speeds for heavier ones.

In our case, even though hydrogen molecules move faster, oxygen molecules have more mass, and thus, both have the same average kinetic energy. This balance is achieved because each molecule's \(v^2\) compensates for its mass to maintain equilibrium.

Thermal Equilibrium

Thermal equilibrium is a state where two or more systems interacting with each other don't exchange heat—they've reached a balanced state at the same temperature. In this state, energy is still in motion, but there is no net change in energy between systems.
This equilibrium underlies our understanding of molecular speeds and kinetic energy.

When gases like hydrogen and oxygen reach thermal equilibrium, their molecules settle into a state where their average kinetic energies are equal. However, because these gases differ in mass, their molecular speeds differ even though they're in equilibrium. This concept is essential to understanding gas behaviors because the energy distribution becomes stable across different gases.

• Each molecule's speed is influenced by its mass and the prevailing temperature.
• Energy is uniformly distributed across the molecules at equilibrium, though their speeds may differ.

These principles allow us to predict behaviors in various mixtures of gases—highlighting why lighter gases move faster even though they share the same kinetic energy as heavier gases.