AP Chemistry Unit 3.5 The Kinetic Molecular Theory of Gases Notes - New Syllabus 2024-2025
AP Chemistry Unit 3.5 The Kinetic Molecular Theory of Gases Notes – New syllabus
AP Chemistry Unit 3.5 The Kinetic Molecular Theory of Gases Notes – AP Chemistry – per latest AP Chemistry Syllabus.
LEARNING OBJECTIVE
Explain the relationship between the motion of particles and the macroscopic properties of gases with:
i. The kinetic molecular theory (KMT).
ii. A particulate model.
iii. A graphical representation.
Key Concepts:
- Kinetic Molecular Theory
- The Maxwell Boltzmann Distribution
- Temperature and Average Kinetic Energy
- The Maxwell–Boltzmann Distribution
Kinetic Molecular Theory
The Kinetic Molecular Theory explains the macroscopic properties of gases (such as pressure, temperature, and volume) in terms of the microscopic motion of gas particles.
Key Assumptions of KMT:
- Gas particles are in constant, random motion and move in straight lines until they collide.
- The volume of individual gas particles is negligible compared to the total volume of the gas.
- Collisions between gas particles and with container walls are perfectly elastic—no energy is lost overall.
- There are no intermolecular attractions or repulsions between gas particles.
- The average kinetic energy of particles depends only on the gas temperature (in kelvin).
Key Idea: KMT provides a molecular-level interpretation of macroscopic gas properties such as pressure (due to collisions with container walls) and temperature (proportional to average kinetic energy).
Example :
Explain, in terms of KMT, why gas pressure increases when temperature increases at constant volume.
▶️ Answer/Explanation
Step 1: Increasing temperature raises the average kinetic energy of gas molecules.
Step 2: Faster-moving particles collide with container walls more frequently and with greater force.
Final Answer: Increased collision frequency and impact force lead to higher pressure, consistent with KMT.
Kinetic Energy and Molecular Velocity
The motion of gas particles can be quantified using the relationship between a particle’s kinetic energy and its velocity:
\( \mathrm{KE = \dfrac{1}{2} m v^2} \)
- \( \mathrm{KE} \): kinetic energy of a single particle
- \( \mathrm{m} \): mass of the particle (kg)
- \( \mathrm{v} \): instantaneous velocity (m/s)
Key Properties:
- All gas particles are in continuous, random motion, but not all move at the same speed.
- The kinetic energy distribution depends on temperature—higher temperature → higher average kinetic energy.
- At a given temperature, lighter particles move faster on average than heavier ones.
Average Molecular Speed Relationships:
The root mean square speed \( \mathrm{v_{rms}} \) represents the average speed of gas molecules:
\( \mathrm{v_{rms} = \sqrt{\dfrac{3RT}{M}}} \)
- \( \mathrm{R} \): gas constant (\(8.314\, \mathrm{J·mol^{-1}·K^{-1}}\))
- \( \mathrm{T} \): absolute temperature (K)
- \( \mathrm{M} \): molar mass of the gas (kg/mol)
Key Idea: The average kinetic energy of all gases is the same at a given temperature, but lighter gases move faster because their mass is smaller.
Example :
At 298 K, which gas has a higher average molecular speed—hydrogen (\(\mathrm{H_2}\)) or oxygen (\(\mathrm{O_2}\))? Justify your answer using the relationship between kinetic energy and molecular velocity.
▶️ Answer/Explanation
Step 1: According to the kinetic molecular theory, the average kinetic energy of all gases at the same temperature is the same.
\( \mathrm{KE_{avg} = \dfrac{1}{2} m v^2} \)
Step 2: Since \(\mathrm{KE_{avg}}\) is constant for both gases at the same temperature, the gas with the smaller molar mass must have a higher velocity to maintain the same kinetic energy.
Step 3: Use the expression for root-mean-square (rms) speed:
\( \mathrm{v_{rms} = \sqrt{\dfrac{3RT}{M}}} \)
Step 4: \(\mathrm{M_{H_2} = 2.0 \, g/mol}\) and \(\mathrm{M_{O_2} = 32.0 \, g/mol}\). Since \( \mathrm{v_{rms}} \propto \dfrac{1}{\sqrt{M}} \), the smaller the molar mass, the greater the molecular speed.
Step 5: Therefore, \(\mathrm{H_2}\) molecules move much faster than \(\mathrm{O_2}\) molecules at the same temperature.
Final Answer: Both gases have the same average kinetic energy at 298 K, but hydrogen (\(\mathrm{H_2}\)) moves faster because it has a smaller molar mass than oxygen (\(\mathrm{O_2}\)).
Temperature and Average Kinetic Energy
The Kelvin temperature of a substance is directly proportional to the average kinetic energy of its particles.
\( \mathrm{KE_{avg} = \dfrac{3}{2}RT} \)
- \( \mathrm{KE_{avg}} \): average kinetic energy per mole of gas particles
- \( \mathrm{T} \): absolute temperature in kelvin
Key Relationships:
- When temperature doubles (in kelvin), the average kinetic energy also doubles.
- Since \( \mathrm{KE \propto T} \), higher temperature → faster molecular motion and higher molecular speeds.
- At 0 K, all molecular motion theoretically ceases (absolute zero).
Key Idea: Temperature is a direct measure of the average kinetic energy of particles in a system. Thermal energy changes manifest as changes in molecular motion.
Example :
Compare the average kinetic energies of helium and xenon gases at 300 K.
▶️ Answer/Explanation
Step 1: According to KMT, average kinetic energy depends only on temperature, not mass.
Step 2: Since both gases are at the same temperature, \( \mathrm{KE_{avg}} \) is identical for helium and xenon.
Step 3: However, helium atoms (lighter) move faster than xenon atoms to have the same kinetic energy.
Final Answer: \( \mathrm{KE_{He} = KE_{Xe}} \), but \( \mathrm{v_{He} > v_{Xe}} \).
The Maxwell–Boltzmann Distribution
The Maxwell–Boltzmann distribution is a probability curve that describes the spread of molecular speeds (or energies) in a gas at a given temperature.
Key Features of the Distribution:
- The curve shows the fraction of molecules moving at each speed.
- Most particles move near an average speed, but some move much slower or faster.
- The curve’s area represents the total number of particles—it remains constant as temperature changes.
Effect of Temperature:
At higher temperatures:
- The curve flattens and broadens.
- The most probable speed (\(v_p\)) and average speed increase.
- A greater fraction of molecules have higher kinetic energies.
At lower temperatures:
- The curve becomes narrower and taller.
- Speeds cluster closer to the mean; fewer high-energy particles exist.
Effect of Molar Mass:
For gases at the same temperature:
- Lighter molecules (smaller M) → faster average speeds.
- Heavier molecules (larger M) → slower average speeds.
The Maxwell–Boltzmann distribution illustrates that gas particle speeds vary widely and depend on both temperature and molar mass. Temperature shifts the curve to higher energies; mass shifts it horizontally (lighter → faster).
Kinetic Molecular Theory and Distribution of Molecular Speeds
| Concept | Key Relationship | Main Implication |
|---|---|---|
| Kinetic Molecular Theory | Gas behavior explained by particle motion | Pressure due to collisions; temperature ↔ energy |
| Kinetic Energy | \( \mathrm{KE = \frac{1}{2} m v^2} \) | At same T, lighter gas → faster speed |
| Temperature–Energy Relation | \( \mathrm{KE_{avg} = \frac{3}{2}RT} \) | Higher T → higher average KE |
| Maxwell–Boltzmann Distribution | Shows range of molecular speeds | Temperature ↑ → broader, flatter curve |
Example :
Describe how the Maxwell–Boltzmann distribution of oxygen gas changes when temperature increases from 300 K to 600 K.
▶️ Answer/Explanation
Step 1: Increasing temperature raises the average kinetic energy of all gas molecules.
Step 2: The distribution curve shifts toward higher velocities and flattens (broader range of speeds).
Step 3: The total area under the curve (number of molecules) remains constant.
Final Answer: At 600 K, the curve is wider and shifted right, showing that more \(\mathrm{O_2}\) molecules move faster than at 300 K.