3.5.A.1 Kinetic Molecular Theory and Maxwell-Boltzmann Distribution:

1. Kinetic Molecular Theory (KMT) Principles:

i. Gas particles are in constant, random motion: Gas molecules move in straight lines until they collide with another molecule or the walls of the container.

ii. Gas particles are very small compared to the distances between them: The volume of individual gas particles is negligible compared to the total volume of the gas.

iii. No forces of attraction or repulsion between gas particles: KMT assumes that gas particles do not interact with each other, either through attraction or repulsion.

iv. Collisions between gas particles are elastic: When gas particles collide, they transfer energy, but the total energy of the system remains constant. There is no loss of energy in the collisions.

v. The average kinetic energy of gas particles is directly proportional to the temperature of the gas: The temperature of the gas is a measure of the average kinetic energy of the particles. As the temperature increases, the average speed and energy of the particles increase as well.

Gas Behavior According to KMT:

• Pressure: Pressure is caused by the constant collisions of gas particles with the walls of the container. More frequent or harder collisions result in higher pressure.
• Temperature: As the temperature increases, the kinetic energy of the particles increases, leading to faster motion and greater pressure (at constant volume).
• Volume: If the gas is in a flexible container, increasing the volume of the container allows gas particles to spread out, reducing pressure (at constant temperature).

2. Kinetic Energy of Particles:

The kinetic energy of gas particles is directly related to their temperature. This relationship is described in Kinetic Molecular Theory (KMT), and it’s important for understanding how gases behave.

i. Relationship Between Temperature and Particle Energy:

a. Kinetic Energy and Temperature: The average kinetic energy of gas particles is directly proportional to the absolute temperature (measured in Kelvin). This means that as the temperature of the gas increases, the particles move faster, and their kinetic energy increases. Mathematically, this is expressed by the formula:

$$\text{Average Kinetic Energy} = \frac{3}{2} k_B T$$

where:

• $k_B$

  is the Boltzmann constant (

  $1.38 \times 10^{-23} \, \text{J/K}$

  ),

• $T$

  is the temperature in Kelvin.

This equation tells us that if the temperature increases, the kinetic energy of the particles increases as well, meaning they move more quickly.

b. Temperature and Particle Speed: As the temperature of a gas rises, the particles move faster because their kinetic energy is increasing. Conversely, as the temperature decreases, the particles slow down because their kinetic energy is lower

ii. Points to Keep in Mind:

a. Direct Proportionality: The kinetic energy of the particles increases when the temperature increases, and decreases when the temperature decreases. This is because higher temperature means more energy is supplied to the particles, making them move faster.

b. Speed and Kinetic Energy: The speed of individual particles also depends on their kinetic energy. At higher temperatures, more particles are moving faster, and at lower temperatures, more particles are moving slower. However, it’s important to note that while the average speed of particles increases with temperature, there will still be a distribution of speeds — not all particles will have the same speed.

3. Maxwell-Boltzmann Distribution:

The Maxwell-Boltzmann distribution describes the distribution of speeds (or kinetic energies) of particles in a gas. It provides insight into how particles at a given temperature are distributed in terms of their energy or speed.

i. Key Concepts of Maxwell-Boltzmann Distribution:

a. Speed Distribution: At any given temperature, not all gas particles move at the same speed. The Maxwell-Boltzmann distribution curve shows the number of particles that have a certain speed, and it gives a statistical view of particle behavior in terms of their speeds.

b. Energy Distribution: Similarly, particles in a gas have a range of kinetic energies, even if they are at the same temperature. The distribution describes how kinetic energy is spread out among the particles.

ii. The Shape of the Maxwell-Boltzmann Distribution Curve:

a. At low temperatures: Most particles have low speeds and therefore low kinetic energy. The curve is skewed toward the lower speed/energy values.

b. At high temperatures: The curve flattens and broadens out, showing that a greater number of particles now have higher speeds and energies. This happens because more energy is available to the particles at higher temperatures.

iii. Effects of Temperature on the Maxwell-Boltzmann Distribution:

a. Shift of the Peak: As the temperature increases, the peak of the distribution curve moves to higher speeds/energies. This means that, on average, the particles will have greater kinetic energy and move faster at higher temperatures.

b. Wider Distribution: With higher temperatures, the distribution becomes wider. This indicates that there is a greater variety of particle speeds, with more particles having both higher and lower speeds compared to a cooler sample.

c. More High-Energy Particles: At higher temperatures, more particles will have enough energy to overcome potential barriers, leading to a greater proportion of particles with higher kinetic energies. In other words, the tail of the curve (representing particles with very high speeds/energies) extends farther out.

iv. Temperature Effects in Detail:

a. Lower Temperature: At lower temperatures, fewer particles have enough energy to move quickly. The distribution is narrower and peaks at a lower speed. Most particles have speeds closer to the average, and fewer particles have high energies.

b. Higher Temperature: When the temperature increases, the particles move faster on average. The distribution becomes broader, and the peak shifts toward higher speeds. More particles are likely to have higher energy levels, and the overall energy distribution is more spread out.

v. Graphical Representation:

a. The x-axis represents the particle speed or kinetic energy.

b. The y-axis represents the number of particles with that particular speed or energy.

At lower temperatures, the curve is sharply peaked at lower speeds, while at higher temperatures, the curve becomes broader and moves toward higher speeds, indicating that more particles are moving faster.

iv. Key Points :

• Temperature increases → The peak of the distribution moves to higher speeds, and the curve spreads out, showing a greater variety of particle speeds.
• Temperature decreases → The peak shifts to lower speeds, and the distribution becomes narrower, with most particles moving more slowly.

4. Gas Properties and KMT:

Kinetic Molecular Theory (KMT) provides a molecular-level explanation for the relationships between pressure, volume, and temperature in gases. By considering the motion and interactions of gas particles, KMT helps explain the behavior of gases and why they follow laws like Boyle’s Law, Charles’s Law, and Gay-Lussac’s Law.

i. Pressure and KMT:

Pressure is the force exerted by gas particles as they collide with the walls of the container.

a. KMT Explanation: Gas particles are constantly moving in random directions. When they collide with the walls of the container, they exert a force. The pressure of a gas is a result of the frequency and force of these collisions. The more frequently particles collide with the container walls, or the harder the collisions, the higher the pressure.

b. Temperature’s effect on pressure: According to KMT, if the temperature of the gas increases, the kinetic energy of the particles increases. This leads to faster-moving particles that collide more frequently and with greater force, thereby increasing pressure (if the volume is held constant).

c. Volume’s effect on pressure: If the volume of the container is reduced, the gas particles have less space to move around, leading to more frequent collisions with the walls. Therefore, the pressure increases when volume decreases, assuming temperature remains constant.

ii. Volume and KMT:

Volume is the space that a gas occupies.

a. KMT Explanation: The volume of a gas is related to how much space the gas particles have to move. When the gas particles are confined to a smaller volume, they have fewer opportunities to spread out, resulting in more frequent collisions with the walls, which increases pressure. Conversely, increasing the volume allows particles more space, leading to fewer collisions and a decrease in pressure.

• Boyle’s Law: This relationship between pressure and volume is explained by KMT: When the volume decreases, gas particles are more crowded and collide more often with the container walls, leading to an increase in pressure (if temperature is constant). Boyle’s Law states:

  $$P_1V_1 = P_2V_2$$

  This means that pressure and volume are inversely proportional when temperature is constant.

iii. Temperature and KMT:

Temperature is a measure of the average kinetic energy of the gas particles.

a. KMT Explanation: Temperature is directly related to the average kinetic energy of gas particles. As temperature increases, the kinetic energy of the particles increases, meaning they move faster. Faster-moving particles collide with the container walls more forcefully and more frequently. As a result, both the pressure and volume are affected by changes in temperature.

• Charles’s Law: If the volume of a gas is held constant, increasing the temperature will cause the gas particles to move faster, resulting in an increase in pressure. If the pressure is held constant, increasing the temperature will cause the gas particles to occupy a larger volume, because faster-moving particles will push the walls of the container outward. This is what Charles’s Law describes:

  $$V_1/T_1 = V_2/T_2$$

  This shows that volume and temperature are directly proportional when pressure is constant.

• Gay-Lussac’s Law: If the volume of the gas is constant, increasing the temperature causes the particles to collide with the walls more forcefully, which increases the pressure. Gay-Lussac’s Law states:

  $$P_1/T_1 = P_2/T_2$$

  This shows that pressure and temperature are directly proportional when volume is constant.

iv. Summary of Relationships Based on KMT:

a. Pressure and Volume: Inversely proportional (Boyle’s Law). When volume decreases, pressure increases, and vice versa, assuming temperature is constant. This is explained by the fact that less volume leads to more collisions between gas particles and the container walls.

b. Pressure and Temperature: Directly proportional (Gay-Lussac’s Law). When temperature increases, pressure increases, assuming volume is constant. Faster-moving particles exert more force and more frequent collisions with the container walls.

c. Volume and Temperature: Directly proportional (Charles’s Law). When temperature increases, volume increases, assuming pressure is constant. Faster-moving particles require more space, causing the gas to expand.

5. Real Gases:

While ideal gas laws work well for gases under most conditions, real gases deviate from ideal behavior, particularly under extreme conditions such as very high pressures or very low temperatures. Real gases do not perfectly follow the assumptions of Kinetic Molecular Theory (KMT), especially the assumptions that:

• Gas particles do not interact with each other.
• Gas particles have negligible volume compared to the space between them.

Let’s explore why real gases deviate from ideal behavior and how the deviations occur under extreme conditions.

i. Deviations at Low Temperature:

At low temperatures, gas particles have lower kinetic energy, meaning they move slower. When they move slower, the forces of attraction between the particles become more significant.

a. Intermolecular Forces: Real gas particles experience attractive forces (e.g., Van der Waals forces or dipole-dipole interactions). These forces cause particles to stick together more at low temperatures, reducing the ability of the gas to expand freely and reducing its pressure compared to what would be expected from the ideal gas law.

b. Reduced Kinetic Energy: As the temperature drops, the particles lose kinetic energy, and these attractive forces cause the particles to be drawn closer together, leading to a smaller volume than predicted by the ideal gas law.

c. Deviations: The real gas behaves differently from the ideal gas because the assumption that there are no attractive forces between particles no longer holds. The pressure of the real gas will be lower than that predicted by the ideal gas law at low temperatures.

ii. Deviations at High Pressure:

At high pressures, gas particles are forced closer together, and the volume of the gas becomes significant compared to the space between the particles.

a. Volume of Gas Particles: The assumption that gas particles have negligible volume is no longer valid at high pressure. As the particles are compressed, the actual volume of the gas particles becomes more important. The volume of the gas will be larger than predicted by the ideal gas law.

b. Intermolecular Forces: At high pressures, gas particles are closer together, so attractive forces between the particles become significant. These forces can reduce the frequency of collisions with the container walls and result in lower pressure than the ideal gas law predicts.

c. Deviations: The real gas will occupy a larger volume and exert a lower pressure than the ideal gas law predicts at high pressures.

iii. Real Gas Behavior and Van der Waals Equation:

To account for the deviations from ideal behavior, the Van der Waals equation is used. It adjusts the ideal gas law to account for the volume of gas particles and intermolecular attractions:

$$\left( P + \frac{a}{V^2} \right) \cdot (V – b) = nRT$$

Where:

• P = Pressure of the gas

• V = Volume of the gas

• n = Number of moles of the gas

• R = Ideal gas constant

• T = Temperature

• a = A constant that accounts for the attractive forces between particles

• b = A constant that accounts for the volume occupied by the gas particles themselves

• The

  $a$

  term corrects for intermolecular forces (attractive forces) that reduce the pressure.

• The

  $b$

  term corrects for the finite volume of gas particles that is significant at high pressures.

iv. Critical Point and Supercritical Fluids:

a. At temperatures and pressures above a certain critical point, the gas cannot be liquefied no matter how much pressure is applied. This is due to the strong intermolecular forces, and the gas behaves like a supercritical fluid, where it has properties of both a gas and a liquid.

b. The behavior of gases near the critical point deviates significantly from ideal gas behavior because the assumptions of no intermolecular forces and no volume for the gas particles no longer hold true.

v. Summary of Deviation Causes:

a. At low temperature: Real gases experience attractive forces between particles, causing deviations such as lower pressure and smaller volume than predicted by ideal gas laws.

b. At high pressure: The volume of the gas particles becomes significant, and intermolecular forces affect gas behavior, leading to lower pressure and larger volume than predicted by the ideal gas law.