Question

The Maxwell speed distribution assumes that the gas is in equilibrium. Thus, if a gas, all of whose molecules were moving at the same speed, were given enough time, they would eventually come to satisfy the speed distribution. But the kinetic theory derivations in the text assumed that when a gas molecule hits the wall of a container, it bounces back with the same energy it had before the collision and that gas molecules exert no forces on each other. If gas molecules exchange energy neither with the walls of their container nor with each other, how can they ever come to equilibrium? Is it not true that if they all had the same speed initially, some would have to slow down and others speed up, according to the Maxwell speed distribution?

Short answer

Answer: The gas molecules can achieve equilibrium and the Maxwell speed distribution through binary collisions. Over time, through a large number of binary collisions, the gas molecules will rearrange themselves into a wide variety of speeds with a distribution determined by their mass and temperature, ultimately leading to the Maxwell speed distribution. This emergence is a result of the conservation of energy and momentum during binary collisions, which are not prohibited in the kinetic theory.

Step-by-step solution

Understanding the Assumptions in Kinetic Theory

The kinetic theory of gases assumes that gas molecules do not exchange energy with the walls of the container (perfectly elastic collisions) and do not exert forces on each other (ideal gas). The speed distribution in this case is governed by the Maxwell speed distribution as it assumes that the gas is in equilibrium. When a gas molecule collides with the wall of the container, its speed may change, but its energy remains constant. If all gas molecules start with the same speed, they would need to change their speeds according to the Maxwell speed distribution to reach equilibrium.

Role of Collisions in Establishing Equilibrium

The gas molecules can achieve the equilibrium and the Maxwell speed distribution through binary collisions. Whenever a collision occurs between two gas molecules, they may exchange energy and momentum without violating the assumptions made in the kinetic theory. The energy and momentum are conserved during these binary collisions.

Emergence of Maxwell Speed Distribution

Over time, through a large number of binary collisions, the gas molecules will rearrange themselves into a wide variety of speeds with a distribution determined by their mass and temperature, ultimately leading to the Maxwell speed distribution. This emergence of the Maxwell distribution is a result of the conservation of energy and momentum during binary collisions, which are not prohibited in the kinetic theory.

Conclusion

In summary, even though the kinetic theory assumes that gas molecules do not exchange energy with the walls or exert forces on each other, the binary collisions between the gas molecules are enough to cause a change in their speeds over time, eventually leading to the emergence of the Maxwell speed distribution and the equilibrium state.

Key concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental concept that explains the behavior of gases at the molecular level. It assumes that gases are composed of a large number of small particles, which are in constant, random motion. The particles are considered to be point masses with no volume and perfectly elastic during collisions.

According to the kinetic theory, the pressure exerted by a gas on the walls of its container results from collisions of the gas particles with the container walls. The temperature of the gas is directly related to the average kinetic energy of its particles. Therefore, even though individual gas molecules can vary in speed and energy due to collisions, the average kinetic energy (and thus temperature) remains constant in a system that is in equilibrium.

While the individual behaviors of gas particles are unpredictable, their average behavior can be described by statistical mechanics, which provides the link between microscopic properties of gas particles and their macroscopic observable properties like pressure, volume, and temperature.

Ideal Gas

An ideal gas is a hypothetical gas that perfectly follows the gas laws under all conditions of temperature and pressure. To qualify as an ideal gas, several strict conditions must be met:

• The molecules of the gas are infinitely small and do not occupy any volume.
• There are no attractive or repulsive forces between the particles of the gas.
• All collisions between gas particles or with the walls of the container are perfectly elastic, meaning there is no loss of kinetic energy.
• The particles are in constant, random motion.
• The average kinetic energy of the gas particles is directly proportional to the temperature of the gas

These conditions are approximations that facilitate the mathematical modeling of gases. Real gases behave like an ideal gas at high temperature and low pressure, where the interactions between molecules are least significant.

Binary Collisions

Binary collisions refer to the encounters between two gas particles in which energy and momentum are exchanged. These collisions are central to understanding how a gas evolves towards an equilibrium state.

In an ideal gas, these collisions are elastic; total kinetic energy and total linear momentum are conserved. The outcomes of these binary collisions can be a simple exchange of velocities between similar particles or a redistribution of energy and momentum for dissimilar particles.

Through multiple binary collisions, a gas with all molecules initially moving at the same speed can evolve to a state where the speeds are distributed according to Maxwell's speed distribution. This illustrates one of the ways in which equilibrium and temperature distribution are achieved in a gas, even without interactions with walls or external forces.

Equilibrium in Gases

Equilibrium in gases refers to the state in which the properties of the gas particles are uniformly distributed and do not change over time. When a gas is in equilibrium, it means that the macroscopic properties such as pressure, temperature, and volume remain constant.

Equilibrium does not imply that the individual particles are at rest, but rather that their collective behavior results in a stable state. The Maxwell speed distribution describes the distribution of speeds of particles in a gas at equilibrium. It is only obtained when the sum total of all microscopic changes due to binary collisions cancels out, resulting in no net change in the macroscopic state of the gas.

It is a dynamic equilibrium where individual particles are constantly exchanging energy through collisions, but the overall energy distribution remains the same. This equilibrium condition is reached regardless of the initial distribution of speeds or energies in the gas particles.

Conservation of Energy and Momentum

The conservation of energy and momentum are key principles in physics that play crucial roles in the kinetic theory of gases. During binary collisions between gas molecules, two fundamental principles apply:

• Conservation of Energy: The total kinetic energy of the two colliding particles before the collision is equal to their total kinetic energy after the collision.
• Conservation of Momentum: The total momentum of the colliding particles remains constant before and after the collision.

These conservation laws ensure that, while individual particles may change their velocities and energies during collisions, the overall energy and momentum of the system remain constant. This is crucial for the gas to reach equilibrium because it allows for the redistribution of energy among particles that results in the statistical distribution described by Maxwell. As this process continues over many collisions, the system settles into a state where the distribution of molecular speeds reflects the temperature of the gas, and equilibrium is achieved without any net change in total energy or momentum.