Question

At any particular time, different particles in the gas (a) have same speed and kinetic energy (b) have same speed but different kinetic energies (c) have different speeds but same kinetic energy (d) have different speeds and hence different kinetic energies.

Short answer

Particles in a gas have different speeds and hence different kinetic energies (d).

Step-by-step solution

Understanding particle behavior in a gas

In a gas, particles are in constant random motion and collide with each other and with the walls of the container. The distribution of speeds of these particles is given by the Maxwell-Boltzmann distribution.

Analyzing each option

For option (a), it is incorrect because, according to the kinetic theory of gases, particles have a range of speeds and therefore a range of kinetic energies. Option (b) is incorrect because if particles have the same speed, they must also have the same kinetic energy since kinetic energy is directly related to the speed of the particles. For option (c), it is incorrect because two particles with the same kinetic energy can have different masses and therefore can have different speeds. Option (d) is the correct choice because in a gas, particles have different speeds, and since kinetic energy depends on both mass and velocity, particles with different speeds will have different kinetic energies.

Key concepts

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution is a fundamental concept in understanding the speeds of particles within a gas. It describes the likelihood of particles achieving a certain speed in a system that is in thermal equilibrium. Imagine a swarm of bees buzzing at different speeds around their hive. Much like these bees, gas particles move at various speeds, and the Maxwell-Boltzmann distribution is like a map that shows us how many particles are 'buzzing' fast, slow, or somewhere in between.

This distribution curves typically resemble a bell shape, skewing towards higher speeds, and then tailing off as the speed increases. At the very essence, it reflects the chaotic and random nature of particle movement in gases. Understanding this concept is crucial for grasping why gases behave as they do on a microscopic level.

Kinetic Energy

Kinetic energy in the context of gases refers to the energy that particles have due to their motion. We can think of it like the energy a child has while running around a playground—the faster they run, the more energy they have. In a similar way, the faster a gas particle moves, the more kinetic energy it possesses. The formula for kinetic energy is given as \[ KE = \frac{1}{2}mv^2 \]where \( m \) is the mass of the particle and \( v \) is its velocity. Because different particles can have different velocities and masses, they will inherently have different amounts of kinetic energy. This variance is part of what creates the Maxwell-Boltzmann distribution's characteristic shape.

Particle Behavior in Gases

In a gas, particles are independent, far apart, and constantly in motion. They exhibit random behavior, moving in all directions and colliding with one another and the container walls. These collisions are perfectly elastic, meaning there is no net loss of kinetic energy. The diversity in particle speed and resulting kinetic energy is a direct consequence of these collisions and the inherent energy transfer involved.

An important takeaway is that no two gas particles are likely to be moving at the same speed at any given time, leading to a variety of kinetic energies. This constant motion and variation in speeds are essential to the properties of gases, such as pressure and temperature, and our comprehension of them arises from analyzing the Maxwell-Boltzmann distribution.