Question

Do all the molecules of a gas strike the walls of their container with the same force? Justify your answer on the basis of the kinetic model of gases.

Short answer

No, not all molecules of a gas strike the walls of their container with the same force due to variations in kinetic energy and speed resulting from their random motion and collisions.

Step-by-step solution

Understand the Kinetic Molecular Theory

The Kinetic Molecular Theory of gases makes certain assumptions about the particles in a gas. These assumptions include that gas particles are in constant, random motion, they collide with each other and the walls of the container without losing energy, and that they exert force upon collision due to their momentum. Furthermore, the distribution of the speeds (and thus kinetic energy) of the molecules follows a statistical distribution.

Analyze Particle Speed and Collisions

Not all gas molecules move with the same speed. Due to random motion and collisions, their speeds follow a distribution (often Maxwell-Boltzmann distribution for ideal gases), meaning there is a variety of kinetic energies at any given moment. Therefore, if the kinetic energies are different, the force exerted by the molecules during collisions with the walls will also be different.

Conclude Based on Forces Exerted

Because kinetic energy and speed of molecules vary, and because force exerted on the walls of the container depends on the molecule's momentum change upon collision (which in turn depends on speed and mass), it can be concluded that not all molecules of a gas strike the walls with the same force.

Key concepts

Maxwell-Boltzmann Distribution

Imagine a crowded dance floor where each dancer moves at a different pace, quickly reflecting the diverse energy levels among them. Similarly, gas molecules within a container exhibit a wide range of speeds, captured by the Maxwell-Boltzmann distribution. This statistical distribution is a cornerstone of the kinetic theory, describing the probability of finding a gas molecule at a particular speed at a given temperature.

At lower speeds, more molecules can be found, forming the rising slope on the graph of the distribution. As speed increases, fewer molecules travel that fast, which corresponds to the descending portion. The most probable speed is where most molecules are 'dancing'—not too fast, not too slow. This reflects the temperature's role: higher temperatures shift the peak of the curve rightward, indicating more high-energy (faster) molecules.

When applying this concept to the exercise, one understands why the forces exerted by gas molecules on container walls vary. The variability in molecular speeds, as furnished by the Maxwell-Boltzmann distribution, directly translates to variability in the momentum and therefore the force upon collision.

Kinetic Energy of Gases

Kinetic energy is the energy of motion. For gas molecules, envision tiny, invisible spheres zooming unpredictably in every direction. This motion defines their kinetic energy, adhering to the formula: \( KE = \frac{1}{2}mv^2 \), where 'm' is the mass and 'v' the speed of a molecule. In a gas, molecules bounce around, clashing into each other, akin to balls in a lottery machine.

The kinetic energy is different for each molecule based on its speed, highlighting the individuality of their 'dance'. With varying kinetic energies comes varying impact when they meet the container; a swift molecule delivers a larger 'punch' to the walls than a sluggard. The step-by-step solution from the exercise demonstrates that kinetic energy's diversity among gas molecules explains the differing forces upon collision with the container walls. Hence, just like dancers on a floor, not every molecule 'steps' with the same force.

Molecular Collisions in Gases

Continuing with our dance floor analogy, each gas molecule's 'dance' involves countless 'spins and twists'—collisions—both with fellow molecules and the container walls. These molecular collisions are perfectly elastic under the kinetic molecular theory, meaning the molecules do not lose kinetic energy when they interact; rather, they exchange it.

In this microscopic ballroom, the collisions dictate the pressure exerted on the container walls. It's a consequence of the continuous, dynamic transfer of momentum between molecules and walls. When a molecule with greater kinetic energy collides with the wall, the change in its momentum is greater, yielding a stronger force. The diversity of these energetic encounters, as itemized in the exercise's solution, confirms that not all gas molecules exert equal force upon impacting the walls.

The dance of molecules is precise but chaotic, where every 'move'—collision—adds up to the overall pressure that is felt as a macroscopic property of the gas.