A gaseous mixture contains 4 molecules with a velocity of \(6 \mathrm{~cm} \mathrm{sec}^{-1}, 5\) molecules with a velocity of \(2 \mathrm{~cm} \mathrm{sec}^{-1}\) and 10 molecules with a velocity of \(3 \mathrm{~cm} \mathrm{sec}^{-1}\). What is the RMS velocity of the gas: (a) \(2.5 \mathrm{~cm} \mathrm{sec}^{-1}\) (b) \(1.9 \mathrm{~cm} \mathrm{sec}^{-1}\) (c) \(3.6 \mathrm{~cm} \mathrm{sec}^{-1}\) (d) \(4.6 \mathrm{~cm} \mathrm{sec}^{-1}\)

The concept of Root Mean Square (RMS) velocity is crucial when considering the motion of particles in a gas. It denotes a form of average velocity, but unlike a simple arithmetic average, RMS velocity takes into account the fact that particles move in various directions and at different speeds.

In essence, RMS velocity is defined as the square root of the average of the squares of the individual velocities of the gas particles. The formula to determine RMS velocity is given by: \[ v_{\text{rms}} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} v_{i}^{2}} \] where \( v_{i} \) represents the velocity of a single particle and \( N \) is the total number of particles.

Because the velocities are squared before they're averaged, the RMS velocity gives greater weight to higher velocities which reflects the kinetic energy of the particles more accurately than a simple average would. After summing and averaging the squared velocities, taking the square root brings the units back to the original velocity units and provides a meaningful average velocity.

When teaching about molecular velocities in gases, it's important to convey that the particles in a gas are in constant random motion. This chaotic movement results from the kinetic energy that the particles possess, which in turn depends on the temperature of the gas.

Gases consist of a large number of molecules moving freely and colliding with each other and the walls of their container. These collisions are perfectly elastic, meaning that there is no net loss of kinetic energy as a result. The velocities of gas molecules follow a distribution, meaning that at any given time, there will be some molecules moving slowly and others moving more rapidly.

Maxwell-Boltzmann Distribution

The distribution of molecular speeds in a gas is described by the Maxwell-Boltzmann distribution, which demonstrates that while there is an average speed, no single speed is most probable. This distribution is dependent on the mass of the particles and the temperature of the gas, reinforcing the idea that understanding molecular velocity is a key concept in thermodynamics and statistical mechanics.

Average velocity of gas particles is a simplified way to understand the collective motion of particles within a gas. Unlike RMS velocity which accounts for the magnitude of all particle velocities, the 'average velocity' often refers to the arithmetic mean of the speeds without considering the direction. This can be misleading in the context of gases because the average of speeds would be zero due to the random motion in all directions.

However, in a gas, when we talk about 'average' in a kinetic sense, we're usually referring to the RMS value because it more accurately conveys physical quantities like kinetic energy and temperature. The RMS velocity is more relevant when discussing properties of gases because it relates to the energy they possess— the higher the RMS velocity, the higher the temperature of the gas, assuming ideal behavior.

It's essential for students to recognize that while 'average velocity' might imply a simple mean of speeds, in the kinetic theory of gases, RMS velocity is the correct measure when discussing average kinetic energy or predicting behaviors like diffusion and effusion rates based on molecular speed.