RMS speed of gas molecules

Gas molecules are one of the most volatile particles that exist in nature. To measure and understand the effect these molecules have on their surroundings and the intermolecular interaction that these molecules have can only be done by a good understanding of their velocities.

Hence, understanding the velocity of these gas molecules using the RMS formula is very important.

Through the root mean square velocities of gas molecules, we can effectively calculate the effusion and diffusion rate since without the root mean square velocity, the aspect of all the molecules in a gas having net-zero value can be rectified.

Kinetic Theory of Gases

The kinetic theory of gases introduces a classical model that describes the thermodynamic behaviour of gases in an easy-to-understand fashion. This theory has become historically significant and using this theory numerous other theories of thermodynamics were established.

In the kinetic theory of gases, gas is a voluminous number of identical submicroscopic particles (atoms or molecules). These submicroscopic particles are assumed to be in a constant state of rapid-fire, arbitrary movement. The size of these particles is assumed to be much lower than the average distance between the particles.

The atoms are enclosed in the vessel that encloses the gas. These atoms suffer arbitrary collisions between themselves and with the enclosing walls of the vessel and all these collisions are assumed to be elastic. The kinetic theory of gases describes the gas as an ideal gas and considers that the other particles have no other inter-particle relation or encounters.

The kinetic theory of gases can be used to explain the macroscopic properties of gases, such as volume, pressure and temperature. It can also be used to describe transitive properties as well, such as density, thermal conductivity and mass diffusivity. The kinetic theory of gases also accounts for affiliated phenomena, similar to the Brownian motion.

Historically, the kinetic theory of gases was the first unequivocal exercise of the ideas of statistical mechanics. The kinetic theory implications to ideal gases provide the assumptions mentioned below:

• The gas comprises veritably small atoms. Their smallness in size is comparable in that the sum of the volume of the individual gas atoms is irrelevant compared to the volume of the vessel that stores the gas. This is original to stating that the average length separating the gas atoms is largely analogized to their size. The elapsed time of collision between atoms and the vessel’s wall is insignificant when analogized to the time between consecutive collisions.
• The number of atoms is so big that a statistical treatment of the problem is well justified. This supposition is occasionally called the thermodynamic limit.
• The fleetly moving atoms frequently collide among themselves and with the walls of the vessel. All these collisions are impeccably elastic, which means the motes are perfect hard spheres.
• Except during collisions, the interactions among atoms are insignificant. They apply no different forces to one another.
• Therefore, the dynamics of atom movement can be treated classically and the equations of movement are time-reversible.

As a simplifying supposition, the atoms are generally accepted to have the same mass as one another. Still, the proposition can be generalised to mass distribution, with each mass type contributing to the gas properties singly of one another in agreement with Dalton’s Law of partial pressures.

Many of the model’s prognostications are the same whether collisions between atoms are included, so they’re frequently disregarded as a simplifying supposition in derivatives.

Root mean square velocity of gas molecule

According to kinetic molecular theory, gas atoms are in a state of constant arbitrary movement, individual atoms move at different velocities, constantly colliding and changing directions. We use velocity to describe the movement of gas atoms, thereby taking into account both speed and direction. 

Although the velocity of gas atoms is constantly changing, the distribution of velocities doesn’t change. We can not gauge the velocity of each individual atom, so we frequently reason in terms of the atoms’ average demeanour. Atoms moving in contrary directions have velocities of contrary signs.

Since a gas’ atoms are in arbitrary movement, it’s presumptive that there will be about as numerous moving in one direction as in the contrary direction, meaning that the average velocity for a collection of gas atoms equals zero; as this value is irrelevant, the average of velocities can be determined using an indispensable system.

By squaring the velocities and taking the square root, we overcome the “ directional” element of velocity and contemporaneously acquire the atoms’ average velocity. Since the value excludes the atoms’ direction, we now relate to the value as the average speed. The root means square velocity provides a measure of the velocity with which atoms in a gas can move around. It can be defined as the result obtained by square rooting the average of all the velocities of the atoms in a gas after squaring the velocities, to remove the directional aspect.

 It’s represented by the equation

vrms=3RTM

Here vrms represents the root mean square velocities,

whereas R represents the Molar Gas constant,

T is the temperature in Kelvin of the gas, 

M is the gas’s molar mass in kilograms per mole.

Both molecular weight and temperature are taken into consideration by the root mean square velocity. These two factors drastically affect the root mean square velocity since the molecular velocity will decide how fast the particles move, more the molecular weight lesser the speed of the molecules. The temperature of the gas also decides how much energy the particles have to move around.

Conclusion

The root mean square velocity is the square root of the normal of the forecourt of the haste. Similarly, it has units of velocity. The reason we use the root mean square velocity rather than the normal is that for a typical gas sample, the net haste is zero since the patches are moving in all directions.

This is a crucial formula, as the haste of the patches is what determines both the prolixity and effusion rates. To summarise:

• All gas atoms move with arbitrary speed and direction. 
• Solving for the average velocity of gas atoms gives us the average velocity  zero, assuming that all atoms are moving inversely in different directions. 
• You can acquire the average speed of gas atoms by taking the root of the square of the average velocities. 
• The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect a material’s kinetic energy.