The molecular velocities like average velocity, root mean square velocity, and most probable velocity are well discussed here:
Average velocity
Average velocity is the arithmetic mean of the various velocities of the molecules. It is symbolically represented by vav or v.
If c1, c2, c3, …………cn are the velocities of individual molecules in a gas and ‘n’ is the total number of molecules present in the gas, then the average molecular velocity is given by
The value of average velocity calculated from Maxwell’s law of distribution is
where R = Gas constant, T = Temperature (in Kelvin scale), and M = Molar mass of the gas
Thus by substituting the values of R, T, π, and M in the above equation, average velocity can be calculated.
Root mean square velocity
The square root of the mean of squares of each gas molecule’s velocity is known as the root mean square velocity (r.m.s). It is denoted by μ or vrms.
If c1, c2, c3, …………cn are the velocities of individual molecules in a gas, and ‘n’ is the total number of molecules present in the gas, then root mean square velocity (r.m.s.) is given by
The Kinetic Gas Equation can be used to calculate the RMS velocity u at a given temperature as
where R = Gas constant, T = Temperature (in Kelvin scale), and M = Molar mass of the gas
Thus, root mean square velocity can be calculated by putting the value of R, T, and M in the above equation.
Most Probable Velocity
Most probable velocity is defined as the velocity possessed by the largest number of molecules in a gas. It is denoted by vmp or α.
According to Maxwell’s law of distribution of molecular velocities, the most probable velocity is given by the expression.
This is the required formula for most probable velocity of gas molecules.
Relation between Average Velocity, Root mean square Velocity and Most Probable Velocity
As we know,
Average velocity is the arithmetic mean of the various velocities of the molecules.
mean velocity of gas molecules is also called average velocity and is defined as the arithmetic mean of the various velocities of the molecules.
average velocity of gas molecules is proportional to the temperature.
The distance traveled by a molecule in a given gas per unit of time is known as molecular velocity.
Three types of velocities are considered in the study of the kinetic molecular theory of gases. They are average velocity, r.m.s, and most probable velocity.
- Raymond A. Serway; Jerry S. Faughn & Chris Vuille (2011). College Physics, Volume 1
- Arun Bahl, B. S. Bahl & G. D. Tuli, Essentials of Physical Chemistry, S. Chand and Company Ltd., New Delhi, 2012.
Average velocity is defined as disarticulation (i.e., change in position) per unit time. This velocity is disarticulation divided by a time period of the disarticulation. Root mean square (RMS) velocity is the (square root of the) average value of the square of its velocity. The first typical velocity is easiest to estimate and term as the most probable velocity.
Average Velocity, r.m.s. Velocity and Most Probable Velocity relationship –
The distribution equation [dn/n = 4π (M/2πRT)3/2 x e– (mc2/2RT) x c2dc]; may be used 10 obtain the average velocity, c , of the molecules.
This is given by the relation:
c = √(8RT/πM)
If we compare the average velocity with r.m.s. velocity may be seen that values are not the same,
Average velocity: C = √(8/3π) x r.m.s. velocity = 0.9213 x r.m.s. velocity
Most probable velocity is the velocity possessed by the largest number of molecules in a gas. Maxwell showed that the most probable velocity is given by the expression.
Cmpv = √(2RT/M)
A value of the most probable velocity may be calculated from the values of R, T, and M. A relation between the r.m s. velocity and the most probable velocity can be established as follows:
Cmpv/Cr.m.s. = [√(2RT/M)/√3RT/M] = √2/3 = 0.8165
Hence, Cmpv = 0.8165 x Cr.m.s
Average velocity
The average velocity of an object is its total disarticulation divided by the entire time taken. In other words, it is the rate at which an object changes its position from one place to another. It is a Vector quantity. The SI unit is meters per second. However, any distance unit per any time unit can be used when necessary, such as miles per hour (mph) or kilometer per hour (kmph).
The average velocity formula: average velocity = (change in position) / (change in time)
If we were given a position function, s(t), which gave the position of an object at time t, then the average velocity between time t sub 1 and t sub 2 is given by the formula:
Vavg = [s(t2) – s(t1)] / (t2 – t1) or, Δd/Δt.
Root mean square velocity
RMS velocity is the square root of the mean of squares of the speed of individual gas molecules. The root-mean-square velocity is that of a wave through subsurface layers of dissimilar interval velocity along a particular ray path and is usually quite a few percents higher than the average velocity. The stacking velocity and the root-mean-square velocity approach equality when source-receiver offset approaches zero and layers are horizontal and isotropic.
Root mean square speed = √(3RT/M)
The most probable velocity
The most probable velocity at which the most molecules in gas travel. The first distinctive velocity is easiest to estimate and term as the most probable velocity. The formula for most probable velocity is, Mp = √(2RT/M).
Fig: Average Velocity, r.m.s. Velocity and Most Probable Velocity – a relationship in equation