Distribution of Velocity of Gases

Root Mean Square Velocity(C_rms)

It is defined as the square root of the mean value of the squares of the velocities of all the molecules at a particular temperature.
example: If a container contains a gas molecules having n molecules and at a particular moment the velocities of the molecules are C₁, C₂, C₃ ... C_n
then-

From kinetic gas equation-
C²=3PV/M=3RT/M
or, C_rms=√3RT/M
where M is the molecular weight of the gas.

Average Velocity (C_avg)

It is defined as the mean of the velocities of all molecules at a particular temperature.
C_avg=√8RT/πM

Most Probable Velocity (C_mpv)

The velocity possessed by the largest number of gas molecules at a particular temperature is called most probable velocity.
C_mpv=√2RT/M
where M is the molecular weight of the gas.

Relationship among the RMSV, AV and MPV

From the equation
C_mpv : C_avg : C_rms=1 : 1.128 : 1.225
Average Velocity=0.921 X C_rms
C_mpv=0.816 X C_rms

Mean Free Path

The gas molecules are always in state of rapid motion colliding with each other. The distance travelled by a gas molecules before colliding with another molecules is called free path and the average length of a large number of such paths is called mean free path. It is denoted by l and is directly proportional to temperature and inversely proportional to pressure.

Collision Number

Collision number is defined as the "number of collisions" per unit time.A collision is an interaction between two or more movable bodies.The collision number is indicated by the symbol of Z. When one molecule involved, the average number of collisions per unit time nearly one second per moles of reactant between reacting molecules is called collision number.

Collision Frequecny

Number of collisions suffered by gas per second per cubic meter of a gas is called collision frequency. It is equal to the ratio of the rms velocity and mean free path of a gas. i.e.
Collision frequency=Number of collision per second
or, Collision frequency=C_rms / mean free path

Equipartition Energy

Law of equipartition energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom.
The RMS velocity of gas molecules can be resolved into its components along x, y and z axis as -
c² = c_x² + c_y² + c_z²
multiplying by 1/2m, we get-
1/2mc² = 1/2mc_x² + 1/2mc_y² + 1/2mc_z²
or, E = E_x + E_y + E_z -----(1) (as K.E.(E) = 1/2mc²)
We know that gas molecules do not have any preferential direction. So, velocities along all three axes are equally probable-
c_x = c_y = c_z
or, E_x = E_y = E_z
or, E = 3E_x -----(2)
Each energy term in component contributes equally to the total energy. This is called the law of equipartition of energy.
We know that-
PV = 1/3 mnc²
PV = 2/3 X 1/2 mnc²=2/3 K.E.
or, K.E.=3 X 1/2 PV
or, E = 3 X 1/2 RT (as PV = RT for one mole)
From equation (2), we have-
3 E_x = 3 X 1/2 RT
so, E_x = E_y = E_z = 1/2 RT
Hence, the average kinetic energy possed by molecules in each component per degree of freedom is 1/2 RT per mole.

Maxwell law of distribution of velocities for gas molecules

We know that gases molecules can not maintain the same velocity for any length of time as it collides with another molecules. However, there must be a fraction of the total number of gas molecules that have a particular velocity at any time. This fraction was obtained by Maxwell and Boltzman using probability considerations.
They have shown that the distribution of molecular velocities depends upon the temperature and molecular weight of gas molecules is given by the expression-
dnc/n = 4π(M/2πRT)^3/2 e^{−(Mc2/2RT).c2dc}
where, dnc is the number of molecules out of a tota number of 'n' molecules which have velocities between c and c+dc, T is temperature and M is the molecular weight of the gas. The ration dn_c/n gives the fraction of the total number of molecules having velocities between c and c+dc.
Dividing both sides in the above equation by dc we get-
1/n(dnc/dc) = 4π(M/2πRT)^3/2e^{−(Mc2/2RT).c2}
The LHS expression give the probability of finding the molecules with velocity c. If the fraction of the total number of molecules for any gas of known molecular weight at any perticular temperature is plotted against the velocities, a distribution curve is looks like-

The curve shows that the fraction of molecules having velocities between c and c+dc is given by the arc under the strip of the curve within the velocity range. The area between two ordinates separated by dc is equal to dnc/n, hence, the area under the whole curve is equal to the total number of molecules and the height of curve says that the fraction of molecules having too low or too high velocities is very small and the majority of the gas molecules have some intermediate velocity with a small range of variation more or less around the peak, known as the most probable velocity.

Effect of Temperature on the distribution of molecular velocities

On increasing the temperature, the motion of the gas molecules becomes rapid as their kinetic energy increases and hence the value of most probable velocity also increases.
As a result, the entire distribution curve becomes flatter and peak shifts to regions of higher velocities as shown in the figure.

V₁ = Most probable velocity at temp. T₁
V₂ = Most probable velocity at temp. T₂
V₂ > V₁
As the gas gets colder, the graph becomes taller and more narrow. Similarly, as the gas gets hotter the graph becomes shorter and wider.

Q. According to Maxwell-Boltzmann distribution law of molecular velocities of gases, the average translational kinetic energy is

a. KT/2 per molecule
b. KT per molecule
c. RT per molecule
d. 3/2KT per molecule

Q. Maxwell's laws of distribution of velocities shows that

a. The number of molecules with most probable velocity is very large
b. The number of molecules with most probable velocity is small
c. The number of molecules with most probable velocity is zero
d. The number of molecules with most probable velocity is exactly equal to 1

Q. According to Maxwell's law of distribution of velocities of molecules, the most probable velocity is

a. greater than the mean velocity
b. equal to the mean velocity
c. equal to the root mean square velocity
d. less than the root mean square velocity