Root-Mean-Square Velocity of Gas Molecules
Detail explanation read from book
Problems
1.Calculate the root mean square(r.m.s) velocity of the molecules of
hydrogen ,oxygen and air at 0 ℃ and atmospheric pressure. Density of
hydrogen is .08988 g/L, oxygen 1.43 g/L, air 1.225 kg/m3
2. Calculate the root mean square velocity of hydrogen molecule in N.T.P
3.Oxygen is confined in a container at 0℃ and normal atmospheric
pressure. The temperature of the gas is then increased until the
pressure is doubled. Neglect any change in volume of the container.
Find r.m.s speed of the molecules at this temperature. The density of
oxygen at N.T.P =1.43gm/L
Mean free path
In a gaseous system, the molecules never move in a straight path
without interruptions. This is because they collide with each other
and change speed and direction. Between every two collisions, a
molecule travels a path length. The mean free path is the average
of all path lengths between collisions
A gas molecule’s mean free path λ is its average path length
between collisions.
The larger the particles or the denser the gas, the more frequent the
collisions are and the shorter the mean free path
𝑇𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑆
𝜆=
=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑁
𝜆=
1
2𝜋𝑑 2
𝑁
𝑉
Here, d= diameter of the molecule
N= Number of molecule
1. Calculate the mean free path air molecules at 0°C and I atm.
pressure. Given the effective diameter of air molecule =2 A0 , the root
mean square speed air molecules at N.T.P. is about 1 x 10 cm/sec and
the number of -molecules per c.c. = 3 x 1019.
2.Calculate the mean free path in CO2 at 27∘C and a pressure
of 10−9 bar. ( molecular diameter = 500 pm )
3.
Degree of freedom
The degree of freedom for a dynamic system is the number of
directions in which a particle can move freely or the total number of
coordinates required to describe completely the position and
configuration
of
the
system.
It is denoted by f or N.
Example:
1. A free particle moving along x-axis needs only one coordinate to
specify it completely. So its degree of freedom is one.
2. Similarly a particle moving over a plane has two degrees of freedom.
3. A particle moving in space has three degrees of freedom.
Suppose if we have N number of gas molecules in the container, then
the total number of degrees of freedom is f = 3N.
Degree of freedom for different atomic particles are given
below.
1.For monoatomic gas = 3
2.For diatomic gas = 5
3.For non-linear triatomic gas = 6
Monoatomic Molecule
Diatomic Molecule
Diatomic Molecule
Equipartition law of energy
Law of equipartition of 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 energy associated with each degree
of freedom per moelcule is ½ kT, where k is the Boltzmann’s constant.
This law helps scientists:
•Understand how gases behave
•Predict how much heat something will hold
•Explain temperature using particle motion
1.What is the average translational kinetic energy of nitrogen molecules at
1600 K?