Many modern technologies, such as computers and smartphones, are built on the principles of Physics Topics such as quantum mechanics and information theory.
Basic Assumptions of Kinetic Theory of Gases
A gas is made of atoms and molecules. The three variables—volume, pressure and temperature, are all consequences of the motion of the molecules. The kinetic theory of gases relates the motion of the molecules to the volume, pressure and temperature of the gas. Actually the kinetic theory of gases explains the macroscopic properties of the gases, though it is a microscopic mode.
Rudolf Clausius and James Clark Maxwell developed the kinetic theory of gases and explained gas laws in terms of motion of the gas molecules.
In order to formulate the kinetic theory of gases some simplifying assumptions are made about the behaviour of the molecules of the ideal gas. The assumptions are-
i) A gas is composed of a large number of molecules. For a particular gas, the molecules are identical. But they are different for different gases.
ii) Every gas molecule behaves as a point mass. So the sum of their volumes is negligible compared to the volume of the container. The intermolecular space in the container is an empty space.
iii) The molecules are in continuous and random motion in all possible directions. The value of molecular velocities varies from zero to infinity.
iv) During random motion, the molecules collide with each other and with the walls of the container. These collisions are perfectly elastic. This means that the velocity changes due to collision, but the net momentum and kinetic energy remain unchanged.
v) The average distance between the molecules is sufficiently large, so that the attractive or repulsive forces between them are negligible except during collision. As a result,
- molecular motion is unrestricted and the gas spreads throughout the inner volume of its container;
- the potential energy of a molecule is negligible, the total energy comes from its kinetic energy only
- between two collisions, a molecule moves with uniform velocity (according to Newton’s first law of motion). The straight line path between two successive collisions is called a free path.
vi) Every collision is instantaneous; the time of a collision is negligible compared to the time taken by a molecule to describe a free path.
vii) The gas is homogeneous and isotropic. This means that the properties of the gas in any small portion are identical to those in any other equivalent portion anywhere inside the container.
A gas obeying the properties outlined in these assumptions is called an ideal gas or a perfect gas. Real gases show some deviations from these properties. Real gases available to us are good approximation of an ideal gas at low pressure and high temperature.
The kinetic theoretical definitions of mass and volume of a gas are obtained directly from the above assumptions:
- The mass of a gas is defined as the sum of the masses of the constituent molecules.
- The volume of a gas Is defined as the Inner volume of the gas container.
Mean free path: The straight line path described by a molecule between two collisions is called a free path. The gas molecules move randomly. As a result, the lengths of the free paths vary in an irregular manner. So, to get a concrete picture, the idea of a mean free path becomes essential. It is the average distance that a molecule can travel between two successive collisions.
Definition: The mean value of the distance travelled by a molecule between two successive collisions is called the mean free path (λ).
In other words, if a molecule suffers N number of collisions with other molecules when it travels through a total distance d, then the mean free path is, λ = \(\frac{d}{N}\).
An expression for the free path: Let σ = diameter of each molecule of the gas; n = number of molecules per unit volume = number density of the molecules.
Let us consider a particular molecule of the gas moving with a velocity v at an instant of time. It will collide with all molecules whose centres come at a distance of σ or less along its line of motion. Now we choose a cylinder of radius σ and of length v [Fig.]. Clearly,
- the axis of this cylinder is the path described by the chosen molecule in unit time, and
- all other molecules, whose centres come within this cylinder, collide with the chosen molecule in that unit time.
Volume of cylinder = cross-sectional area × length = πσ2v; and, the number of molecules having centres in this volume = πσ2vn. So, the number of collisions per unit time, N = πσ2vn.
The mean free path of the molecules is, therefore,
λ \(=\frac{\text { path described by a molecule }(d)}{\text { number of collisions }(N)}\) = \(\frac{v}{\pi \sigma^2 v n}\)
i.e., λ = \(\frac{1}{\pi \sigma^2 n}\) ……. (1)
This equation (1) shows that the mean free path (λ) of gas molecules is
i) inversely proportional to the number of gas molecules (n) in unit volume of the gas and
ii) inversely proportional to the square of molecule diameter (σ).
So, λ ∝ \(\frac{1}{n}\) and λ ∝ \(\frac{1}{\sigma^2}\)
i.e., λ ∝ \(\frac{1}{\sigma^2 n}\) or, λ = \(\frac{k}{\sigma^2 n}\)
The constant k in equation (1) is \(\frac{1}{\pi}\). However, it has been estimated by different scientists by different other methods. The estimates give different results, but the value of k is always slightly greater or slightly less than 1. Then, the approximate expression for the mean free path is λ = \(\frac{1}{\sigma^2 n}\).