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  Kinetic theory of Gases and Application, Physics tutorial

Kinetic Theory of Gases:

Kinetic theory of gases makes transition between microscopic world of molecules and macroscopic world of quantities such as temperature and pressure. It begins out with a some basic hypothesizes concerning molecular behavior, and infers how behavior manifests itself on the macroscopic level. One of the most significant results of kinetic theory is derivation of the ideal gas law that not only is very helpful and significant.

We can summarize kinetic theory of gases with four basic postulates:

Gases are composed of molecules: We can treat molecules as point masses which are perfect spheres. Molecules in the gas are extremely far apart, so that space between every individual molecule is several orders of magnitude greater than diameter of molecule.

Molecules are in constant random motion: There is no general pattern leading either magnitude or direction of velocity of molecules in the gas. At any given time, molecules are moving in several different directions at several different speeds.

Movement of molecules is directed by Newton's Laws: In accordance with Newton's First Law, every molecule moves in the straight line at the steady velocity, not interacting with any of other molecules except in the collision. In the collision, molecules exert equal and opposite forces on one another.

Molecular collisions are perfectly elastic: Molecules don't lose any kinetic energy when they collide with one another.

Kinetic theory projects the picture of gases as small balls which bounce off one another whenever they come in contact. This is, of course, only an estimate, but it becomes extraordinarily accurate estimate for how gases behave in real world.

Pressure Exerted By Gas:

To compute pressure exerted by the gas, we have to make some essential assumptions; they are as follows:

  • The gas comprises of molecules that are identical and can be treated as small, hard elastic spheres moving at random in all directions with all possible velocities.
  • Molecules are treated as mass points, i.e., their volume is negligible in comparison to volume of container.
  • During the motion, molecules collide with each other and with walls of container.
  • Collisions are considered to be perfectly elastic; there is no loss of kinetic energy when collision takes place.
  • Though molecules are continually colliding with each other, they don't affect.

Let the molecules of the gas moving at random in the container. Molecules are repeatedly colliding with each other and with walls of container. It is supposed that all collisions are elastic. When the molecule collides with wall, a change of momentum takes place. Change in momentum is caused by force exerted by wall on molecule. Molecule applies the equal but opposite force on wall. Pressure applied by gas is because of the sum of all the collision forces.

Root Mean Square Velocity (R.M.S) of Gas Molecules:

Expression for pressure can be written as, 1/3ρC2 ‾

Therefore the root-mean-square velocity of all the gas molecules can be defined as

Therefore C2 ‾ =3P/ρ

Therefore  √C2 ‾ = √3Pρ

As a result if we know pressure (P) and its density ρ of gas. We can compute r.m.s. velocity of gas molecules.

Distribution of Molecular Speeds:

This explains how the speeds of molecules are distributed in the given closed system at particular temperature. Actual speeds differ from low to high values. At the given temperature, variation follows what is called as Maxwellian distribution.

Boltzmann constant:

Boltzmann constant, (symbol k), the fundamental constant of physics taking place in almost every statistical formulation of both classical and quantum physics. Constant is named after Ludwig Boltzmann, 19th-century Austrian physicist, who considerably contributed to foundation and development of statistical mechanics, the branch of theoretical physics. Having dimensions of energy per degree of temperature, Boltzmann constant has the value of 1.380650 × 10-23 joule per kelvin (K), or 1.380650 × 10-16 erg per kelvin.

Physical importance of k is that it gives the measure of amount of energy (that is, heat) corresponding to random thermal motions of molecules of the substance. For the classical system at equilibrium at temperature T, average energy per degree of freedom is kT/2. In simplest example of the gas comprising of N noninteracting atoms, every atom has three translational degrees of freedom (it can move in x-, y-, or z-directions), and so total thermal energy of gas is 3NkT/2.

Internal Energy of a Gas:

Internal energy is stated as energy related with random, disordered motion of molecules. It is divided in scale from macroscopic ordered energy related with moving objects; it refers to invisible microscopic energy on atomic and molecular scale. For instance, room temperature glass of water sitting on the table has no clear energy, either potential or kinetic. But on microscopic scale it is the seething mass of high speed molecules wandering at hundreds of meters per second. If water were tossed across room, this microscopic energy wouldn't essentially be changed when we superimpose ordered large scale motion on water as a whole.

Monatomic gas:

Monatomic is the combination of words mono and atomic, and signifies single atom. It is generally applied to gases: the monatomic gas is one in which atoms are not bound to each other. Every chemical elements will be monatomic in gas phase at adequately high temperatures.

Diatomic gas:

Diatomic molecules are molecules composed of two atoms chemically bonded together. Atoms can be of same element (homonuclear molecules), or of different elements (heteronuclear molecules). Diatomic molecules having different elements are hydrogen chloride (HCl), carbon monoxide (CO) and nitrogen monoxide (NO).

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