# Kinetic Theory of Gases

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Ideal Gas:

• A gas which obeys gas laws at all temperatures and pressures is called as an ideal gas or Perfect gas. e.g. an ideal gas does not liquify even at low temperature but continues to obey Charle’s law and finally occupies no volume at –273oC or 0o K.
• Ideal gas also consists of molecules in a continuous state of motion with neither attraction and repulsion between them. Molecules of ideal gas collide with each other without any net loss of kinetic energy.
• For an ideal gas, PV = nRT. This relation is known as the ideal gas equation. Where,  P = Pressure of the gas, V = Volume of the gas, n = Number of moles of the gas, R = Universal gas constant, T = Absolute temperature of the gas
• In reality, there exists some negligible force of attraction and repulsion between the molecules of the gas. Besides, there is some loss of kinetic energy in the collision of the molecules of the gas. Hence ideal gas is an imaginary or hypothetical concept.
• A gas is said to non-ideal or real gas if it obeys gas laws only at low pressures and high temperature.

• Equal volumes of all gases under the same conditions of temperature & pressure contain the equal number of molecules.

#### Explanation:

• Let two gases gas A and Gas B be taken in two containers having equal volume (V). Let the temperature of both the gasses be the same (T).
• Let the pressure of the two gases be same (P). By Avogadro’s law under such conditions of equal pressure, equal volume & equal temperature, the number of molecules of gas A in the container should be equal to the number of molecules of gas B in the container.

#### Importance of Avogadro’s Hypothesis:

• It differentiates between atoms and molecules of gasses.
• It modified Dalton’s atomic theory.
• It explains gay lussac’s law of combining volume.
• It helps in determination of the atomic mass of elements.
• It established that the number of molecules per unit volume is same for all gases at a fixed temperature and pressure.
• It established that at N.T.P.one gram mole of any gas occupies 22.4 dm3 by volume. one mole of a gas contains 6.023 X 1023 molecules of gas.
• It gives the relation between vapour density & molecular weight.

Molecular weight = 2 × vapour density.

#### Evidences of the Molecules of a Gas are Always in Constant Motion:

• Diffusion of gases.
• Indefinite expansion of gases.
• Gases exert pressure on the walls of container.
• Brownian like motion in gases.

#### Assumptions of Kinetic Theory of Gases:

• A gas consists of a large number of extremely small molecules which are exactly identical in all respects.
• The molecules are rigid and perfectly elastic spheres of very small diameters.
• The intermolecular forces between gas molecules are negligible.
• The molecules are in a  state of random motion  i.e.  they move with all possible velocities in all possible directions.
• During their random motion the molecules collide with each other and with the walls of the container and these collisions are supposed to be perfectly elastic, (i.e. there is no loss of kinetic energy during these collisions).
• Between successive collisions, molecules move with uniform velocity in straight paths and these paths are called free paths.
• All free paths are not equal. Average of free paths is called mean free path.
• Actual volume of the molecules is negligible compared to the total volume of the gas. Therefore, molecules can be treated as geometrical points.
• The number of molecules per unit volume of a gas remains constant.
• At constant temperature, the average kinetic energy of the gas molecules remains constant. The average kinetic energy of the molecules of a gas depends only on the absolute temperature of the gas.
• The time of impact i.e. the time interval during which collision occurs is very small compared to the time interval between successive collisions.

### Terminology of Kinetic Theory of Gases:

#### Mean Free Path of Gas Molecule:

• The molecules of a gas are always moving in random motion i.e. in all possible directions with all possible velocities. This is also called as molecular chaos. Therefore they constantly collide with one another and with the walls of the container. • Between two successive collisions, a molecule travels in a straight line. The distance covered by a molecule between two successive collisions is called the free path.
• All the free paths are not equal. Therefore their average value is considered. The average distance covered by a molecule between successive collisions is called its mean free path. It is denoted by ‘λ’. Its S.I. unit is m but the practical unit is angstrom.
• If λ1, λ2,l3, ….., λN are the free paths, then mean free path is given by Where N is a number of collisions.

#### Mean or Average Velocity:

• Mean velocity of a molecule of a gas is defined as the arithmetic mean of the velocities of the molecules of the gas at given temperature.
1. Let C1, C2, C3, …. ,CN be the velocities of N molecules of a gas, then the mean velocity of molecules of gas is given by • As the molecules of gases are in random motion, i.e. they can move in any direction with any possible velocity, by the probability theory the mean velocity of gas molecules should be zero.
• Its S.I. unit is m/s.

#### Mean Square Velocity of Gas Molecules:

• Mean square velocity of a molecule of a gas is defined as the arithmetic mean of squares of the velocities of the molecules of the gas at given temperature.
• LetC1, C2, C3, …. ,CN be the velocities of N molecules of a gas, then the mean velocity of molecules of gas is given by • Its S.I. unit is m²/s².

#### Root Mean Square Velocity of Gas Molecules:

• The square root of the mean of the squares of the velocities of the molecules of a gas is called root mean square (r.m.s.) velocity of the molecules of a gas.
• Let C1, C2, C3, …. ,CN be the velocities of   N   molecules of a gas, then the r.m.s. velocity of molecules of gas is given by • It is clear that r.m.s.  Velocity cannot be zero.

#### Example – 01:

• In the following table, ni represents the number of molecules of a gas and Ci represents their speed in m/s Calculate the average and R.M.S. speeds of the molecules.
 ni 2 4 8 6 3 Ci 1 2 3 4 5 Ans: Average speed of molecule is 3.173 m/s and r.m.s. speed is 3.369 m/s

#### Example – 2:

• Find the average velocity, mean square velocity and root mean square velocity of six molecules having velocities 4, 5, 8, -6, -4, and 10 m/s respectively. Ans : average speed = 2.83 m/s, mean square speed = 42.83 m2/s2; rms speed = 6.544 m/s

#### Example – 3:

• The velocities of seven molecules are 1, 2, 3, 4, 5, 6, 7 km/s respectively. Find the mean square velocity of the molecule. Ans : Mean square velocity = 20 km2/s2

#### Example – 4:

• Find the r.m.s. velocity of three molecules having velocities 10, 20, 30 km/s. Ans : r.m.s. speed = 21.60 km/s

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