# Kinetic Theory Of Gases, And Properties Of Vapour

The experiments of James Joule during the 1840s proved beyond doubt that mechanical work and heat are equivalent. They also delivered the final blow to the erstwhile caloric theory of heat. The collapse of caloric theory inspired three gentlemen — Rudolf Clausius, Clerk Maxwell and Ludwig Boltzmann — to develop the kinetic theory of gases from a theoretical standpoint. The experimental proof of kinetic theory came a few decades later, in 1909, chiefly through the investigation of Brownian motion by Albert Einstein and Jean Perrin. Although originally developed for gases, kinetic theory can be extended to explain the thermal phenomena in solids and liquids.

The total energy of a system usually consists of external energy and internal energy. External energy is made up of potential energy due to an external conservative force and kinetic energy due to the motion of the system as a whole relative to its surroundings. Internal energy is made up of intermolecular potential energy, energy of molecular motion relative to the centre of mass of the system, and intramolecular energy. The energy of molecular motion (which can be translation, rotation and vibration) is the disordered part of internal energy and known as thermal energy. Even though it is impossible to determine how much internal energy a system possesses, we are interested only in the change in internal energy between two states and there are alternative methods to determine that.

The kinetic model of an ideal gas proposes a microscopic model in which any pure gas is made up of a large number of identical minute particles called "molecules". The molecules move randomly in the container in any direction at any speed between zero and infinity. As they move, molecules collide with one another and with the walls of the container. These collisions are all perfectly elastic and of negligible duration compared to the time spent between two successive collisions.

Using the kinetic model, the pressure exerted by an ideal gas is:

`P = Mvrms2 / 3V = ρvrms2 / 3`

where ρ is the density of the gas and vrms is the root mean square speed or RMS speed of the molecules of the gas. It follows that RMS speed can be expressed as:

`vrms = √(3 RT / M0) `

where M0 is the molar mass and T is the Kelvin temperature of the gas; R is the universal gas constant. Typically, the RMS speeds of hydrogen and oxygen gas molecules are 1836 m/s and 461 m/s respectively at 0°C.

It must be clearly understood that all molecules of a gas do not move with RMS speed. Many of the molecules move at speeds higher than RMS speed, while many others move at lower speeds. The distribution of molecular speeds is governed by a complex equation known as Maxwell distribution function. In fact, three different speeds are associated with Maxwell speed distribution curve. They are RMS speed, average speed and most probable speed. vrms > vav > vmp

The distance travelled by a molecule between two successive collisions is known as the free path and the average value of this quantity is called mean free path. The number of collisions taking place per unit time is called collision frequency. The reciprocal of collision frequency is called mean free time.

A gas which obeys the equation of state, PV = nRT, in all circumstances is known as an ideal gas. Strictly speaking, no real gas obeys the equation of state. But helium, hydrogen, nitrogen and oxygen which have extremely low boiling points behave like an ideal gas at room temperature and low pressure. Among several modified equations proposed for real gases, the best known is van der Waals' equation:

`(P + n2a /V2)(V - nb) = nRT`

In the expression, P, V and T have usual meanings; n is number of moles of the gas; a and b are positive constants characteristic of the gas.

When a space contains the maximum amount of vapour it can hold at a given temperature, the vapour is said to be saturated vapour. A vapour whose concentration is less than the maximum possible is said to be unsaturated vapour. The saturation vapour pressure (s.v.p. in short) of a liquid is the maximum pressure exerted by its vapour at a given temperature. In this context, two types of phase diagram are discussed: pressure-temperature phase diagram and pressure-volume phase diagram. Terms such as critical point and triple point are also introduced.

Because of continuous evaporation from the seas, rivers, moist earth etc, the atmosphere always contains varying amounts of water vapour. Hygrometry deals with the measurement of the amount of water vapour in a given volume of air. If air is gradually cooled to a temperature at which the s.v.p. of water is just equal to the actual pressure of water vapour, that temperature is called the dew point. If the temperature of air falls below the dew point, some vapour condenses and appears as tiny drops of water, which we call dew. The dew point varies with location and time.

The absolute humidity is defined as the mass of water vapour present per unit volume of air. A more useful term is relative humidity, which is the ratio of the mass of water vapour present in a given volume of air to the mass of water vapour required to saturate the volume. Regnault's hygrometer is a useful device to measure the relative humidity of air.

#### Basic level videos

Basic Concepts Of Kinetic Theory Of Heat, And Kinetic Model Of An Ideal Gas 32:29 Basic
200 3
Pressure Exerted By An Ideal Gas 1:38:17 Basic
400 5
Translational Kinetic Energy Of A Gas 1:02:19 Basic
400 4
More Problems On Pressure And Translational Kinetic Energy Of A Gas 1:20:54 Basic
300 4
Deduction Of Gas Laws From Kinetic Theory Of Gases 29:47 Basic
300 3
Distribution Of Molecular Speeds 1:01:33 Basic
300 4
Mean Free Path 1:40:1 Basic
300 3
Brownian Motion, And Deviation From Ideal Gas Behaviour 1:13:07 Basic
400 3
Problems On Brownian Motion, And Deviation From Ideal Gas Behaviour 1:05:30 Basic
400 5
Saturated Vapour And Its Properties 1:06:33 Basic
300 3
Phase Diagrams 44:25 Basic
300 3
Hygrometry 1:10:33 Basic
300 3
Problems On Hygrometry 57:18 Basic
300 3

#### Advanced level Videos Note: (CE) Stands for Problems from Competitive Examination Papers

Advanced-level Problems On Kinetic Theory Of Gases And Properties Of Vapour I 1:11:47
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Advanced-level Problems On Kinetic Theory Of Gases And Properties Of Vapour II 1:11:47
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Advanced-level Problems On Kinetic Theory Of Gases And Properties Of Vapour III 1:31:46
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Advanced-level Problems On Kinetic Theory Of Gases And Properties Of Vapour IV 1:10:40
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