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# Molecular-kinetic theory of ideal gases

## 1.

Physics 1Voronkov Vladimir Vasilyevich

## 2. Lecture 5

MOLECULAR-KINETIC THEORY OFIDEAL GASES

THE MOLECULAR BASIS OF THERMAL

PHYSICS

EVAPORATION AND BOILING

COLLISIONS AND TRANSPORT

PHENOMENA

## 3. Main assumptions for Ideal Gas Model

1.2.

3.

4.

5.

The number of molecules in the gas is large, and the average

separation between them is large compared with their

dimensions. This means that the molecules themselves occupy

a negligibly small volume in the container. The molecules are

considered to be point-like.

The molecules obey Newton’s laws of motion, but as a whole

they move randomly. That is any molecule can move in any

direction with any speed. At any given moment, one of the

molecules move at high speeds, and others move at low

speeds.

The molecules interact only by short-range forces during

elastic collisions, no long-range forces on among molecules.

The molecules make elastic collisions with the walls.

All molecules of the gas are identical.

## 4. MOLECULAR-KINETIC THEORY OF IDEAL GASES

An ideal gas is in acubical container with

sides of length d. We

see i-th molecule, its

velocity Vi, x-component

of velocity is Vxi. Then

is time for travelling

from one side of the

container to another.

## 5.

The change of the i-th moleculemomentum after collision is:

Then the short-term force acting on the

molecule during collision with the wall is:

The long-term force acting on the

molecule in the x-direction is:

## 6.

Using previous expressions we can find xcomponent of the long-term average forceexerted by the wall on the molecule as:

Then by a molecule on the wall:

So the net force of all molecules on the wall:

## 7.

Root mean square (rms) value of x-component ofvelocities of N molecules:

Velocity consists of 3 equal components:

So the total force exerted on the wall is:

Then the pressure on the square wall is:

## 8.

N is the number of moleculesThis result indicates that the pressure of

a gas is proportional to the number of

molecules per unit volume and to the

average translational kinetic energy of

the molecules

## 9. Molecular interpretation of temperature

So we haveExperimentally found the equation of state for

an ideal gas:

So we can define temperature as:

It tells us that temperature is a direct

measure of average molecular kinetic energy.

## 10. Theorem of equipartition of energy

We can transform the last expression intoGeneralization of this result is known as the theorem

of equipartition of energy:

Each degree of freedom contributes 12 k BT to

the energy of a system, where possible

degrees of freedom in addition to translation

arise from rotation and vibration of

molecules.

## 11. Root-mean square speed of molecules

Using the equation of state for an Ideal gashere n is the number of moles

we can find that

## 12. Internal Energy

In the molecular-kinetic model internalenergy of a gas equals the sum of

kinetic energies of every molecule of

the gas:

N is the number of molecules

## 13. Equation of State for an Ideal Gas

Found experimentally:n is the number of moles of the gas

R is the universal gas constant:

## 14. The Boltzmann Distribution Law

We found average kinetic energy of amolecule. But all molecules move

chaotically, collide with each other. So

arises question what is distribution of

molecules depending on their energy.

The answer is given by statistical

mechanics:

## 15. The Boltzmann Distribution Law

Where n0 is defined such that n0dE is thenumber of molecules per unit volume having

energy between E = 0 and E = dE.

The Boltzmann law states that the probability

of finding the molecules in a particular energy

state varies exponentially as the negative of

the energy divided by kBT:

n is the number density

## 16. Maxwell–Boltzmann speed distribution function

If N is the total number of molecules,then the number of molecules with

speeds between V and V+dv is

dN = NV dV.

## 17. Maxwell–Boltzmann speed distribution function

dN = NV dV.## 18. Gas molecules velocities

Root mean squareMean

Most probable

## 19. Evaporation

We know that liquids evaporate when they’re belowboiling temperature. The speed distribution curve for

molecules in a liquid is described by Maxwell–

Boltzmann function. The molecules that escape the

liquid by evaporation are those that have sufficient

energy to overcome the attractive forces of the

molecules in the liquid phase. Consequently

molecules left behind in the liquid phase have a

lower average kinetic energy; as a result, the

temperature of the liquid decreases. Hence,

evaporation is a cooling process.

## 20.

• .• In order to evaporate, a mass of water must

collect the large heat of vaporization, so

evaporation is a potent cooling mechanism.

Evaporation heat loss is a major climatic factor

and is crucial in the cooling of the human body.

## 21. Saturation Vapor Pressure

Ordinary evaporation is a surface phenomenon some molecules have enough kinetic energy toescape. If the container is closed, an equilibrium is

reached where an equal number of molecules return

to the surface. The pressure of this equilibrium is

called the saturation vapor pressure.

## 22.

• The process of evaporation in aclosed container will proceed until

there are as many molecules

returning to the liquid as there are

escaping. At this point the vapor is

said to be saturated, and the

pressure of that vapor is called the

saturated vapor pressure.

• Since the molecular kinetic energy is greater at higher

temperature, more molecules can escape the surface and the

saturated vapor pressure is correspondingly higher. If the liquid

is open to the air, then the vapor pressure is seen as a partial

pressure along with the other constituents of the air. The

temperature at which the vapor pressure is equal to the

atmospheric pressure is called the boiling point.

## 23. Evaporation vs Boiling

• Ordinary evaporation is a surface phenomenon since the vapor pressure is low and since thepressure inside the liquid is equal to atmospheric

pressure plus the liquid pressure, bubbles of water

vapor cannot form.

• But at the boiling point, the saturated vapor pressure

is equal to atmospheric pressure, bubbles form, and

the vaporization becomes a volume phenomena.

## 24. Boiling Point

• The boiling point is definedas the temperature at which

the saturated pressure of a

liquid is equal to the

surrounding atmospheric

pressure.

• For water, the vapor pressure

reaches the standard sea

level atmospheric pressure of

760 mmHg at 100°C.

• Since the vapor pressure increases with

temperature, it follows that for pressure greater than

760 mmHg (e.g., in a pressure cooker), the boiling

point is above 100°C and for pressure less than 760

mmHg (e.g., at altitudes above sea level), the boiling

point will be lower than 100°C.

## 25. Collisions

is the collision cross section. Then thevolume it sweeps is

. So this

molecule suffers N=nV collisions (here n is the

number density).

## 26.

Then we can find mean collision timeConsidering movement of the target

molecules:

N is the number of molecules

n is the number density

## 27.

Mean free path is an average distancebetween collisions:

## 28. Tortuous path of a gas molecule

A randomly movingmolecule has such

displacement:

The displacement of a gas

molecule is proportional to

the square root of the

time.

## 29. Transport Phenomena

By means of collisions that molecules can carryphysical properties (momentum, energy,

concentration, etc.) through a gas. If there is local

heating, that extra thermal energy can be carried to

other parts of the gas through successive collisions.

If some molecules with a particular odour or colour

are introduced in one location, they will spread

through the gas through collisions, even without

convection. The movement of such properties is

called transport phenomena (of thermal energy,

change of concentrations, and so forth), and, more

particularly, the movement of molecules by random

collisions is called diffusion.

## 30. Some terms

The critical temperature of a gas is that temperatureabove which the gas will not liquefy, regardless of the amount

of pressure applied to it.

The saturated vapor pressure of a substance is the

additional pressure exerted by vapor molecules on the

substance and its surroundings under a condition of

saturation.

Boiling is defined as vaporization within the body of a liquid

when its vapor pressure equals the pressure in the liquid.

The absolute humidity is defined as the mass of water per

unit volume of air.

The relative humidity can be computed from saturatedvapor-pressure tables according to the following definition:

Relative humidity = actual vapor pressure/ saturated vapor

pressure.

## 31. Van der Waals Gas

n is the number of moles of the gasTerm

represents long-range attraction

between molecules.

Term b - the volume taken up by one mole of

molecules. It accounts for the strong repulsion

between molecules at a characteristic radius.

## 32.

## 33. NOTE

In this lecture quantity n is used in 2different ways, each time noting the used

meaning:

n can be the number of moles of the gas

n can be the density number, n=N/V