Kinetic Theory of Kinetic Theory of Gases Gases

CM2004 CM2004 States of Matter: States of Matter: Gases Gases

A Theory for 10A Theory for 102323 Particles Particles• In classical theory a

particle’s next move depends upon (equated to) its position, velocity and force acting on it

• Trying to solve such equations for a mole of gas with 1023 particles each with x,y,z coordinates and different speeds is almost impossible

So we theoretically describe the kinetic system on average in terms of a large set of no-volume “points”, which do not attract or repel each other

Pressures on AveragePressures on Average

On average the speed term is best represented by <v> as given in the Maxwell-Boltzmann distribution.

Furthermore a particle is equally likely to hit any one of the 6 available walls of the box. Hence:

“Mean-square speed”

Microscopic EnergiesMicroscopic Energies

Can be reformulated as:

<k> is called the average kinetic energy per particle

Macroscopic Energies and Macroscopic Energies and Boyle’s LawBoyle’s Law

N0<k> is the Total Kinetic Energy of one mole and is called Ek, the macroscopic energy:

PV=nRT

So TEMPERATURE is a direct measure of the INTERNAL

ENERGY of moving gas particles

Internal EnergiesInternal Energies

T2>T1

COLD HOT

Each particle moves with an average kinetic energy of:

Root Mean Square Speeds Root Mean Square Speeds These (vRMS)represent a single chosen speed to associate with every gas particle, as if they were all moving at this rate.

START END

Molar Mass

Thermal Energy: Energy Thermal Energy: Energy at a Definite Temperatureat a Definite Temperature

Kinetic Energy of 1 mole is:

Define Boltzmann’s constant:

Because:

Then Kinetic Energy of 1 particle is:

Equipartition of EnergyEquipartition of Energy

The EQUIPARTITION theorem states that a molecule gains ½ kBT of thermal energy for each DEGREE OF FREEDOM (i.e. x,y, z directions). So the total is ³/2 kBT

Quantifying Collision RatesQuantifying Collision Rates

Collision Rate (Z*) per face of

cubep = 2mv x Z/6A

Z = 6pvA/ 2mv2

Z = pvA/(kBT)

A is termed, , the collision cross-section

v is termed crel the relative mean speed

NOTE:

But, mv2 = 3kBTTOTAL pressure in the cube volume, where

Z=6Z*

Relative Mean Speeds, cRelative Mean Speeds, crelrel

Same Direction

Direct Approach

Typical “on average”

approach

Mean Free Path,Mean Free Path,The average distance between collisions is called the

MEAN FREE PATH,

Hence if a molecule collides with a frequency, Z, it spends a time, 1/Z in free flight between collisions and therefore travels a distance of [(1/Z) x c]

= c/Z Z = p crel /(kBT)

= c kB T/p crel

crel = 2½ c

Therefore:

and

= kB T/2½p

=d/2)2

d is the

collision

diameter

Maxwell-Boltzmann and vMaxwell-Boltzmann and vRMSRMS

Probability that particle has specific energy,

INCREASING TEMPERATURE

MORE PARTICLES MOVE FASTER

PopulationsPopulations

We shall return to the importance of Maxwell-Boltzmann Distributions in CM3006 next year

Molecules and atoms consist of many “micro” states and the higher the temperature the higher the probability

that “excited” states become populated

Important Equations (1)Important Equations (1)

Important Equations (2)Important Equations (2)

Z = p crel /(kBT) = kB T/ 2½ p