Kinetic Theory of Gases CM2004 States of Matter: Gases.
-
date post
21-Dec-2015 -
Category
Documents
-
view
230 -
download
2
Transcript of Kinetic Theory of Gases CM2004 States of Matter: Gases.
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