kinetic theory of gases cm2004 states of matter: gases
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Kinetic Theory of Kinetic Theory of Gases Gases
CM2004 CM2004 States of Matter: States of Matter: Gases Gases
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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
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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”
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Microscopic EnergiesMicroscopic Energies
Can be reformulated as:
<k> is called the average kinetic energy per particle
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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
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Internal EnergiesInternal Energies
T2>T1
COLD HOT
Each particle moves with an average kinetic energy of:
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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
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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:
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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
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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*
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Relative Mean Speeds, cRelative Mean Speeds, crelrel
Same Direction
Direct Approach
Typical “on average”
approach
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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
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Maxwell-Boltzmann and vMaxwell-Boltzmann and vRMSRMS
Probability that particle has specific energy,
INCREASING TEMPERATURE
MORE PARTICLES MOVE FASTER
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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
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Important Equations (1)Important Equations (1)
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Important Equations (2)Important Equations (2)
Z = p crel /(kBT) = kB T/ 2½ p