Ch 19 Kinetic Theory of Gases
Ch 19 Problems: 7, 19, 43, 55
Gases
• Introduce gas into a container, the gas will fill
the volume of the container.
• Gases are described by their pressure,
volume, temperature, and amount. These
quantities are related by an equation of state.
• These are macroscopic properties.
• We will work with ‘ideal gases’.
Ideal Gas
• In an ideal gas, the atoms or molecules move randomly
• exert no long range forces on each other
• The particles are point-like, they have negligible volume.
• Real gases behave like ideal gas when the density is low
enough.
Ideal gas gives describes limit of behavior of real gas.
Avogadro’s Number
• Since gases usually consist of very large
amounts of particles, we want a convenient
unit for the amount of particles.
• moles
• NA = 6.02x1023 particles/mole
Ideal Gas Law
Tells us how the pressure, volume and
temperature are related.
PV = nRT
R = Universal Gas constant 8.31 J/(mol K)
n = number of moles
P = pressure
V = volume
T = temperature
If instead of using the number of moles, we use
the number of molecules, N.
n = N/NA
Now the Gas Law becomes:
PV = NkT
k = Boltzmann’s constant
k = R/NA = 1.38x10-23 J/k
nR = Nk
Isothermal process
isothermal process, no change in temperature
U=Q-W
U=0
Q=W
PV = nRT
P = nRT/V
vf
W
vf
pdV
vi
vi
vf
nRT
dV nRT ln
V
vi
From P-V diagram we can se that the work
done by a constant volume process is zero.
In an isobaric process W = p V
Kinetic Theory of Gases
Using model of ideal gas, try to study gas in microscopic
terms.
Using assumptions:
1) number of molecules is large, and average
separation between is large compared to their
dimensions.
2) molecules obey Newton’s Laws, but as a whole they
move randomly
3) molecules interact only through short-range forces
during elastic collisions
4) molecule collide elastically with the walls
5) the molecules are identical
Molecular Interpretation
of Temperature
Because the molecules are moving, they have kinetic energy.
The higher the temperature, the faster the average speed if
the gas molecules.
Introducing concept of rms (root mean squared)
The vrms is a new way to describe the speed of the gas
molecules. vrms is not the average speed, but is similar.
table 19-1 lists some values of vrms
3RT
vrms
M – molar mass
M
For N molecules
Since ideal gas is made up of pointlike molecules,
kinetic energy is only from translation.
K ave
K ave
K ave
K ave
1 2
mvrms
2
1 3RT
( m)
2
M
3RT
2N A
3
kT
2
Mean Free Path
The individual molecules don’t travel very far in
straight lines.
If they did, because of their high rms speeds,
the molecules would get very far in a short
period of time.
Since they collide with each other and follow
randomlike paths (see fig 19-5), it takes a while
before the net displacement gets large. This is
why it takes some time before you can smell
perfume from across a room.
Mean Free Path
The mean free path gives the average distance
a particle travels between collisions.
1
2 d 2N /V
d = diameter or particle
Distribution of Molecular Speeds
Not all the molecules travel at same speed.
Plot of probability that particle has speed v versus speed is
given by Maxwell’s speed distribution law.
P (v )
0
4
M
2 RT
P(v)dv 1
3/ 2
2
ve
Mv2 /( 2 RT )
This shows that sum of all the probabilities
for each velocity adds up to 1.
See fig 19-8 for graphs Notice that there is a new curve for each
temperature. Notice that the most probable speed, vrms, and
vave are not the same.
Molar specific heats of ideal gas.
Eint =(3/2) nRT
monatomic ideal gas
Eint =(3/2) nR T
If volume is constant Q = nCv T
1st Law of T.D.
Eint = Q – W
Eint = nCv T – W
V constant, W = 0
Eint = nCv T
combined with Eint =(3/2) nR T gives:
Cv = (3/2) R
Constant Pressure
Q = nCp T
Eint = Q – W
W = p V = nR T
Eint = nCv T
nCv T = nCp T – nR T
Cp = Cv + R
For monatomic gas: Cv = (3/2)R Cp = (5/2)R
Adiabatic Expansion of Ideal Gas
pV = constant
= Cp/Cv
TV -1 = constant
Free expansions are special type of adiabatic expansion.
These relations do not work for free expansions. In a
free expansion T is constant.
pV = nRT = constant
piVi = pfVf