Transcript Powerpoint
Lecture 25
Goals:
• Chapters 18, micro-macro connection
• Third test on Thursday at 7:15 pm.
Physics 207: Lecture 25, Pg 1
Nitrogen molecules
near room temperature
Percentage of
molecules
15
10
100-1200
000-1100
00-1000
800-900
700-800
600-700
500-600
400-500
300-400
200-300
100-200
0-100
5
(m/s)
Physics 207: Lecture 25, Pg 2
Atomic scale
What is the typical size of an atom or a small
molecule?
A) 10-6 m
B) 10-10 m
C) 10-15 m
r
r ≈1 angstrom=10-10 m
Physics 207: Lecture 25, Pg 3
Mean free path
Average distance particle moves between collisions:
1
l=
4 2p ( N /V )r 2
N/V: particles per unit volume
The mean free path at atmospheric pressure is:
λ=68 nm
Physics 207: Lecture 25, Pg 4
Pressure of a gas
m
vx
vx
Physics 207: Lecture 25, Pg 5
Consider a gas with all
molecules traveling at
a speed vx hitting a
wall.
If (N/V) increases by a factor of 2, the pressure would:
A) decrease
B) increase x2
C) increase x4
If m increases by a factor of 2, the pressure would:
A) decrease
vx
B) increase x2
C) increase x4
If vx increases by a factor of 2, the pressure would:
A) decrease
B) increase x2
C) increase x4
Physics 207: Lecture 25, Pg 6
P=(N/V)mvx2
Because we have a distribution of speeds:
P=(N/V)m(vx2)avg
For a uniform, isotropic system:
(vx2)avg= (vy2)avg= (vz2)avg
Root-mean-square speed:
(v2)avg=(vx2)avg+(vy2)avg+(vz2)avg=Vrms2
Physics 207: Lecture 25, Pg 7
Microscopic calculation of pressure
P=(N/V)m(vx2)avg
=(1/3) (N/V)mvrms2
PV = (1/3) Nmvrms2
Physics 207: Lecture 25, Pg 8
Micro-macro connection
PV = (1/3) Nmvrms2
PV = NkBT (ideal gas law)
kBT =(1/3) mvrms2
The average translational kinetic energy is:
εavg=(1/2) mvrms2
εavg=(3/2) kBT
Physics 207: Lecture 25, Pg 9
The average kinetic energy of the molecules of an ideal gas at 10°C
has the value K1. At what temperature T1 (in degrees Celsius) will the
average kinetic energy of the same gas be twice this value, 2K1?
(A) T1 = 20°C
(B) T1 = 293°C
(C) T1 = 100°C
Suppose that at some temperature we have oxygen molecules
moving around at an average speed of 500 m/s. What would be the
average speed of hydrogen molecules at the same temperature?
(A) 100 m/s
(B) 250 m/s
(C) 500 m/s
(D) 1000 m/s
(E) 2000 m/s
Physics 207: Lecture 25, Pg 10
Equipartition theorem
Things are more complicated when energy can be stored in other
degrees of freedom of the system.
monatomic gas: translation
solids: translation+potential energy
diatomic molecules: translation+vibrations+rotations
Physics 207: Lecture 25, Pg 11
Equipartition theorem
The thermal energy is equally divided among all possible energy
modes (degrees of freedom). The average thermal energy is (1/2)kBT
for each degree of freedom.
εavg=(3/2) kBT (monatomic gas)
εavg=(6/2) kBT (solids)
εavg=(5/2) kBT (diatomic molecules)
Note that if we have N particles:
Eth=(3/2)N kBT =(3/2)nRT (monatomic gas)
Eth=(6/2)N kBT =(6/2)nRT (solids)
Eth=(5/2)N kBT =(5/2)nRT (diatomic molecules)
Physics 207: Lecture 25, Pg 12
Specific heat
Molar specific heats can be directly inferred from the thermal energy.
Eth=(6/2)N kBT =(6/2)nRT (solid)
ΔEth=(6/2)nRΔT=nCΔT
C=3R (solid)
The specific heat for a diatomic gas will be larger than the specific
heat of a monatomic gas:
Cdiatomic=Cmonatomic+R
Physics 207: Lecture 25, Pg 13
Entropy
A perfume bottle breaks in the corner of a room. After some time,
what would you expect?
A)
B)
Physics 207: Lecture 25, Pg 14
very unlikely
The probability for each particle to be on the left half is ½.
probability=(1/2)N
Physics 207: Lecture 25, Pg 15
Second Law of thermodynamics
The entropy of an isolated system never decreases. It can only
increase, or in equilibrium, remain constant.
The laws of probability dictate that a system will evolve towards the
most probable and most random macroscopic state
Thermal energy is spontaneously transferred from a hotter system to
a colder system.
Physics 207: Lecture 25, Pg 16