Transcript Chapter 4

View on Cold in 17th Century
…while the sources of heat were obvious – the sun, the
crackle of a fire, the life force of animals and human beings –
cold was a mystery without an obvious source, a chill
associated with death, inexplicable, too fearsome to
investigate.
“Absolute Zero and the Conquest of Cold” by T. Shachtman
• Heat
“energy in transit” flows from hot to cold: (Thot > Tcold)
• Thermal equilibrium “thermalization” is when Thot = Tcold
•Arrow of time, irreversibility, time reversal symmetry breaking
Zeroth law of thermodynamics
A
C
If two systems are separately
in thermal equilibrium with a
third system, they are in
thermal equilibrium with each
other.
Diathermal
wall
B
C
C can be considered the
thermometer. If C is at a
certain temperature then A
and B are also at the same
temperature.
Simplified constant-volume gas thermometer
Pressure (P = gh) is the thermometric property that
changes with temperature and is easily measured.
Temperature scales
• Assign arbitrary numbers to two convenient
temperatures such as melting and boiling points of
water. 0 and 100 for the celsius scale.
• Take a certain property of a material and say that
it varies linearly with temperature.
X = aT + b
• For a gas thermometer:
P = aT + b
Pressure
Gas Pressure Thermometer
o
-273.15 C
Ice point
Steam point
LN2
-300
-200
-100
0
100
o
Temperature ( C)
200
Pressure
Gas Pressure Thermometer
Celsius scale
P = a[T(oC) + 273.15]
o
-273.15 C
Ice point
Steam point
LN2
-300
-200
-100
0
100
o
Temperature ( C)
200
Phase diagram of water
Near triple point can have ice, water, or vapor on making arbitrarily
small changes in pressure and temperature.
Concept of Absolute Zero
(1703)
Guillaume Amonton first derived mathematically
the idea of absolute zero based on Boyle-Mariotte’s
law in 1703.
For a fixed amount of gas in a fixed volume,
p = kT
Amonton’s absolute zero ≈ 33 K
Other Types of Thermometer
•Metal resistor : R = aT + b
•Semiconductor : logR = a - blogT
•Thermocouple : E = aT + bT2
Low Temperature Thermometry
Platinum resistance thermometer
150
R ()
100
50
0
0
50
100
150
200
T (K)
250
300
350
400
CERNOX thermometer
R ( )
10000
1000
100
0
100
200
T (K)
300
400
International Temperature Scale of 1990
microstate
Prob. (microstate)
Macrostates: n,m
Macrostate: n-m
hhhh
1/16
4, 0
4
thhh
1/16
3, 1
2
hthh
1/16
3, 1
2
hhth
1/16
3, 1
2
hhht
1/16
3, 1
2
tthh
1/16
2, 2
0
thth
1/16
2, 2
0
htht
1/16
2, 2
0
hhtt
1/16
2, 2
0
htth
1/16
2, 2
0
thht
1/16
2, 2
0
httt
1/16
1, 3
-2
thtt
1/16
1, 3
-2
ttht
1/16
1, 3
-2
ttth
1/16
1, 3
-2
tttt
1/16
0, 4
-4
16 different configurations (microstates), 5 different macrostates
Microcanonical ensemble:
• Total system ‘1+2’ contains 20 energy quanta and 100 levels.
• Subsystem ‘1’ containing 60 levels with total energy x is in
equilibrium with subsystem ‘2’ containing 40 levels with total
energy 20-x.
• At equilibrium (max), x=12 energy quanta in ‘1’ and 8 energy
quanta in ‘2’
Ensemble: All the parts of a thing taken together,
so that each part is considered only in relation to
the whole.
The most likely macrostate the system will
find itself in is the one with the maximum
number of microstates.
E1
E2
1(E1)
2(E2)
d ln 1 d ln  2
1


dE1
dE2
k BT
Most likely macrostate the system will find
itself in is the one with the maximum
number of microstates. (50h for 100 tosses)
Number of Microstates ()
1.2e+029
1e+029
8e+028
6e+028
4e+028
2e+028
0
0
20
40
x
Macrostate
60
80
100
Microcanonical ensemble: An ensemble of
snapshots of a system with the same N, V, and E
A collection of systems that
each have the same fixed energy.
E
(E)
Canonical ensemble: An ensemble of snapshots
of a system with the same N, V, and T (red box
with energy  << E. Exchange of energy with
reservoir.
E-
(E-)

 I(  )
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Log10 (P())
Canonical ensemble: P()  (E-)1  exp[-/kBT]
• Total system ‘1+2’ contains 20 energy quanta and 100 levels.
• x-axis is # of energy quanta in subsystem ‘1’ in equilibrium with ‘2’
• y-axis is log10 of corresponding multiplicity of reservoir ‘2’