P,V,q

Lecture 2 – The First Law (Ch. 1)
Wednesday January 9th
•Statistical mechanics
•What will we cover (cont...)
•Chapter 1
•Equilibrium
•The zeroth law
•Temperature and equilibrium
•Temperature scales and thermometers
Reading:
All of chapter 1 (pages 1 - 23)
1st homework set due next Friday (18th).
Homework assignment available on web page.
Assigned problems: 2, 6, 8, 10, 12
Statistical Mechanics
What will we cover?
Probability and Statistics
PHY 3513 (Fall 2006)
Number of students
4
Mean 78%
Median 81%
3
B
C+
B+
A
2
D
1
0
50 55 60 65 70 75 80 85 90 95 100
Score (%)
Probability and Statistics
Probability distribution function
PHY2048 - Fall 2002
18
Mean 63±0.5
Sigma 27.5±1.5
Area 470±33
Number of students
16
Gaussian statistics:
12
10
8
6
4
2
  x  x 2 
1
f  x 
exp  

2
2 
 2

Input parameters:
14
458 students
00
0
20
40
60
80
Final score (%)
Quality of teacher and level of difficulty
Abilities and study habits of the students
100
Probability and Statistics
Probability distribution function
PHY2048 - Fall 2002 (test 2)
Gaussian statistics:
Number of students
120
100
80
522 students
60
40
20
  x  x 2  0
1
f  x 
exp  

2
2 
 2

Input parameters:
Mean 3.03±0.09
Sigma 3.41±0.32
Area 561±75
0
1
2
3
4
5
6
7
8
Score (out of 8)
Quality of teacher and level of difficulty
Abilities and study habits of the students
The connection to thermodynamics
Maxwell-Boltzmann speed distribution function
 m 
f  v   4 

2

kT


1/ 2
 2kT 
vm  

 m 
3/ 2

v 2 exp mv 2 / 2kT

Input parameters:
Temperature and mass (T/m)
1/ 2
 8kT 
v 

 m 
1/ 2
vrms
 3kT 


m


Equation of state:
1
2
1 2 2
2
PV  Nmv   N  mv   N   NkT
3
3
2
 3
Probability and Entropy
Suppose you toss 4 coins. There are 16 (24)
possible outcomes. Each one is equally
probably, i.e. probability of each result is
1/16. Let W be the number of
configurations, i.e. 16 in this case, then:
1
p1  ;
W
Ptot   pi  W  p1  1
i
Boltzmann’s hypothesis concerning
entropy:
S  kB ln W
where kB = 1.38 × 1023 J/K is Boltzmann’s
constant.
The bridge to thermodynamics through Z
Z   exp   E j / kT ; js represent different configurations
j
  1/ kT
Quantum statistics and identical particles
Indistinguishable events
Heisenberg
uncertainty
principle
The indistinguishability of identical particles has a profound effect on
statistics. Furthermore, there are two fundamentally different types of
particle in nature: bosons and fermions. The statistical rules for each
type of particle differ!
The connection to thermodynamics
Maxwell-Boltzmann speed distribution function
 m 
f  v   4 

2

kT


1/ 2
 2kT 
vm  

 m 
3/ 2

v 2 exp mv 2 / 2kT
Input parameters:
Temperature and mass (T/m)
1/ 2
 8kT 
v 

 m 
1/ 2
vrms
 3kT 


m


Consider T  0

# of bosons
Bose particles (bosons)
Internal energy = 0
Entropy = 0
11
10
9
8
7
6
5
4
3
2
1
0
Energy
Fermi-Dirac particles (fermions)
# of fermions
Internal energy ≠ 0
Free energy = 0
Entropy = 0
Pauli exclusion principle
1
0
Particles are indistinguishable
EF
Energy
Applications
Specific heats:
Insulating solid
Diatomic molecular gas
Fermi and Bose gases
The zeroth & first Laws
Chapter 1
Thermal equilibrium
Pi, Vi
System 2
System 1
Pe, Ve
Heat
If Pi = Pe and Vi = Ve, then system 1 and systems 2 are already in
thermal equilibrium.
Different aspects of equilibrium
Mechanical equilibrium:
Piston
1 kg
1 kg
Already in thermal
equilibrium
Pe, Ve
gas
When Pe and Ve no longer change (static)  mechanical equilibrium
Different aspects of equilibrium
Chemical equilibrium:
nl ↔ nv
Already in thermal
and mechanical
equilibrium
nl + nv = const.
P, nv, Vv
vapor
liquid
P, nl, Vl
When nl, nv, Vl & Vv no longer change (static)  chemical equilibrium
Different aspects of equilibrium
Chemical reaction:
A + B ↔ AB # molecules ≠ const.
Already in thermal
and mechanical
equilibrium
A, B & AB
When nA, nB & nAB no longer change (static)  chemical equilibrium
Different aspects of equilibrium
If all three conditions are met:
•Thermal
•Mechanical
•Chemical
Then we talk about a system being thermodynamic equilibrium.
Question:
How do we characterize the equilibrium state of a system?
In particular, thermal equilibrium.....
The Zeroth Law
a)
b)
A
C
B
VA, PA VC, PC
C
VB, PB VC, PC
“If two systems are separately in thermal equilibrium with
a third system, they are in equilibrium with each other.”
c)
A
B
VA, PA VB, PB
The Zeroth Law
a)
b)
A
C
VA, PA VC, PC
B
C
VB, PB VC, PC
“If two systems are separately in thermal equilibrium with
a third system, they are in equilibrium with each other.”
•This leads to an equation of state, q  f(P,V ), where the
parameter, q (temperature), characterizes the equilibrium.
•Even more useful is the fact that this same value of q
also characterizes any other system which is in thermal
equilibrium with the first system, regardless of its state.
More on thermal equilibrium
q characterizes (is a measure
of) the equilibrium.
Each equilibrium is
unique. Erases all
information on history.
•Continuum of different mechanical equilibria (P,V)
for each thermal equilibrium, q.
•Experimental fact: for an ideal, non-interacting gas,
PV = constant (Boyle’s law).
•Why not have PV proportional to q ; Kelvin scale.
Equations of state
•An equation of state is a mathematical relation between
state variables, e.g. P, V & q.
•This reduces the number of independent variables to two.
General form: f (P,V,q ) = 0
Example:
PV = nRq
or q = f (P,V)
(ideal gas law)
•Defines a 2D surface in P-V-q state space.
•Each point on this surface represents a unique equilibrium
state of the system. q
f (P,V,q ) = 0
Equilibrium state
Temperature Scales
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
An experiment that I did in PHY3513
T
79
0
-195.97
20
Pressure (arb. units)
P
17.7
13.8
3.63
T = aP + b
Value Error
b
-267.2 2.8
a
19.5 0.2
 = 2.2
10
Data
Linear fit
0
-300 -250 -200 -150 -100 -50
o
Temperature ( C)
0
50
100
Pressure
The ‘absolute’ kelvin scale
0
T(K) = T(oC) + 273.15
Triple point
of water:
273.16 K
100
200
300
400
Temperature (K)
Other Types of Thermometer
•Thermocouple:
E = aT + bT2
•Metal resistor:
R = aT + b
•Semiconductor:
logR = a  blogT
Low Temperature Thermometry
How stuff works

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