Lecture 2 – The First Law (Ch. 1) Wednesday January 9 th

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Lecture 2 – The First Law (Ch. 1) Wednesday January 9 th. 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)

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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)

Probability and Statistics

Probability distribution function

Gaussian statistics:

Input parameters: Quality of teacher and level of difficulty

Abilities and study habits of the students

Probability and Statistics

Probability distribution function

Gaussian statistics:

Input parameters: Quality of teacher and level of difficulty

Abilities and study habits of the students

The connection to thermodynamics

Maxwell-Boltzmann speed distribution function

Input parameters:

Temperature and mass (T/m)

Equation of state:

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:

Boltzmann’s hypothesis concerning entropy:

where kB = 1.38 × 10-23 J/K is Boltzmann’s constant.

The bridge to thermodynamics through Z

js represent different configurations

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

Input parameters:

Temperature and mass (T/m)

Consider T 0

Bose particles (bosons)

Internal energy = 0

Entropy = 0

# of bosons

11

10

9

8

7

6

5

4

3

2

1

0

Energy

Fermi-Dirac particles (fermions)

Internal energy ≠ 0

Free energy = 0

Entropy = 0

# of fermions

Pauli exclusion principle

1

EF

0

Energy

Particles are indistinguishable

Applications

Specific heats:

Insulating solid

Diatomic molecular gas

Fermi and Bose gases

Thermal equilibrium

System 2

Heat

System 1

Pi, Vi

Pe, Ve

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

equilibrium

Pe, Ve

gas

When Peand Ve no longer change (static)  mechanical equilibrium

Different aspects of equilibrium

Chemical equilibrium:

nl↔ nv nl + nv = const.

P, nv, Vv

and mechanical

equilibrium

vapor

P, nl, Vl

liquid

When nl, nv, Vl & Vv no longer change (static)  chemical equilibrium

Different aspects of equilibrium

Chemical reaction:

A + B↔ AB# molecules ≠ const.

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

C

VA, PA

VC, PC

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

B

C

VA, PA

VC, PC

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

• 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.

q characterizes (is a measure of) the equilibrium.

Each equilibrium is unique. Erases all information on history.

Equations of state

• Defines a 2D surface in P-V-qstate space.
• Each point on this surface represents a unique equilibriumstate of the system.

q

f(P,V,q) = 0

Equilibrium state

• An equation of stateis 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 or q = f(P,V)

Example: PV = nRq (ideal gas law)

Gas Pressure Thermometer

Ice point

Steam point

LN2

Celsius scale

P = a[T(oC) + 273.15]

An experiment that I did in PHY3513

P T

17.7 79

13.8 0

3.63 -195.97

The ‘absolute’ kelvin scale

T(K) = T(oC) + 273.15

Triple point

of water:

273.16 K

Other Types of Thermometer

• Thermocouple: E = aT + bT2
• Metal resistor: R = aT + b
• Semiconductor: logR = a-blogT

Low Temperature Thermometry

How stuff works