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 ) 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 Input parameters:Quality of teacher and level of difficulty Abilities and study habits of the students 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:

The connection to thermodynamics Maxwell-Boltzmann speed distribution function Equation of state: Input parameters: Temperature and mass (T/m)

Probability and Entropy Suppose you toss 4 coins. There are 16 (2 4 ) 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 k B = 1.38 × 10  23 J/K is Boltzmann’s constant.

The bridge to thermodynamics through Z j s 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

Energy # of bosons Bose particles (bosons) Internal energy = 0 Entropy = 0

Energy # of fermions 1 0 Fermi-Dirac particles (fermions) Pauli exclusion principle EFEF Internal energy ≠ 0 Free energy = 0 Entropy = 0 Particles are indistinguishable

Applications Insulating solidDiatomic molecular gas Specific heats: Fermi and Bose gases

The zeroth & first Laws Chapter 1

Thermal equilibrium System 1 System 2 Heat P i, V i P e, V e If P i = P e and V i = V e, then system 1 and systems 2 are already in thermal equilibrium.

Different aspects of equilibrium 1 kg Mechanical equilibrium: P e, V e Piston gas Already in thermal equilibrium When P e and V e no longer change (static)  mechanical equilibrium

P, n l, V l P, n v, V v Different aspects of equilibrium Chemical equilibrium: Already in thermal and mechanical equilibrium liquid vapor n l ↔ n v n l + n v = const. When n l, n v, V l & V v no longer change (static)  chemical equilibrium

A, B & AB Different aspects of equilibrium Chemical reaction: A + B ↔ AB # molecules ≠ const. Already in thermal and mechanical equilibrium When n A, n B & n AB 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 ACCB a)b) V A, P A V C, P C V B, P B “If two systems are separately in thermal equilibrium with a third system, they are in equilibrium with each other.” AB c) V A, P A V B, P B

The Zeroth Law ACCB a)b) V A, P A V C, P C V B, P B “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,   f(P,V ), where the parameter,  (temperature), characterizes the equilibrium. Even more useful is the fact that this same value of  also characterizes any other system which is in thermal equilibrium with the first system, regardless of its state.

More on thermal equilibrium  characterizes (is a measure of) the equilibrium. Continuum of different mechanical equilibria (P,V) for each thermal equilibrium, . Experimental fact: for an ideal, non-interacting gas, PV = constant (Boyle’s law). Why not have PV proportional to   ; Kelvin scale. Each equilibrium is unique. Erases all information on history.

Equations of state An equation of state is a mathematical relation between state variables, e.g. P, V & . This reduces the number of independent variables to two. General form: f (P,V,   ) = 0 or  = f (P,V) Example:PV = nR  (ideal gas law)  Defines a 2D surface in P-V-  state space. Each point on this surface represents a unique equilibrium state of the system. f (P,V,   ) = 0 Equilibrium state

Temperature Scales

P = a[T( o C) ] Gas Pressure Thermometer Celsius scale Steam point Ice point LN 2

PT An experiment that I did in PHY3513

T(K) = T( o C) The ‘absolute’ kelvin scale Triple point of water: K

Other Types of Thermometer Thermocouple: E = aT + bT 2 Metal resistor: R = aT + b Semiconductor: logR = a  blogT How stuff works How stuff works Low Temperature Thermometry Low Temperature Thermometry