Chapter 19 Statistical thermodynamics: the concepts Statistical Thermodynamics Kinetics Dynamics T, P, S, H, U, G, A... { r i},{ p i},{ M i},{ E i} … How to translate mic into mac?

The job description

Brute force approach does not work! 1 mol  particles  ~ equations needed to be solved!!! ……… Bad news: We cannot afford it! Good news: We do not need that detailed description! How?

Thanks to them Josiah Willard Gibbs James Clark Maxwell Ludwig Boltzmann Cheng-Ning Yang Tsung-Dao Lee Van Hove Landau Arrhennius Enrico Fermi Bose Paul Dirac Langevin Einstein Ising MottAnderson Bardeen …

THE magic word Statistical We, the observers, are macroscopic. We only need average of microscopic information. Spatial and temporal average

T, P, S, H, U, G, A... { r i},{ p i},{ M i},{ E i }... Still lots of challenges (opportunities) herein!

Contents The distribution of molecular states 19.1 Configuration and weights 19.2 The molecular partition function The internal energy and the entropy 19.3 The internal energy 19.4 The statistical entropy The canonical partition function 19.5 The canonical ensemble 19.6 The thermodynamic information in partition function 19.7 Independent molecules

Assignment for Chapter 19 Exercises: 19.1(a), 19.2(b), 19.4(a) Problems: 19.6(a), 19.9(b), 19.11(b), 19.15(a) 19.3, 19.7, 19.14, 19.18, 19.22

The distribution of molecular states EEE E These particles might be distinguishable Distribution = Population pattern

Enormous possibilities! EE E E E E E E E E E E E E

Distinguishable particles E E E EE

Principle of equal a priori probabilities All possibilities for the distribution of energy are equally probable. An assumption and a good assumption.

They are equally probable EE E E E E E E E E E E E E

E E E E E They are equally probable

Configuration and weights {5,0,0,...} The numbers of particles in the states

{3,2,0,...}

One configuration may have large number of instantaneous configurations

{N-2,2,0,...} How many instantaneous configurations? N(N-1)/2

E 18!/3!/4!/5!/6! {3,4,5,6}

Configuration and weights W is huge! 20 particles: {1,0,3,5,10,1}  W= How about particles with {2000,3000,4000,1000}?

Stirling’s approximation:

W max {n i}max {n i } There is an overwhelming configuration W

Equilibrium configuration The dominating configuration is what we actually observe. All other configurations are regarded as fluctuation. The dominating configuration is the configuration with largest weight. Constant total number of molecules: Constant total energy :

Find the distribution with largest lnW

Lagrange’s method of undetermined multipliers z=f(x,y) with g(x,y)=c 1, h(x,y)=c 2 To find the maximum of z with constraints g and h, we May use

Boltzmann distribution Boltzmann constant

The molecular partition function (nondegenerate case) E

The molecular partition function (degenerate case) E

Angular momentum

The Rotational Energy Levels (Ch 16) Around a fixed-axis Around a fixed-point (Spherical Rotors)

Example: Linear Molecules (rigid rotor) E E1E1 E2E2 E3E3 E4E4 E5E5

Exercises E E1E1 E2E2 E3E3 E4E4 E5E5 q=? E=0,g=1 E=ε,g=2 q=?

The physical interpretation of the molecular partition function q is an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system. E, T E, T=0 E, ∞

Example: Uniform ladder of energy levels (e.g., harmonic vibrator)

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1 E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1 E=0,g=1 E=ε,g=1

Approximations and factorizations Exact, analytical partition functions are rare. Various kinds of approximations are employed: dense energy levels independent states (  factorization of q) …

Dense energy levels One dimensional box:

Independent states (  factorization of q) Three-dimensional box: Thermal wavelength (Translational partition function)

Why q, the molecular partition function, so important? It contains all information needed to calculate the thermodynamic properties of a system of independent particles (e.g., U, S, H, G, A, p, Cp, Cv …) It is a kind of “thermal wavefunction”. (Remember the wavefunction in quantum mechanics which contains all information about a system we can possibly acquire)

Find the internal energy U from q Total energy of the system: At T=0, U=U(0)

A two-level system E=0,g=1 E=ε,g=1 U=?

A two-level system E=0,g=1 E=ε,g=1 W=? Exercise

A two-level system E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

The value of β For monatomic perfect gas, Translational partition function:

The statistical entropy When no work is done, For the most probable configuration: dN=0

Heat does not change energy levels

Work changes energy levels

Boltzmann formula T  0, W=1, S  0 (Third law of thermodynamics )

E=0,g=1 E=ε,g=1 The two-level system

Negative temperature! E=0,g=1 E=ε,g=1 E=0,g=1 E=ε,g=1 T>0, N + <N- T N- Examples: laser, maser, NMR etc.

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1

E=0,g=1 E=ε,g=1 Positive temperature Negative temperature

U and S =1

Example: Simple harmonic oscillator

E=0,g=1 E=ε,g=1 Exercise: The two-level system U=?, S=?

The canonical partition function Q Independent system vs interacting system Molecular partition function q Ensemble: an imaginary collection of replication of the actual system with a common macroscopic parameter. Canonical ensemble: an imaginary collection of replications of the actual system with a common temperature. (N, V,T) Microcanonical ensemble: an imaginary collection of replications of the actual system with a common energy. (N, V,E) Grand canonical ensemble: an imaginary collection of replications of the actual system with a common chemical potential. (μ, V,T) (close system) (isolated system) (open system)

The number of states with energy between E i and E i+1

The canonical partition function Q The most probable configuration:

Find internal energy U from Q The total energy of the ensemble: The average energy: The fraction of members of the ensemble in a state with energy Ei: (classroom exercise)

Find entropy S from Q The total weight : (5 points bonus!!!)

Independent molecules (classroom exercise)

(a)For distinguishable independent molecules: (b) For indistinguishable independent molecules: E E

Monatomic gas (Sackur-Tetrode equation)

Using the Sackur-Tetrode equation Calculate the standard molar entropy of gaseous argon at 25C

Sackur-Tetrode equation As the container expands, X increeases  more states accessible for the system  S increases.