Chapter 2 Statistical Thermodynamics

1- Introduction - The object of statistical thermodynamics is to present a particle theory leading to an interpretation of the equilibrium properties of macroscopic systems. - The foundation upon which the theory rests is quantum mechanics. - A satisfactory theory can be developed using only the quantum mechanics concepts of quantum states, and energy levels. - A thermodynamic system is regarded as an assembly of submicroscopic entities in an enormous number of every-changing quantum states. We use the term assembly or system to denote a number N of identical entities, such as molecules, atoms, electrons, photons, oscillators, etc.

1- Introduction The macrostate of a system, or configuration, is specified by the number of particles in each of the energy levels of the system. Nj is the number of particles that occupy the jth energy level. If there are n energy levels, then A microstate is specified by the number of particles in each quantum state. In general, there will be more than one quantum state for each energy level, a situation called degeneracy.

1- Introduction In general, there are many different microstates corresponding to a given macrostate. The number of microstates leading to a given macrostate is called the thermodynamic probability. It is the number of ways in which a given macrostate can be achieved. - The thermodynamic probability is an “unnormalized” probability, an integer between one and infinity, rather than a number between zero and one. For a kth macrostate, the thermodynamic probability is taken to be ωk.

1- Introduction - A true probability pk could be obtained as where Ω is the total number of microstates available to the system.

2- Coin model example and the most probable distribution - We assume that we have N=4 coins that we toss on the floor and then examine to determine the number of heads N1 and the number of tails N2 = N-N1. - Each macrostate is defined by the number of heads and the number of tails. A microstate is specified by the state, heads or tails, of each coin. We are interested in the number of microstates for each macrostate, (i.e., thermodynamic probability). - The coin-tossing model assumes that the coins are distinguishable.

Macrostate Label Macrostate Microstate k Pk k N1 N2 Coin 1 Coin 2 Coin 3 Coin 4 1 4 H 1/16 2 3 T 4/16 6 6/16 5

2- Coin model example and the most probable distribution The average occupation number is where Nik is the occupation number for the kth macrostate. For our example of coin-tossing experiment, the average number of heads is therefore

2- Coin model example and the most probable distribution k 1 4 2 6 3 Figure. Thermodynamic probability versus the number of heads for a coin-tossing experiment with 4 coins. Suppose we want to perform the coin-tossing experiment with a larger number of coins. We assume that we have N distinguishable coins. Question: How many ways are there to select from the N candidates N1 heads and N-N1 tails?

2- Coin model example and the most probable distribution The answer is given by the binomial coefficient Example for N = 8 N1 k 1 8 2 28 3 56 4 70 5 6 7 Figure. Thermodynamic Probability versus the number of heads for a coin-tossing experiment with 8 coins. The peak has become considerably sharper

2- Coin model example and the most probable distribution What is the maximum value of the thermodynamic probability (max) for N=8 and for N=1000? The peak occurs at N1=N/2. Thus, Equation (1) gives For such large numbers we can use Stirling’s approximation:

2- Coin model example and the most probable distribution For N = 1000 we find that max is an astronomically large number

2- Coin model example and the most probable distribution Figure. Thermodynamic probability versus the number of heads for a coin-tossing experiment with 1000 coins. (the most probable distribution is a macrostate for which we have a maximum number of microstates) The most probable distribution is that of total randomness The “ordered regions” almost never occur; ω is extremely small compared with ωmax.

2- Coin model example and the most probable distribution For N very large Generalization of equation (1) Question: How many ways can N distinguishable objects be arranged if they are divided into n groups with N1 objects in the first group, N2 in the second, etc? Answer:

3- System of distinguishable particles The constituents of the system under study (a gas, liquid, or solid) are considered to be: a fixed number N of distinguishable particles occupying a fixed volume V. We seek the distribution (N1, N2,…, Ni,…, Nn) among energy levels (ε1, ε2,…, εi,…, εn) for an equilibrium state of the system.

3- System of distinguishable particles We limit ourselves to isolated systems that do not exchange energy in any form with the surroundings. This implies that the internal energy U is also fixed Example: Consider three particles, labeled A, B, and C, distributed among four energy levels, 0, ε, 2ε, 3ε, such that the total energy is U=3ε. Tabulate the 3 possible macrostates of the system. Calculate ωk (the number of microstates), and pk (True probability) for each of the 3 macrostates. c) What is the total number of available microstates, Ω, for the system

3- System of distinguishable particles Macrostate Label Specification Microstate Thermod. Prob. True k N0 N1 N2 N3 A B C ωk pk 1 2 3ε 3 0.3 ε 2ε 6 0.6 0.1

3- System of distinguishable particles The most “disordered” macrostate is the state of highest probability. this state is sharply defined and is the observed equilibrium state of the system (for the very large number of particles.)

4- Thermodynamic probability and Entropy In classical thermodynamics: as a system proceeds toward a state of equilibrium the entropy increases, and at equilibrium the entropy attains its maximum value. In statistical thermodynamics (our statistical model): system tends to change spontaneously from states with low thermodynamic probability to states with high thermodynamic probability (large number of microstates). It was Boltzmann who made the connection between the classical concept of entropy and the thermodynamic probability: S and Ω are properties of the state of the system (state variables).

4- Thermodynamic probability and Entropy Consider two subsystems, A and B Subsystem A Subsystem B The entropy is an extensive property, it is doubled when the mass or number of particles is doubled. Consequence: the combined entropy of the two subsystems is simply the sum of the entropies of each subsystem:

4- Thermodynamic probability and Entropy One subsystem configuration can be combined with the other to give the configuration of the total system. That is, Example of coin-tossing experiment: suppose that the two subsystems each consist of two distinguishable coins. Macrostate Subsystem A Subsystem B ( N1 , N2 ) Coin 1 Coin 2 ωkA ωkB pkA pkB ( 2 , 0 ) H 1 1/4 ( 1 , 1 ) T 2 2/4 ( 0 , 2 )

4- Thermodynamic probability and Entropy Thus Equation (4) holds, and therefore Combining Equations (3) and (5), we obtain The only function for which this statement is true is the logarithm. Therefore Where k is a constant with the units of entropy. It is, in fact, Boltzmann’s constant:

5- Quantum states and energy levels In quantum theory, to each energy level there corresponds one or more quantum states described by a wave function Ψ(r,t). When there are several quantum states that have the same energy, the states are said to be degenerate. The quantum states associated with the lowest energy level are called the ground states of the system; those that correspond to higher energies are called excited states. Figure. Energy levels and quantum states.

5- Quantum states and energy levels - For each energy level εi the number of quantum states is given by the degeneracy gi (or the degeneracy gi is the number of quantum states whose energy level is εi). The energy levels can be thought of as a set of shelves at different heights, while the quantum states correspond to a set of boxes on each shelf.

5- Quantum states and energy levels Example: Consider a particle of mass m in a one-dimensional box with infinitely high walls. (The particle is confined within the region 0 ≤ x ≤ L.) Within the box the particle is free, subjected to no forces except those associated with the walls of the Container. Time-independent Shrödinger equation:

5- Quantum states and energy levels 0 < x < L: V = 0 General solution: Using k = nπ/L, we find that the particle energies are quantized:

5- Quantum states and energy levels The integer n is the quantum number of the one-dimensional box. - Stationary states for the particle in the box: For each value of the quantum number n there is a specific wavefunction n(x) describing the state of the particle with energy εn. For the case of a three-dimensional box with dimensions Lx, Ly, and Lz , the energy becomes Any particular quantum state is designated by three quantum numbers nx, ny, and nz.

5- Quantum states and energy levels If Lx = Ly = Lz = L, then ni is the total quantum number for states whose energy level is εi. - εi depend only on the values of ni2 and not on the individual values of the integers (nx,ny,nz). - The volume V of a cubical box equals L3, so L2 = V2/3 and hence as V decreases, the value of εi increases.

5- Quantum states and energy levels Table. First three states of a three-dimensional infinite potential well Level Energy State (nx,ny,nz) ni2 gi i = 1 Ground state (1,1,1) 3 1 i = 2 First excited state (1,1,2); (1,2,1); (2,1,1) 6 i = 3 Second excited state (1,2,2); (2,1,2); (2,2,1) 9

6- Density of Quantum States When the quantum numbers ni are large, the energy levels εi are very closely spaced and that the discrete spectrum may be treated as an energy continuum. Consequence: We can regard the n’s and the ε’s as forming a continuous function rather than a discrete set of values. We are interested in finding the density of states g(ε) We have: Dropping the subscripts in equation above, using , and solving for n2, we obtain

6- Density of Quantum States According to equation above, for a given value of ε, the values of nx, ny, nz that satisfy this equation lie on the surface of a sphere of radius R. g(ε)dε is the number of states whose quantum numbers (nx,ny,nz) lie within the infinitesimally thin shell of the octant of a sphere with radius proportional to the square root of the energy.

6- Density of Quantum States Evidently, n(ε) is the number of states contained within the octant of the sphere of radius R; that is,

6- Density of Quantum States The last equation takes into account the translational motion only of a particle of the system. But quantum particles may have spin as well. Thus we must multiply the above equation by a spin factor γs: where γs = 1 for spin zero bosons and γs = 2 for spin one-half fermions. Bosons are particles with integer spin. Fermions are particles with half-integer spin. For molecules, states associated with rotation and vibration may exist in addition to translation and spin.