1. Nash Chapter 1, "The Statistical Viewpoint"

• Multiple body systems (N = body count) sharing energy can distribute it among themselves in many ways, each a microstate of the whole. Microstates indistinguishable from one another by virtue of the identity of the bodies are called configurations. Configurations are distiguished from one another by virtue of unique populations, {n_i}, over energy states, _i. The statistical weight, W, (importance) of a configuration is given by the number of microstates comprising it.

• W = N! / n_i! describes the number of microstates in a configuration as all possible relabelling of bodies except (see the denominator) those which relabel bodies in the same quantum state, i.e., wavefunction. Degeneracy among quantum states merely means several adjacent n_i will share the same _i.

• While the total number of ways to "find" the system configured would be the sum of all the configuration W's, for enormous N, one W* dominates. Configurations of W near W* lie within 1/N^½ of W* which is so close that they all have virtually the same thermodynamic properties. Said another way, the uncertainty or breadth of thermodynamic values is on the order of 1/N^½. It is this wonderful narrowness which permits statistical mechanics to succeed where particle-following mechanics must fail.

• Maximizing ln(W) over all possible configurations {n_i} is equivalent to maximizing W itself since dW/dn_i = 0 where d[ln(W)]/dn_i = 0. It permits us to employ Stirling's Approximation, ln(X!) = XlnX - X, when X is truly huge. Since for molar N and essentially finite rotation, vibration, and electronic quantum states, all of the n_i will be huge as well, we have no problem with Stirling.
  • But for the essentially infinite translational states, it is rare that n_i is greater than 0, not a huge number. Instead, we group essentially degenerate translational neighboring states together to boost their populations, _j , into the range where Stirling again applies.

• For any mode, then, application of conservation of bodies, n_i=N, and conservation of energy, n_ii=E, finally yields (via Lagrange Multipliers): n_i = e e^_i or n_i = N e^-_i/kT / z where Partition Function z = e^-_i/kT after the above conservation conditions are exercised and thermodynamical average energies of ½kT are presumed.

• Boltzmann's identification of statistical entropy, S = k ln(W*), can then be differentiated to show that dS = k dE/kT = q_rev/T as anticipated from macroscopic thermodynamics. The same consequences for T follow in both dynamics: systems in thermal contact spontaneously converge in the same T (since W is maximal then).

Pink Nash Chapter 2 "The Partition Function"

• z, the sum over states, actually adds each state's availability at the given temperature to arrive at a weighted total. Higher T gives e^-_i/kT weights near their highest value (1 each) while large energy spacings (e.g., electronic ones) yield very, very small contributions (by virtue of that minus sign).

• Chemically different systems will be in equilibrium with one another by the ratio of their available states, that is, by the ratio of their partition functions, z⁰, if they share the same value of ₀. If not, the _i in each system can be scaled with the offset of its ₀ from any convenient reference point, and the offsets can be collected into a single term, e^-₀/kT, which multiplies the z⁰ ratio assuming all had ₀=0.

• Since independent energies add to the total energy, and e^{sum a} = product e^a, the total z for many types of energy is just the product of z for each type, i.e., z_TOT = z_TRANS z_ROT z_VIB z_EL

• As we often would rather sum up energy levels rather than (possibly degenerate) states, the degeneracy of each level, g_i, scales the contribution of that level like: z = g_i e^-_i/kT

• If the bodies are truly distinguishable, as the molecules in lattice positions in a solid are distinguished each by its location, the number of ways of finding all of them Z = z^N since any different state occupied by one is a new system state for the collection. So the z's multiply to Z.

• If they're indistinguishable, the above leads to different "states" of the whole which really can never be told apart. Instead, then, we "label" each molecule with something else unique. If not its lattice location, then its triple of (x,y,z) translational quantum numbers; this works since it is vanishingly rare for any two gas molecules to share that triple! (There are just too many translational states.) But distributing those labels reduces Z by the factor N! (the # of ways you can do it). Thus indistinguishable particle partition functions are Z=z^N/N! instead.

• Since from the Boltzmann distribution, you can show that E = - N d[ln(z)]/d application of the definitions of those terms gives E = + kT² [ d lnZ / dT ]_V

• Similar application of S= k lnW to the Boltzmann Distribution yields S = k lnZ + E/T

• Recalling the physicist's Free Energy, A = E - TS, shows that this state function which governs equilibrium (when V rather than P is fixed) has the simplest statistical expression of all: A = - kT lnZ

• With these formulae and the product of different mode's partition functions, all macroscopic thermodynamical values can be calculated if the energy levels (and degeneracies) are known.

Pink Nash Chapter 3 "Evaluation of Partition Functions"

• Substituting the ideal gas translational _nx = n_x² h² / 8 m L_x² into the previous expressions gives E_TRANS = (3/2) RT where R = kN_Av.

  Likewise A = NkT [1 + ln(z/N)] which proves P=RT/V once the perfect differential of the work function (A) is remembered to be

  dA = - SdT - PdV which stems from the fundamental relation of thermodynamics, dE=TdS-PdV.

• Rotation shares with translation the possiblity of integrating the sum-over-states as a good approximation to the sum. But this is only true when RT is large relative to the energy gaps. That cannot be true at low enough T, and the temperature at which many levels of the mode begin to become available is called the (whatever mode) temperature, _mode.

  For linear molecule rotation, _ROT is roughly B/k where B is the rotational quantum state constant, _J = B J(J+1). At much higher T than B/k, many states involved permit the (easier) integration, giving

  z_ROT = kT/B unless symmetry reduces the orientation of the molecule which look unique. Each symmetry equivalent is an overcount since we can't tell 'em apart yet again!

  The count of all rotations which leave the molecule indistinguishable is the sum of all rotational symmetry elements (plus the identity, the "zero" rotation) and is called the symmetry factor, . The actual rotational partition function becomes

  z_ROT = kT/B and E_ROT = RT.

• If the molecule isn't linear, it has 3 (rather than 2) rotation axes, and E_ROT = (3/2) RT.

• For vibration, the energy levels are all equally-spaced, and the sum over states has a closed form expression z_VIB = 1 / [ 1 - e^-hv/kT ] where v is the vibration frequency for each vibration mode, some of which may be degenerate. Group theory can be called upon to identify the degeneracies (as well as the spectroscopic methods likely to show transitions between vibrational states).

  Since hv is usually huge compared to kT, z_VIB is often negligibly different from 1. But high T can offset that!

• Finally, the equilibrium constant, K_P, can be shown to be the ratio of z⁰ for each reactant (raised to its stoichiometric coefficient) times the zero-point correction factor, e^+D₀/kT, where D₀, being measured down from atomization commands that plus sign instead of the expected minus one.

  Here means what it usually does in thermodynamics, the difference in the quantities, scaled by their stoichiometric coefficients.

Nash Chapter 2 "First Principle of Thermodynamics and the Term q"

• The fundamental expression of thermodynamics
• The definition of enthalpy, H
• Hess's Law
• C_V and C_P and the T variation of E and H
• Adiabatic flames (P fixed) and explosions (T fixed)

Nash Chapter 3 "Second Principle of Thermodynamics, and the term S"

• Reversibility and work
• Isothermality and heat
• C_whatever / T as the T dependence of S
• Phase changes and H and S