Statistical explanations of thermodynamic properties

Thermodynamic properties follow the specified distribution

For a canonical system with temperature-independent state energies, U = Σⱼ pⱼEⱼ = −(∂ln Q/∂β)V,N. Helmholtz energy is A = −k_B T ln Q, and entropy is S = −k_B Σⱼ pⱼ ln pⱼ = k_B ln Q + U/T. These expressions use the same whole-system states and energy basis. A visual impression of disorder is not the definition used in this calculation.

For independent identical localized units, Q = qᴺ allows total properties to scale with N. A mole of such units uses R = N_A k_B for the molar formulas. The independence and fixed spectrum are assumptions; interactions, temperature-dependent effective levels or changing phase can require a different model.

A fixed gap gives a simple population calculation

Take a nondegenerate ground level 0 and excited level ε = k_B θ, where θ is the energy gap expressed as a temperature in kelvin. The single-unit sum is q = 1 + exp(−θ/T). The excited probability is p₁ = exp(−θ/T)/q. For one mole of independent units, Uₘ = Rθp₁ and Aₘ = −RT ln q.

The molar entropy is Sₘ = R[ln q + (θ/T)p₁], and Cᵥ,ₘ = R(θ/T)²p₁(1 − p₁). At θ = T = 300 K, p₁ ≈ 0.26894, Uₘ ≈ 671 J mol⁻¹, Sₘ ≈ 4.84 J mol⁻¹ K⁻¹ and Cᵥ,ₘ ≈ 1.63 J mol⁻¹ K⁻¹. The displayed model assumes a fixed gap and independent distinguishable sites.

The energy spectrum controls the heat-capacity limit

For the two-level model, the excited population approaches zero as T approaches zero. At very large T it approaches 1/2, so energy approaches its bounded midpoint and heat capacity tends to zero. Entropy tends to R ln 2 per mole. This finite spectrum does not have the universal heat capacity of a classical quadratic coordinate.

An independent harmonic oscillator instead has infinitely many equally spaced levels. With its ground energy subtracted, q = 1/[1 − exp(−θ/T)] and Uₘ = Rθ/[exp(θ/T) − 1]. Its heat capacity approaches R per mole of modes when T greatly exceeds θ within the mathematical harmonic model. A real vibrating molecule can become anharmonic or dissociate before an extrapolated high-temperature limit is meaningful.

Fluctuations and a shifted zero test the basis

For the canonical fixed-spectrum model, Var(E) = k_B T² Cᵥ, with variance and heat capacity belonging to the same actual system. For one two-level unit, Var(E) = ε²p₁(1 − p₁). For N independent units both the total variance and total heat capacity scale by N. A per-unit variance cannot be computed by inserting a molar heat capacity without the corresponding conversion.

Adding the same temperature-independent constant to every state energy multiplies all statistical weights by the same factor. Normalized probabilities therefore stay unchanged. U and A shift by that constant on the corresponding basis, while S and Cᵥ stay unchanged. A temperature-dependent shift or a changed level spacing is a different situation.

Worked examples

Try it yourself

1. At θ/T = ln 3 for a nondegenerate two-level unit, what is the excited probability?

   Show the answer and reasoning

   1/4.

   The weights are 1 and 1/3; divide the latter by their sum 4/3.

2. What happens to the two-level model’s heat capacity as T/θ becomes arbitrarily large?

   Show the answer and reasoning

   It tends to zero.

   The finite spectrum’s energy approaches a bounded average.

3. What is the harmonic model’s mathematical high-temperature Cᵥ,ₘ for one mole of complete modes?

   Show the answer and reasoning

   R.

   A complete classical harmonic coordinate has both kinetic and potential quadratic contributions.

4. Does a common constant energy shift change the normalized populations?

   Show the answer and reasoning

   No.

   The common weight factor cancels in normalization.

What to carry forward

• Derive properties from one consistent state distribution.
• Separate per-unit, molar and whole-system quantities.
• Heat-capacity limits depend on the model spectrum.
• A common constant energy zero affects U and A but not populations or Cᵥ.