12. Boltzmann distribution.

Question 1

Explain the meaning of the Boltzmann distribution, explaining by reference of a diagram and give the related formula

Answer 1

At thermal equilibrium, the distribution of particles among the available energy states will take the most probable distribution consistent with the total available energy and total number of particles.
Every specific state of the system has equal probability.

→ unlikely that any one particle will get an energy far above the average (i.e., far more than its share)

Boltzmann distribution function: n_i/n₀ = e^(-(ε_i - ε₀)/kT) = e^(-Δε/kT)

⇒ if T increases: more particles in higher E state

Question 2

Describe the significance (= application) of Boltzmann distribution in reference to gaseous systems. Be able to draw a diagram supporting your explanations

Answer 2

The rules for gaseous systems are based on the Boltzmann distribution and general laws of thermodynamics.

→ average E_kin determined by T

deviation explained by Maxwell-Boltzmann function:

Question 3

Describe the significance (= application) of Boltzmann distribution in reference to systems with structural order

Answer 3

Order is based on interactions (bonds) between the constituting atoms or molecules.

BUT: _according to the Boltzmann distribution: _

depends the probability of breaking bonds by thermal fluctuations on the relation:

Δε = E_bond ⇔ kT n_broken/n_intact = n/n₀

Question 4

Explain the relation of the general concept of bond formation to Boltzmann distribution

Answer 4

E_{pot(repulsion)} = B/r^m

E_{pot(attraction)} = -A/rⁿ

BUT: m > n

→ The concrete expressions for A, B and
the values of n and m are depend on the form of interactions → determine the atom bond distances

Question 5

Relate Boltzmann distribution to the atmospheric pressure

Answer 5

Density of air in the atmosphere (= atmospheric pressure) decreases with the altitude (h) by the

Barometric formula:

p(h)/p(0) = e^-(mgΔh)/kT

Resembles the probability that particles of higher E are located in the higher altitude.

The comparison of E_pot (= of particles is related to mgh (+E_thermic, but negligible), therefore mgΔh substituted into the formula.

Population ratio n_h/n_o = p(h)/p(0) according to the ideal gas law.

Question 6

Relate Boltzmann distribution to the equilibrium rates

Answer 6

Equilibrium rate of chemical reactions:

K_eq = k_AB/k_BA = e^-(ε_A-ε_B)/kT

The k_AB and k_BA rates express the ratio of reactants to overcome the barrier_{AB & BA}. Therefore the equilibrium constant K_eq = 1 if k_AB and k_BA have the same value (both reactions occur at the same rate).