Boltzmann Distribution

1. The thermal energy of a substance is described by its temperature. Simple equations can convert absolute temperature (measured in degrees Kelvin) to standard energy units (joules).

Energy per mole: RT

and R is the ideal gas constant, 8.314 J/K*mol

Energy per molecule: kT

Where k is Boltzmann’s constant, 1.380 * 10^-23 J/K

Note: since RT is the thermal energy per mole of a substance, this term is found in many equations. For example the standard state free energy of a chemical reaction, ∆G^o = -RT ln(K)

2. The Boltzmann distribution describes the probability of finding a molecule at a particular energy level relative to a ground state when the temperature is T. It is a way of converting energies to probabilities.

(P₁ / P₀) = e^-∆E/RT

Where P₁ and P₀ are the probabilities of finding a molecule in the ground state (state 0) and the desired state (state 1) respectively

and ∆E is the change in energy from state 0 to state 1.

Note: since the Boltzmann probability equation is exponential, not linear, the frequency of objects at a given energy level can change greatly with only a small change in temperature or in the chosen energy level.

3. Examples using the Boltzmann distribution:

At room temperature, the quantity RT is 8.314 * 273 = 2270 J/mol

The change in energy due to gravity of an object moving from one height to another is ∆E = mg∆h. Where m is the mass of the object, g is the standard gravity of Earth (9.8 m/s²), and ∆h is the change in height. This means that low-mass objects, such as molecules, have a low change in energy per change in height and are thus likely to be found at a great range of heights, whereas high-mass objects such as chairs are not.

The energy of a hydrogen bond is typically around 21 kJ/mol, for an OH-O hydrogen bond. Using the Boltzmann equation, we get: e^-21000/2270 ≈ e^-10 = 4.5*10^-5

This is the ratio of the number of broken bonds compared to unbroken bonds at any moment in time.