Boltzmann Factor Calculator

The Boltzmann factor answers the question of how much more likely it is for a system to be in one state versus another at a given temperature.

If two energy values and a temperature are entered the tool will return the ratio of the two probabilities. If one of these three values is missing the tool calculates that value. It's possible to determine what temperature produces a given occupancy, or to find out the underlying energy difference.

Boltzmann distribution

A system that is in contact with a heat bath of temperature T will not necessarily remain in the lowest energy state. Thermal energy constantly excites the system and allows it to reach higher energy states as well. The probability of reaching any particular energy level decreases exponentially with increasing energy.

This is called the Boltzmann distribution, also known as the Gibbs distribution.

Here E is the energy of the state, T is the absolute temperature in Kelvin and Z is the partition function. The partition function is a normalization constant that ensures that the sum of all probabilities is equal to one.

It is worth emphasizing what it does not contain. The probability depends only on the energy of the state and does not depend on any other property of the state. Two states with the same energy have the same probability, no matter how different they may look.

The formula for the Boltzmann factor is:

Calculating Z is laborious as it requires all states of the system. So ratios are taken since Z can be cancelled out.

This exponential term is the Boltzmann factor. Since the equation contains only the difference of two energies, a lower energy state can be set to zero and the other state measured relative to it.

Two direct conclusions can be drawn from this: states with the same energy have the same occupancy probability and at positive temperatures the state with lower energy is always more probable than the other state.

The constant in the denominator is the Boltzmann constant, which was precisely defined by the 2019 revision of the SI.

This makes a connection between the macroscopic property of temperature and the energy of individual particles. It can be obtained by dividing the gas constant R by the Avogadro constant NA.

Why we use electron volts and what kBT actually means:

Expressed in Joules the Boltzmann constant is a difficult number to understand and associated energy values are similar. The electronvolt solves this problem. One eV is equal to 1.602176634 x 10^-19 J, which is the amount of energy gained by an electron when it traverses one volt.

Using eV makes calculations more transparent. The product of kB and T, i.e. thermal energy, serves as a yardstick for the respective energy differences.

Room temperature is about 25.7 meV, and many everyday physical phenomena can be explained by this value. Energy differences in the range of a few meV are overwhelmed by thermal energy, while an energy difference of one eV is almost completely "frozen out".

Example calculation

Suppose one state has energy E1 = 0.1 eV and another state has energy E2 = 0.2 eV. Both are at the freezing point of temperature T = 273.15 K which is equivalent to 0 degrees Celsius.

First, thermal energy is calculated.

The exponential term is then calculated which represents the energy difference in units of thermal energy.

The probability of a lower energy state occurring is about 70 times more likely than the higher energy state. Expressed as a ratio between these two states that equates to 98.6 percent versus 1.4 percent.

If the same two states are heated to room temperature, this ratio falls to about 49; if they're cooled down to liquid nitrogen temperatures, the ratio exceeds three million.

Interpretation of the exponential term.

The overall outcome is determined by a dimensionless number that divides the energy difference by thermal energy. This calculation tool shows the energy difference in units of heat.

If the heat units are about five or more above this is virtually the same as a frozen state. If they fall below one heat unit then the system remains in an active thermal mixing state.

Degeneracy: When multiple states belong to one energy level.

Energy levels often contain several states with the same energy. An atomic energy level with total angular momentum J contains 2J+1 states, each of which carries full Boltzmann weight.

When the Degeneracy option is enabled this calculator uses a formula to calculate that number of states.

The weights g1 and g2 are the number of states. Although they change the result, they do not affect the concept of energy itself because they are not contained in the exponent.

This is the format used by spectroscopists. The intensity of a spectral line depends on the number of atoms occupying a particular energy level and if five states occupy an energy level this will contribute to the value by a factor of five.

Backward calculation:

If a cell is left blank, then the calculation tool will calculate the value for that cell.

If the temperature is released and a ratio is entered, the calculation tool will return the temperature that leads to this occupancy distribution. The opposite also applies:

If an energy level is released and a temperature and ratio are entered, the calculation tool will return the corresponding energy difference.

A second type of back-calculation is often used in spectroscopy and semiconductor research. If the occupation ratios are measured and the temperature is known, then the energy difference follows automatically.

There are also combinations that have no solution. If state 1 has a higher energy level, then negative temperature is required if the ratio is greater than one. On the other hand, the ratio is exactly equal to one only when both energies are the same. So this calculator leaves this column blank and explains why.

Where does the Boltzmann factor come in?

Semiconductors:

The carrier concentrations both above and below the band gap follow the same exponential relationship. This is why silicon devices behave so differently at 77K than they do at 400K. The thermal voltage kT/q is about 25.85 mV (at 300 K) and is essentially the Boltzmann factor in an electrical engineering context.

Chemical reaction kinetics:

The Arrhenius law is a Boltzmann factor with the activation energy used in the exponent. The fraction of molecules with enough energy to trigger a reaction corresponds exactly to the occupancy of a high-energy level.

Spectroscopy and astrophysics:

The relative intensity of spectral lines allows the temperature to be determined for gases that can never be studied directly. The Saha equation and Boltzmann's equation allowed scientists, for the first time, to determine the temperature of stars from their spectra alone.

Magnetism and NMR:

The spin states that are split by a magnetic field will be distributed in equal proportions. At room temperature there will only be few particles in the lower spin state - a few parts per million. This is the whole signal used by an MRI scanner.

Common errors:

The most common mistake is using Celsius or Fahrenheit for the exponential terms. Since the formulas require an absolute temperature, this calculator first converts the chosen temperature unit to kelvin before doing any further calculations.

The next mistake is to mix two different energy units. Before you subtract values from each other, both energies must be converted into the same unit, as otherwise subtraction makes no sense. Here, the function for unit conversion takes on this task.

A third mistake is to interpret a factor as a probability. The factor does not mean that there are 70 or 70 percent, but it indicates that a lower energy state is 70 times more likely. Converted into percentage values this means 98.6 percent.

It is important to note that this is a ratio of two states and not an absolute probability. To obtain the actual probability one would need to use a partition function for all states in the system which requires a separate calculation.