﻿ Boltzmann factor

# Boltzmann factor

The Boltzmann factor with

Plays a central role in theoretical thermodynamics (statistical physics ).

States as the Boltzmann statistics, that the probability of finding a particle filled with a state of the power is proportional to the Boltzmann's factor:

When many particles are at the temperature in thermodynamic equilibrium.

Within a given energy interval may be located a plurality of states, so that the probability of observing a particle of this energy is given by the product of the Boltzmann factor, the energy state density and the width of the energy interval:

The Boltzmann factor is derived from purely statistical considerations and is essentially independent of the physical mechanisms (eg, interactions) within a thermodynamic system.

## Derivation of the exponential form

Assumption: The probability that a state is occupied with energy in thermal equilibrium is given by a continuous function. The ratio of occupation of any two states is then a function that can depend only on the energy difference due to the arbitrary choice of the zero point energy:

Let us now consider three states, so is therefore

This functional equation is solved only by the exponential function with one free parameter:

Thus,

And it follows for the shape of this function, the final

The meaning of the parameter turns out when using this equation, the total energy of a system is calculated from many mass points and set equal to the value that is valid for the one -atom ideal gas. result:

## Application Examples

### Barometric formula

Is the potential energy of a gas molecule with a mass of the air at height. The probability it is to be found in this amount proportional to

### Arrhenius equation

At the start of a chemical reaction, the molar activation energy is required. The rate constant of a chemical reaction is proportional to

### Vapor pressure curve

The transition from the liquid into the gas phase requires the molar heat of vaporization ( enthalpy would be more precise ). The saturation vapor pressure is proportional to

• Thermodynamics
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