Boltzmann factor
The Boltzmann factor with
- The energy
- The Boltzmann constant
- The absolute temperature
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