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Photon gas

Overview

In physics, a photon gas is a gas

Gas

This page is about the physical properties of gas as a state of matter. For the uses of gases, and other meanings, see Gas .A gas is one of four states of matter. Near absolute zero, a substance exists as a solid...

-like collection of photon

Photon

In physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s, which has many of the same properties of a conventional gas like hydrogen

Hydrogen

Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. At standard temperature and pressure, hydrogen is a colorless, odorless, nonmetallic, tasteless, highly flammable diatomic gas with the molecular formula H₂...

or neon

Neon

Neon is the chemical element that has the symbol Ne and atomic number 10. Although a very common element in the universe, it is rare on Earth. A colorless, inert noble gas under standard conditions, neon gives a distinct reddish-orange glow when used in discharge tubes and neon lamps...

- including pressure, temperature, and entropy. The most common example of a photon gas in equilibrium is black body radiation.

A massive ideal gas

Ideal gas

An ideal gas is a theoretical gas composed of a set of randomly-moving point particles that interact only through elastic collisions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At...

with only one type of particle is uniquely described by three state functions such as the temperature

Temperature

In physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...

, volume

Volume

The volume of any solid, liquid, gas, plasma, theoretical object, or vacuum is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space...

, and the number of particles

Particle number

The particle number, N, is the number of constituent particles in a thermodynamical system. The particle number is a fundamental parameter in thermodynamics and it is conjugate to the chemical potential....

.

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Encyclopedia

In physics, a photon gas is a gas

Gas

This page is about the physical properties of gas as a state of matter. For the uses of gases, and other meanings, see Gas .A gas is one of four states of matter. Near absolute zero, a substance exists as a solid...

-like collection of photon

Photon

In physics, a photon is an elementary particle, the quantum of the electromagnetic field and the basic "unit" of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s, which has many of the same properties of a conventional gas like hydrogen

Hydrogen

Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. At standard temperature and pressure, hydrogen is a colorless, odorless, nonmetallic, tasteless, highly flammable diatomic gas with the molecular formula H₂...

or neon

Neon

Neon is the chemical element that has the symbol Ne and atomic number 10. Although a very common element in the universe, it is rare on Earth. A colorless, inert noble gas under standard conditions, neon gives a distinct reddish-orange glow when used in discharge tubes and neon lamps...

- including pressure, temperature, and entropy. The most common example of a photon gas in equilibrium is black body radiation.

A massive ideal gas

Ideal gas

An ideal gas is a theoretical gas composed of a set of randomly-moving point particles that interact only through elastic collisions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At...

with only one type of particle is uniquely described by three state functions such as the temperature

Temperature

In physics, temperature is a physical property of a system that underlies the common notions of hot and cold; something that feels hotter generally has the higher temperature. Temperature is one of the principal parameters of thermodynamics...

, volume

Volume

The volume of any solid, liquid, gas, plasma, theoretical object, or vacuum is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space...

, and the number of particles

Particle number

The particle number, N, is the number of constituent particles in a thermodynamical system. The particle number is a fundamental parameter in thermodynamics and it is conjugate to the chemical potential....

. However, for a black body, the energy distribution is established by the interaction of the photons with matter, usually the walls of the container. In this interaction, the number of photons is not conserved. As a result, the chemical potential

Chemical potential

Chemical potential, symbolized by μ, is a quantity first described by the American engineer, chemist and mathematical physicist Josiah Williard Gibbs...

of the black body photon gas is zero. The number of state functions needed to describe a black body state is thus reduced from three to two (e.g. temperature and volume).

Thermodynamics of a black body photon gas

In a gas with massive particles, the energy of the particles is distributed according to a Maxwell-Boltzmann distribution. This distribution is established as the particles collide with each other, exchanging energy (and momentum) in the process. In a photon gas, there will also be an equilibrium distribution, but photons do not collide with each other (except under very extreme conditions) so that the equilibrium distribution must be established by other means. The most common way that an equilibrium distribution is established is by the interaction of the photons with matter. If the photons are absorbed and emitted by the walls of the system containing the photon gas, and the walls are at a particular temperature, then the equilibrium distribution for the photons will be a black body

Black body

In physics, a black body is an idealized object that absorbs all electromagnetic radiation that falls on it. No electromagnetic radiation passes through it and none is reflected. Because no light is reflected or transmitted, the object appears black when it is cold. However, a black body emits a...

distribution at that temperature.

A very important difference between a gas of massive particles and a photon gas with a black body distribution is that the number of photons in the system is not conserved. A photon may collide with an electron in the wall, exciting it to a higher energy state, removing a photon from the photon gas. This electron may drop back to its lower level in a series of steps, each one of which releases an individual photon back into the photon gas. Although the sum of the energies of the emitted photons are the same as the absorbed photon, the number of emitted photons will vary. It can be shown that, as a result of this lack of constraint on the number of photons in the system, the chemical potential

Chemical potential

Chemical potential, symbolized by μ, is a quantity first described by the American engineer, chemist and mathematical physicist Josiah Williard Gibbs...

of the photons must be zero for black body radiation.

The thermodynamics of a black body photon gas may be derived using quantum mechanical arguments

Gas in a box

In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions...

. The derivation yields the spectral energy distribution u which is the energy per unit volume per unit frequency interval:

where h is Planck's constant, c is the speed of light, ν is the frequency, k is Boltzmann's constant, and T is temperature.
Integrating over frequency and multiplying by the volume (V ) gives the internal energy

Internal energy

In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electric energy of atoms...

of a black body photon gas:

The derivation also yields the (expected) number of photons N:
where is the Riemann zeta function

Riemann zeta function

In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann who introduced it in 1859, is a prominent function of great significance in number theory because of its relation to the distribution of prime numbers...

. Note that for a particular temperature, the particle number varies with the volume in a fixed manner, adjusting itself to have a constant density of photons.

The following table summarizes the thermodynamic state functions for a black body photon gas.

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Internal energy

In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules and the potential energy associated with the vibrational and electric energy of atoms...

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! align="center" |Particle number

Particle number

The particle number, N, is the number of constituent particles in a thermodynamical system. The particle number is a fundamental parameter in thermodynamics and it is conjugate to the chemical potential....

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! align="center" |Chemical potential

Chemical potential

Chemical potential, symbolized by μ, is a quantity first described by the American engineer, chemist and mathematical physicist Josiah Williard Gibbs...

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! align="center" |Pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

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! align="center" |Entropy

Entropy

Entropy is a concept of information maintaining great importance in physics, chemistry, and information theory...

! align="center" |
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! align="center" |Enthalpy

Enthalpy

In thermodynamics and molecular chemistry, the enthalpy is a thermodynamic property of a thermodynamic system. It can be used to calculate the heat transfer during a quasistatic process taking place in a closed thermodynamic system under constant pressure...

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! align="center" |Helmholtz free energy

Helmholtz free energy

In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

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! align="center" |Gibbs free energy

Gibbs free energy

In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from an isothermal, isobaric thermodynamic system...

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Isothermal transformations

As an example of a thermodynamic process involving a photon gas, consider a cylinder with a movable piston. The interior walls of the cylinder are "black" in order that the temperature of the photons can be maintained at a particular temperature. This means that the space inside the cylinder will contain a blackbody-distributed photon gas. Unlike a massive gas, this gas will exist without the photons being introduced from the outside - the walls will provide the photons for the gas. Suppose the piston is pushed all the way into the cylinder so that there is an extremely small volume. The photon gas inside the volume will press against the piston, moving it outward, and in order for the transformation to be isothermic, a counter force of almost the same value will have to be applied to the piston so that the motion of the piston is very slow. This force will be equal to the pressure times the cross sectional area (A ) of the piston. This process can be continued at aconstant temperature until the photon gas is at a volume V₀ . Integrating the force over the distance (x ) travelled yields the total work done to create this photon gas at this volume
where the relationship V=Ax has been used. Defining
The pressure is
Integrating, the work done is just
The amount of heat that must be added in order to create the gas is
where H₀ is the enthalpy at the end of the transformation. It is seen that the enthalpy is the amount of energy needed to create the photon gas.

Photon gas

Thermodynamics of a black body photon gas

Isothermal transformations

See also