By treating radiation in a cavity as a gas of photons whose energy εand momentum k are related b

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By treating radiation in a cavity as a gas of photons whose...

By treating radiation in a cavity as a gas of photons whose energy $\varepsilon$ and momentum $k$ are related by the expression $\varepsilon=c k$, where $c$ is the velocity of light, show that the pressure $p$ exerted on the walls of the cavity is one-third of the energy density. With the above result prove that when radiation contained in a vessel with perfectly reflecting walls is compressed adiabatically it obeys the equation $$ P V^\gamma=\text { constant } . $$ Determine the value of $\gamma$.

By treating radiation in a cavity as a gas of photons whose energy $\varepsilon$ and momentum $k$ are related by the expression $\varepsilon=c k$, where $c$ is the velocity of light, show that the pressure $p$ exerted on the walls of the cavity is one-third of the energy density.

With the above result prove that when radiation contained in a vessel with perfectly reflecting walls is compressed adiabatically it obeys the equation
$$
P V^\gamma=\text { constant } .
$$

Determine the value of $\gamma$.

Key Concepts

Adiabatic Exponent (?) and Equation of State

The adiabatic exponent, ?, is the ratio of specific heats (or equivalently, it characterizes the relation P V^? = constant during an adiabatic process). For a photon gas, due to the unique dependence of its energy density on temperature and the relation between pressure and energy density, the value of ? is derived to be 4/3. This exponent is a key characteristic in describing the thermodynamic behavior of radiation under adiabatic compression or expansion.

Radiation Pressure and Energy Density

Radiation pressure arises from the momentum transfer of photons when they collide with the walls of a container. By considering the directional distribution of the photon momenta, it can be shown that the pressure exerted by a photon gas is one-third of its energy density. This connection is fundamental in connecting microscopic particle properties to macroscopic pressure measurements.

Adiabatic Process in Thermodynamics

An adiabatic process is one in which no heat is exchanged with the surroundings. For a photon gas confined within perfectly reflecting walls, any compression or expansion occurs without heat transfer, and the change in internal energy is solely due to the work done on or by the gas. This leads to a specific relation between the pressure and volume of the system.

Energy–Momentum Relationship for Photons

For photons, the energy is directly proportional to their momentum through the relation ? = ck, where c is the speed of light. This fundamental relation is critical in linking the microscopic behavior of photons to macroscopic observable quantities such as pressure and energy density.

Photon Gas

A photon gas is a collection of photons that exhibits gas-like behavior despite photons being massless. In thermodynamic equilibrium, the photons in a cavity can be described statistically in a manner analogous to an ideal gas, with their energy distribution governed by quantum statistics and the energy–momentum relationship.

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