(a) In the big-bang theory of the universe, the radiation...

(a) In the big-bang theory of the universe, the radiation energy initially confined in a small region adiabatically expands in a spherically symmetric manner. The radiation cools down as it expands. Derive a relation between the temperature $T$ and the radius $R$ of the spherical volume of radiation, based purely on thermodynamic considerations. (b) Find the total entropy of a photon gas as a function of its temperature $T$, volume $V$, and the constants $k, \hbar, c$.

(a) In the big-bang theory of the universe, the radiation energy initially confined in a small region adiabatically expands in a spherically symmetric manner. The radiation cools down as it expands. Derive a relation between the temperature $T$ and the radius $R$ of the spherical volume of radiation, based purely on thermodynamic considerations.
(b) Find the total entropy of a photon gas as a function of its temperature $T$, volume $V$, and the constants $k, \hbar, c$.

Key Concepts

Entropy of a Photon Gas

Entropy quantifies the degree of disorder or the number of accessible microstates of a system. For a photon gas, the entropy can be expressed as a function of temperature, volume, and fundamental constants such as Boltzmann’s constant, Planck’s constant, and the speed of light. Deriving the entropy involves statistical mechanics methods and the recognition that, even for radiation that is not particle conserving, the extensive variable of entropy exhibits scaling behavior that reflects the underlying quantum properties of the photon gas.

Blackbody Radiation

Blackbody radiation refers to the spectrum of electromagnetic radiation emitted by a body that absorbs all incident radiation, reaching a state of thermal equilibrium. The energy density of a photon gas in thermal equilibrium is given by the Stefan-Boltzmann law, which relates the energy density to the fourth power of temperature. This law is foundational in deriving the thermodynamic relations for a radiation-dominated system.

Thermodynamic Relations in Expanding Systems

Thermodynamic analysis of an expanding system involves applying principles like energy conservation and the first law of thermodynamics (dU + pdV = 0 for an adiabatic process). For a radiation-dominated universe with a spherical symmetry, the relation between energy density, pressure, volume, and temperature is exploited to determine how temperature evolves as the system expands. This approach is essential for deriving relations such as T ? 1/R in a radiation-dominated expansion.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the environment. In thermodynamics, this implies that any change in the internal energy of a system is solely due to work done by or on the system. For an expanding radiation-filled sphere, the adiabatic condition means that the entropy remains constant, and this constraint is used to derive relationships between thermodynamic variables such as temperature and volume (or radius in spherical symmetry).

Photon Gas

A photon gas is a collection of photons that behave as a gas, and its properties can be described using thermodynamic concepts. Since photons have no rest mass and always move at the speed of light, their energy is directly related to temperature, with the energy density scaling as the fourth power of the temperature. This characteristic is crucial in understanding the behavior of radiation during cosmological expansion.

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