Use the formula P=-(∂U / ∂V)S, N to show that the pressure of a photon gas is 1 / 3 times the en

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Use the formula P=-(∂U / ∂V)S, N to show that the pressure...

Use the formula $P=-(\partial U / \partial V)_{S, N}$ to show that the pressure of a photon gas is $1 / 3$ times the energy density $(U / V)$. Compute the pressure exerted by the radiation inside a kiln at $1500 \mathrm{K}$, and compare to the ordinary gas pressure exerted by the air. Then compute the pressure of the radiation at the center of the sun, where the temperature is 15 million $\mathrm{K}$. Compare to the gas pressure of the ionized hydrogen, whose density is approximately $10^{5} \mathrm{kg} / \mathrm{m}^{3}$

Key Concepts

Photon Gas

A photon gas is a collection of photons in thermal equilibrium that behave according to Bose-Einstein statistics with zero chemical potential. It represents electromagnetic radiation treated as a gas, and its thermodynamic properties can be analyzed using principles of quantum statistical mechanics and blackbody radiation.

Radiation Pressure

Radiation pressure is the pressure exerted by electromagnetic radiation when it transfers momentum to matter or itself. For a photon gas, the pressure is derived from the thermodynamic relation P = -(?U/?V) at constant entropy and number, which, because of the isotropic nature of photon momentum, results in a pressure that is one-third of the energy density (U/V).

Thermodynamic Relationships

The relation P = -(?U/?V)_{S, N} is a fundamental thermodynamic identity that links the internal energy U of a system with the pressure it exerts when the volume V changes under constant entropy and particle number. In the context of a photon gas, this derivative leads directly to the 1/3 relationship between the radiation pressure and energy density.

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the energy density of radiation (or the power per unit area emitted by a black body) is proportional to the fourth power of the temperature (T^4). This law is crucial for determining the absolute values of energy density and hence radiation pressure in systems such as kilns or stellar interiors, linking temperature precisely to energy and pressure.

Comparison of Pressure Contributions

The comparison of radiation pressure with ordinary gas pressure is significant in understanding different physical environments. In many scenarios, particularly at high temperatures like those found in stellar interiors, the radiation pressure can match or exceed gas pressure. Conversely, in lower temperature settings such as in atmospheric conditions, gas pressure typically dominates, highlighting the importance of understanding the relative magnitudes of these pressures in various physical contexts.

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