Electromagnetic radiation in an evacuated vessel of volume \(V\) at equilibrium with the walls at temperature \(T\) (blackbody radiation) behaves like a gas of photons having internal energy \(U=a V T^{4}\) and pressure \(P=1 / 3 a T^{4}\), where \(a\) is Stefan's constant. (a) Plot the closed curve in the \(P-V\) plane for a Carnot cycle using blackbody radiation. (b) Derive explicitly the efficiency of a Carnot engine which uses blackbody radiation as its working substance.

Blackbody radiation refers to electromagnetic radiation emitted by a body in thermal equilibrium. Such a blackbody absorbs all incidents' radiation regardless of frequency or angle. A perfect example is a theoretical construct, but it closely describes the radiation emitted by stars and planets.

In this exercise, blackbody radiation behaves similarly to a gas composed of photons with specific properties. The internal energy of this radiation is given as \( U = a V T^4 \), where \( a \) is Stefan's constant, \( V \) is volume, and \( T \) is the absolute temperature. The pressure exerted by this gas is \( P = \frac{1}{3} a T^4 \). These relationships are key to understanding how blackbody radiation operates within a Carnot cycle.

Blackbody radiation serves as an ideal working substance for thermodynamic cycles because of its predictable behavior. As we move through the cycle's stages (isothermal and adiabatic), these equations help dictate energy exchanges and pressure changes during different phases.

Thermodynamic efficiency is a critical metric in evaluating how well an engine converts heat into work. This efficiency is represented by the symbol \( \eta \) and is inherently less than 100% due to the second law of thermodynamics.

For a Carnot engine, which operates between two heat reservoirs at different temperatures (\( T_H \) and \( T_L \)), the efficiency is given by:
\below is the formula: \texpression: \[ \eta = 1 - \frac{T_L}{T_H} \]

This formula implies that efficiency depends only on the temperatures of the hot and the cold reservoirs. It doesn't depend on the working substance, which in our case, is blackbody radiation. In our specific example, we use the relationship provided, like the heat absorbed \( Q_H \) during isothermal expansion and the heat rejected \( Q_L \) during isothermal compression, to show that the efficiency derived aligns with the general Carnot efficiency formula.

Consequently, this highlights that Carnot efficiency is universal, making it the gold standard for measuring the performance of real-world engines, even those utilizing unconventional working substances like blackbody radiation.

Isothermal and adiabatic processes are two fundamental types of thermodynamic processes that play crucial roles in the Carnot cycle.

• Isothermal Process: During an isothermal process, the temperature of the system remains constant. For our Carnot cycle, this happens twice - during the expansion at the high temperature \( T_H \) and during the compression at the low temperature \( T_L \). Since the temperature stays constant, the product \( PV \) remains constant based on the ideal gas law. In our case, we use the relation \( P = \frac{1}{3} a T^4 \) to determine how pressure and volume change during these stages.
• Adiabatic Process: A process is adiabatic if no heat is exchanged with the surroundings. This means that any change in the system's internal energy is due solely to work done by or on the system. There are two adiabatic processes in a Carnot cycle: an expansion phase following the high-temperature isothermal process and a compression phase preceding the low-temperature isothermal process. During these stages, both temperature and pressure change without heat exchange, driven purely by the thermodynamic properties of blackbody radiation.

Understanding these processes is crucial for plotting the Carnot cycle on a \( P-V \) diagram and calculating engine efficiency.

The combination of isothermal and adiabatic changes ensures maximum theoretical efficiency by minimizing irreversible entropy changes during the cycle.