Photo-Carnot engine

The internal energy of the photo-Carnot engine is proportional to the volume (unlike the ideal-gas equivalent) as well as the 4th power of the temperature (see Stefan–Boltzmann law) using ${\displaystyle a={\frac {4\sigma }{c}}}$  :

${\displaystyle U=V\varepsilon aT^{4}\,.}$

The radiation pressure is only proportional to this 4th power of temperature but no other variables, meaning that for this photo-Carnot engine an isotherm is equivalent to an isobar:

${\displaystyle P={\frac {U}{3V}}={\frac {\varepsilon aT^{4}}{3}}\,.}$

Using the first law of thermodynamics (${\displaystyle dU=dW+dQ}$ ) we can determine the work done through an adiabatic (${\displaystyle dQ=0}$ ) expansion by using the chain rule (${\displaystyle dU=\varepsilon aT^{4}dV+4\varepsilon aVT^{3}dT}$ ) and setting it equal to ${\displaystyle dW_{V}=-PdV=-{\frac {1}{3}}\varepsilon aT^{4}dV\,.}$

Combining these ${\displaystyle dW_{V}=dU}$ gives us ${\displaystyle -{\frac {1}{3}}TdV=VdT}$ which we can solve to find ${\displaystyle T^{3}V={\text{const}}\,}$ , or equivalently ${\displaystyle PV^{4/3}={\text{const}}\,.}$

Since the photo-Carnot engine needs a quantum coherence in the gas which is lost during the process, the rebuild of coherency takes more energy than is produced with the machine.

The efficiency of this reversible engine including the coherency must at most be the Carnot efficiency, regardless of the mechanism and so ${\displaystyle \eta \leq {\frac {T_{H}-T_{C}}{T_{H}}}=1-{\frac {T_{C}}{T_{H}}}\,.}$

• Carnot heat engine
• Radiometer

• Marlan O. Scully; M. Suhail Zubairy; G. S. Agarwal; Herbert Walther (2003-02-07). "Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence". Science. 299 (5608): 862–864. Bibcode:2003Sci...299..862S. doi:10.1126/science.1078955. PMID 12511655. S2CID 120884236.
• Zubairy, M. Suhail (2002). "The Photo-Carnot Cycle: The Preparation Energy for Atomic Coherence". Quantum Limits to the Second Law: First International Conference on Quantum Limits to the Second Law. AIP Conference Proceedings. Vol. 643. pp. 92–97. Bibcode:2002AIPC..643...92Z. doi:10.1063/1.1523787.