Derivation
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a = \frac {4\sigma}{c}} :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P = \frac{U}{3V} = \frac{\varepsilon aT^{4}}{3} \,.}
Using the first law of thermodynamics (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle dU = dW + dQ}) we can determine the work done through an adiabatic (
) expansion by using the chain rule (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle dU = \varepsilon aT^{4} dV + 4\varepsilon aVT^{3} dT}) and setting it equal to 
Combining these Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle dW_V = dU} gives us Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle -\frac{1}{3} T dV = V dT} which we can solve to find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle T^{3} V= \text{const} \,}, or equivalently Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \eta \le \frac{T_H - T_C}{T_H} =1-\frac{T_C}{T_H}\,.}
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