Editing Second law of thermodynamics (section)

=== Second law statements, such as the Clausius inequality, involving radiative fluxes ===
The Clausius inequality, as well as some other statements of the second law, must be re-stated to have general applicability for all forms of heat transfer, i.e. scenarios involving radiative fluxes. For example, the integrand (đQ/T) of the Clausius expression applies to heat conduction and convection, and the case of ideal infinitesimal blackbody radiation (BR) transfer, but does not apply to most radiative transfer scenarios and in some cases has no physical meaning whatsoever. Consequently, the Clausius inequality was re-stated<ref>{{Cite journal |last=Wright |first=S.E. |date=December 2007 |title=The Clausius inequality corrected for heat transfer involving radiation |url=http://dx.doi.org/10.1016/j.ijengsci.2007.08.005 |journal=International Journal of Engineering Science |volume=45 |issue=12 |pages=1007–1016 |doi=10.1016/j.ijengsci.2007.08.005 |issn=0020-7225}}</ref> so that it is applicable to cycles with processes involving any form of heat transfer. The entropy transfer with radiative fluxes (<math display> \delta S_\text{NetRad} </math>) is taken separately from that due to heat transfer by conduction and convection (<math display> \delta Q_{CC} </math>), where the temperature is evaluated at the system boundary where the heat transfer occurs. The modified Clausius inequality, for all heat transfer scenarios, can then be expressed as,<math display="block"> \int_\text{cycle} (\frac{\delta Q_{CC}}{T_b} + \delta S_\text{NetRad}) \le 0 </math>

In a nutshell, the Clausius inequality is saying that when a cycle is completed, the change in the state property S will be zero, so the entropy that was produced during the cycle must have transferred out of the system by heat transfer. The <math display> \delta </math> (or đ) indicates a path dependent integration.

Due to the inherent emission of radiation from all matter, most entropy flux calculations involve incident, reflected and emitted radiative fluxes. The energy and entropy of unpolarized blackbody thermal radiation, is calculated using the spectral energy and entropy radiance expressions derived by Max Planck<ref>{{Cite journal |last=Planck |first=Max |date=1914 |title=Translation by Morton Mausius, The Theory of Heat Radiation |journal=Dover Publications, NY}}</ref> using equilibrium statistical mechanics,<math display="block">
K_\nu = \frac{ 2 h }{c^2} \frac{\nu^3}{\exp\left(\frac{h\nu}{kT}\right) - 1}, 
</math>
<math display="block">
L_\nu = \frac{ 2 k \nu^2 }{c^2} ((1+\frac{c^2 K_\nu}{2 h \nu^3})\ln(1+\frac{c^2 K_\nu}{2 h \nu^3})-(\frac{c^2 K_\nu}{2 h \nu^3})\ln(\frac{c^2 K_\nu}{2 h \nu^3}))
</math>
where ''c'' is the speed of light, ''k'' is the Boltzmann constant, ''h'' is the Planck constant, ''ν'' is frequency, and the quantities ''K''<sub>v</sub> and ''L''<sub>v</sub> are the energy and entropy fluxes per unit frequency, area, and solid angle. In deriving this blackbody spectral entropy radiance, with the goal of deriving the blackbody energy formula, Planck postulated that the energy of a photon was quantized (partly to simplify the mathematics), thereby starting quantum theory.
	
A non-equilibrium statistical mechanics approach has also been used to obtain the same result as Planck, indicating it has wider significance and represents a non-equilibrium entropy.<ref>{{cite journal |last1=Landsberg |first1=P T |last2=Tonge |first2=G |date=April 1979 |title=Thermodynamics of the conversion of diluted radiation |url=http://dx.doi.org/10.1088/0305-4470/12/4/015 |journal=Journal of Physics A: Mathematical and General |volume=12 |issue=4 |pages=551–562 |doi=10.1088/0305-4470/12/4/015 |bibcode=1979JPhA...12..551L |issn=0305-4470}}</ref> A plot of ''K''<sub>v</sub> versus frequency (v) for various values of temperature (''T)'' gives a family of blackbody radiation energy spectra, and likewise for the entropy spectra. For non-blackbody radiation (NBR) emission fluxes, the spectral entropy radiance ''L''<sub>v</sub> is found by substituting ''K''<sub>v</sub> spectral energy radiance data into the ''L''<sub>v</sub> expression (noting that emitted and reflected entropy fluxes are, in general, not independent). For the emission of NBR, including graybody radiation (GR), the resultant emitted entropy flux, or radiance ''L'', has a higher ratio of entropy-to-energy (''L/K''), than that of BR. That is, the entropy flux of NBR emission is farther removed from the conduction and convection ''q''/''T'' result, than that for BR emission.<ref>{{Cite journal |last=Wright |date=2001 |title=On the entropy of radiative heat transfer in engineering thermodynamics |journal=Int. J. Eng. Sci. |volume=39 |issue=15 |pages=1691–1706|doi=10.1016/S0020-7225(01)00024-6 }}</ref> This observation is consistent with Max Planck's blackbody radiation energy and entropy formulas and is consistent with the fact that blackbody radiation emission represents the maximum emission of entropy for all materials with the same temperature, as well as the maximum entropy emission for all radiation with the same energy radiance.