Exergy as Energy Quality

Kinetic energy, electrical energy, chemical and nuclear energy can all be converted fully into heat energy, while heat energy can only be partially converted back again. This is captured in the 2nd Law of Thermodynamics. We can thus say that heat energy is of lower quality compared with the other forms. More generally, the term exergy is used as a measure of quality of energy of fundamental importance for all forms of life and society as ability to do work.

We can make this more precise by recalling that the quality of heat energy comes to expression in radiative and conductive heat transfer from a body B1 of temperature $T_1$ to a neighbouring body B2 of lower  temperature $T_2<T_1$ in basic cases according to Stefan-Boltzmann's Law or Fourier's Law:

• $Q = (T_1^4-T_2^4)$            (SB)
• $Q = (T_1-T_2)$                    (F)

with $Q$ heat energy per time unit. Heat energy of higher temperature thus can be considered to have higher quality than heat energy of lower temperature, which of course also plays role in conversion of heat energy to other forms of energy. The maximal efficiency of a heat engine operating between $T_1$ and $T_2$ and transforming heat energy to mechanical work, is equal to $\frac{T_1-T_2}{T_1}$ displaying the higher quality of $T_1$.

Heat energy at high temperature is the major source for useful mechanical work supporting human civilisation, while heat energy at lower temperatures appears as a useless loss e g in the cooling of a gasoline engine.

But what is the real physics behind (SB) and (F)? This question was addressed in a previous post viewing (F) to be a special case of (SB) with the physics behind (SB) displayed in the analysis of Computational Blackbody Radiation. 

The essence of this analysis is a high-frequency cut-off $\frac{T}{h}$ allowing a body of temperature $T$ to only emit frequencies $\nu <\frac{T}{h}$, where $h$ is a constant. This allows a body B1 of temperature $T_1$ to transfer heat energy to a body B2 of lower temperature $T_2$ via frequencies $\frac{T_2}{h}<\nu <\frac{T_1}{h}$, which cannot be balanced by emission from B2.  

High frequency cut-off increasing linearly with temperature represents Wien's displacement law (W), giving improved exergy with increasing temperature.

The high-frequency cut-off can be seen as an expression of finite precision limiting the frequency being carried and emitted by an oscillating atomic lattice in coordinated motion, with frequencies above cut-off being carried internally as heat energy as uncoordinated motion. 

Higher temperature thus connects to higher quality heat energy or better exergy. The standard explanation of this basic fact is based on statistical mechanics, which is not physical mechanics. 

PS Radiative heat transfer without high-frequency cut-off would boil down to (F), while (SB) is what is observed, which gives support to (W).