Possible Explanation for Why thermal energy tends to be a linear relation of temperature:

An Explanation that has little to do with entropy! 

Thermal Energy

Simply defined thermal energy is heat. A better definition is that thermal energy consists of a spectrum of thermal photons and/or phonons, i.e. those whose wavelengths are sufficiently long that they interact with condensed matter, becoming either intramolecular or intermolecular vibrations. As such, thermal energy for the most part consists of a spectrum of infrared wavelengths. However, depending upon the substance and temperature, the spectrum of thermal energy may also include microwave and/or visible & UV light. 

Rayleigh-Jeans Approximation

For blackbody radiation

When hv<<kT, then the energy density (@) as defined by a constant times temperature to the fourth power, can be obtained by using the Rayleigh-Jeans approximation:

           @=aT      eqn 1                                                                                                         

where @ is the energy density and “a” is the Rayleigh-Jeans constant. The units for “a”  are “energy per volume per degree”, which in SI units would be written as: J/Km³. Note: The Rayleigh-Jeans approximation for 6000 K is sketched in Fig 1.4.3.

Remember that our prime concern in thermodynamics is the thermal radiation i.e. the part of the radiation spectrum, which interacts with matter as thermal energy, e.g. spectrum of relatively long wavelengths. Specifically frequencies around the infrared.

In Table 1.4.1, hv is compared to the value of kT, for two infrared frequencies and two microwave frequencies, all at room temperature (20^oC = 293 K), followed by at 1000^oC (1,273 K) and finally 5,700 K i.e. the approximate temperature of our Sun.

The majority of what we consider to be thermal energy is any spectrum consisting of mid to long infrared frequencies. When considering just thermal energy; the Rayleigh-Jeans approximation cannot be valid for blackbody radiation from a body unless that body is extremely hot i.e. the Rayleigh-Jeans approximation starts becoming valid for all infrared radiation from blackbodies whose temperature is above several thousand degrees Kelvin.

At our Sun’s temperature (T=5,700 K): hv approximately equals kT, when wavelength = 2500 nm. Thus for all wavelengths greater than 2500nm, hv<<kT, andthe Rayleigh-Jeans approximation is valid!! Accepting that wavelength greater than 2500nm constitutes the majority of wavelengths that behave as thermal energy, then we can conclude that the Sun’s thermal radiation density incident upon Earth can be approximated as being directly proportional to the temperatures that we measure. As can be seen in Fig 1.4.3, the apex for temperatures near that of our Sun is in the short wavelength infrared: By Wein’s Laws it is at 3.5x10¹⁴ Hz.

The above helps explain why most thermodynamic relations are directly proportional to temperature? Especially if we realize that the Earth’s atmosphere (can include Earth’s surface and waters) not only acts as a thermal blanket holding in the Sun’s heat, but it also acts as a heat bath surrounding many of our experiments i.e. our biggest experimental heat bath has its thermal energy density directly proportional to temperature.

The fact that the thermal energy density within condensed matter must be in equilibrium with its surroundings. And that generally relates to the thermal energy density within our atmosphere that is related to the thermal radiation density from our Sun, which is directly proportional to the temperatures that we measure. This all cannot be denied.

Asserting that most of the thermal energy here on Earth consists of wavelengths from the microwave through infrared portion of the Sun’s spectrum, surely fits with our understanding of thermal energy. Importantly, having the thermal energy density directly proportional to temperature bodes well with most of our empirically determined thermodynamic relations. The implication being that the density of thermal energy, which is readily absorbed by condensed matter, and/or exchanged through molecular vibrations/collisions would generally be directly proportional to the temperature that we measure. Always remember that systems of matter here on Earth tend to be in thermal equilibrium with their surroundings, and it is in these surroundings wherein the thermal energy density is governed by our Sun’s rays.

Interestingly, Planck’s work showed that the Rayleigh-Jeans approximation is valid, when: hv<<kT, which was based upon Rayleigh-Jeans methodology but Planck further realized that the energy of photons must be quantized. By quantizing photons and realizing that the number of photons decreases with increasing frequency, Planck advert the ultra-violet catastrophe. Note: More on blackbody, radiancy, Rayleigh-Jeans approximation, UV castastrophe and thermal energy is provided in Appendix B.1

Limitations of Temperature in Thermal Radiation

We have discussed that the thermal energy density here on Earth is proportional to temperature with the primary reason being the thermal energy density from our Sun’s blackbody radiation makes it so, Rayleigh-Jeans Approx.  We can see limitations of this statement in Fig 1.4.6 where sketches for the power density of blackbody radiation per wavelength are shown. We can see that our Sun’s (T=6000) thermal energy density has linear functionality but lower temperatures do not

The primary reason that systems here on Earth follow this functionality is our atmosphere, and planet Earth (land and oceans) all behaves as a massive heat bath/sink. Interestingly, at room temperature (300 K) the thermal energy (infrared) dominates the blackbody radiation. We can conclude that matter here absorbs the incoming thermal energy from our Sun, whose energy density is proportional to temperature. It is then re-radiated by that matter as blackbody radiation with a peak around 9 micrometers i.e. an infrared spectrum; see T=300 K in Fig. 1.4.6 (note: Earth’s blackbody radiation peaks at 9.7 micrometers). Obviously, our Sun’s temperature combined with the temperature in which we reside has profound implication to our perception of thermodynamics. Remember the radiated thermal energy is generally infinitesinmally small, when compared to the energy associated with matter.

Fig 1.4.7 shows that the power density per wavelength for temperature of 1800 K to 250 K.  For temperatures above 300 degrees the infrared part of the spectrum approximates a simple linear decreasing function in relation to the power density per increasing unit of wavelength. For temperatures below 250 K this apparently is not the case.

To further complicate matters the power density per unit wavelength increases logarithmically with temperature. This helps explain why when dealing with a blast furnace i.e. 1200 K, the radiated thermal energy is no longer infintesimally small in comparision to the thermal energy associated with matter. It accepted as being proportional to temperature to the fourth power.

Our conceptualizations of the thermal energy should behave somewhat differently at high temperatures, as well as when temperatures approach absolute zero wherein microwaves dominate over infra-red. All that we can say at this point is that we should expect the functionality between temperature and the thermal energy density may have variations at both high and low temperatures.

It must be emphasized that we associate a temperature with radiated thermal energy, while traditional thermodynamics does not.

NOTE

It should be stated that solar flux will need to be added to this discussion. Things like UV are prevented from our atmosphere by ozone, and visble light often degrades into thermal energy. So there is much more to consider than has beemn discussed herein. hopefully some day I will get to it.  

Closing Remarks

It was discussed that the energy associated with kinematics of matter tends to be significantly greater than that of surrounding/radiated thermal radiation. Accordingly, traditional theory would be seemingly validated by experimentation. However, thermal radiation in freespace exists and has relevance to thermodynamics. Herein we determined that the thermal (predominately infrared) energy density of our Sun’s blackbody radiation could be approximated by some linear function of temperature, i.e. the Rayleigh-Jeans approximation is valid hence thermodynamic relations tend to be directly proportional to temperature. This does not provide us with an understanding of entropy, rather it is simply the beginning of our new improved perspective.

Certainly, this linear proportionality between thermal energy density and temperature might not exist for all temperature regimes. For example at the high temperatures of blast furnaces and at the low temperatures approaching absolute zero, our expectation is that the linear functionality between thermal energy density and temperature will falter. In other words our perception of thermodynamics may be unique to our Earth’s position in relation to our Sun & solar systems that are in thermal contact/equilibrium with our planet/atmosphere.

It is also of interest that the UV catastrophe starts just after the location where our Sun’s blackbody radiation curve reaches its apex (see Fig 1.4.3). Making UV, too high of a frequency to contribute to our experienced thermal energy density, and still have it simply proportional to T. Seemingly, this reinforces the assertions rendered in this section.

   Copyright Kent W. Mayhew