Sunshine and Entropy

[Note to regular readers: There is nothing about woodpeckers or any other kind of bird in this post.]

I have a question: What is the entropy of sunlight?

This seemingly esoteric question has been of interest to me for decades. In intermittent searching over that time I have not actually found a satisfying answer to this question, surprisingly. It would seem this would be one of the fundamental quantities in ecological energetics, as it is what makes life on earth possible.

Entropy is a thermodynamic concept that measures the disorder of a system. It also is connected directly to that portion of a source of energy that cannot be used for work. Work in this sense is defined narrowly and precisely: it is a force acting on a mass over a distance, or the freely interchangeable equivalent thereof. In the day-to-day world, works consists of pretty much everything we do; examples include making heat flow from cold to hot (the opposite of its natural tendency), moving mass uphill, pushing objects forward against inertia and friction, making air molecules vibrate to form sound, etc.

Biological activity consists of using work on a microscopic scale to build complexity and reduce entropy. By the unbreakable laws of Thermodynamics, the total entropy of a system can never decrease. If you decrease entropy here, it must increase by at least as much somewhere else. There is no way around this; there is no penalty for breaking this law because it simply and absolutely cannot be broken. So how does the ecosphere accumulate this vast structure and order, representing a vast decline in entropy on a global scale? Simple. It feeds on the photons from the sun.

Sunlight strikes the earth. All sorts of fancy things happen to this energy, and then it eventually leaves the earth again. It arrives mostly in the form of visible light; it leaves mostly in the form of infrared radiation. Overall the total number of calories that enter and leave the ecosphere in a given time frame are balanced. What has changed is the nature of this energy. The incoming photons are higher in energy that the outgoing ones; the same amount of energy embodied in high energy photons has lower entropy than it does when it is converted to low energy photons, at least in part because it takes more low energy photons to carry the same energy (more photons = more things to keep track of = less order). So the outgoing radiation carries more entropy than the incoming energy did; all this extra entropy leaving the planet is what allows living things on earth to reduce entropy locally. There in fact is an "away:" deep space. That is where we throw all this excess entropy.

The question of the entropy of the incoming sunlight has come up recently in a different context. As I mentioned earlier, an energy source that has higher entropy is less "useful" than one with lower entropy: a smaller fraction of the high entropy source can be converted to work. This is relevant now in the context of solar energy and the prospects for replacing fossil fuel energy sources with it. Fossil fuels represent a fairly low-entropy energy source; this is part of why they are so useful. They consist of complex molecules of the sort that have already dispersed a lot of their disorder and become more organized, orderly forms of matter. My intuition has always told me that sunlight must certainly be a higher entropy source -- all those photons flying willy-nilly as compared to those elaborate molecules sitting around in complex forms. If true this would mean that a calorie of sunlight would not be equivalent to a calorie of petroleum, in terms of the work it could do -- work such as, for instance, generating electricity. But, I had not been able to find the numbers in terms that made sense.

Some recent online discussions may have finally led me to an answer that I think is correct. The last piece mayhave been kicked in to place by a blog post from Stuart Staniford, rebutting posts by John Michael Greer about the weaknesses of solar energy. Interestingly, it was what I believe to be an error in Staniford's analysis that pointed out what I think may be the correct answer. From here on out the thermodynamics will get steadily thicker; soon there will be actual equations. The physics-phobic have been warned.

The fundamental question is this: Sunlight is a diffuse, rather than concentrated energy source, unlike fossil fuels. The amount of sunlight it takes to equal the energy content of a gallon of gasoline is a surprisingly large number; I'll leave it to interested readers to search this one out on their own. Staniford's position is the most widespread one: The diffuse nature of sunlight is just a technical problem. Once we collect it by various means, a calorie of sunlight is as useful as any other calorie. It is basic heat energy that can be used to drive all the things we use fossil fuels for, and just as well. Greer's thesis is that this is not actually true; that sunlight is inherently less useful than fossil fuels.

There was confusion in Greer's writings between energy quantity and energy quality, and Staniford took him to task on this. In more precise terms, energy quantity is the total amount of heat energy available; energy quality is the fraction of this energy that can be converted into work in a given environment -- note that quality is dependent on environment and is not an absolute. This quality issue is intimately related to entropy. The relation between useful energy and total energy generally looks like this:

Usable energy = H - T*S

Here H = heat energy (enthalpy in the language of chemists), T = absolute temperature (measured above absolute zero), and S = entropy. The temperature dependence of this equation is important.

What Staniford presented was the standard Carnot cycle heat engine ( by the way, any terms you might find unfamiliar will have Wikipedia pages dedicated to them -- Wiki knows everything). This is a theoretical ideal engine that converts heat into work with maximum possible efficiency. As it turns out, the two most critical variables determining how much work you get from the energy are the temperature of the input to the heat engine, and the temperature of the environment into which the heat is dissipated. The maximum possible efficiency of conversion of energy in to work is given as:

1 - (T_e/T_s)

Here T_e is the temperature of the environment, and T_s is the temperature of the source, sometimes referred to/approximated as the "flame temperature.". This is a maximum possible efficiency; real engines generally function at significantly lower efficiencies. But it does provide a key piece of information: by determining how much of the energy is "useless," it tells you the entropy content of the energy source. The useless energy is equal to T*S (temperature times entropy). So now we come to the big question:

What is the "temperature" of sunlight?

Sunlight, as it does not consist of particles with mass, does not have a "temperature" in the normal sense (kinetic energy in the form of vibrating molecules or atoms). But it does consist of particles with kinetic energy; in fact particles that have nothing but kinetic energy. Staniford in his analysis used the "blackbody temperature" of sunlight as his value, which is 5500K. Blackbody radiation is the thermal radiation given off by everything. Its frequency and amount increase with the temperature of the object -- microwaves from very cold things, x-rays from very hot things, and various forms of "light" (infrared, visible, ultraviolet) from things in between. The increase in the amount of this radiation with temperature is dramatic, increasing as the fourth power of T (double the temperature, get 16 times as much thermal radiation). The sun emits radiation with a spectrum close to what is expected by an ideal emitter (the theoretical "black body") at a temperature of 5500K. Using this value as the effective "flame temperature" of sunlight, Staniford calculated that the theoretical efficiency for a solar-powered heat engine at "room temperature" (298K) is 94%, much better than fossil fuels.

There is an implicit rationale in the choice of 5500K, the blackbody temperature of sunlight, as the flame temperature here. It comes from the notion of the entropy of the "photon gas" contained within a closed blackbody cavity of a given temperature. This cavity will be filled with photons that are constantly being emitted and destroyed. At steady state it is a bit like a confined gas, with the exception that in a real gas the molecules bounce off the container walls, whereas in the photon gas they are absorbed by the wall, which then emits new photons. This is the one circumstance where I have seen the entropy of photons clearly defined. The photon gas in the cavity has no net useable energy by definition; otherwise you could be creating energy out of nothing and getting work for free. So, since:

H - T*S = 0

then it follows that

S = H/T

This would mean that these photons could do no work at a temperature equal to their source (blackbody) temperature, and for the purposes of a Carnot engine they would have a flame temperature equal to the temperature of their source, and equal to the blackbody temperature of spectrum of light they comprise.

But there's a problem here. When you take a real confined gas, open the box, and allow it to expand into a vacuum, its entropy increases dramatically; at the same time, its temperature decreases dramatically. Why would the same thing not happen with the "photon gas?" Why would a cloud of "free-range" photons pouring out in an expanding sphere unfettered into the cosmos have the same entropy and effective temperature as "caged" photons locked in a little black box? Sticking with the gas analogy, they most certainly would not. Their entropy would be substantially higher and steadily increasing with distance from the source, and their capacity to do work would be steadily dropping -- even when you account for dilution. One joule of photons hurtling away from the sun 93 million miles away should NOT have the same free energy, useful energy, exergy, whatever you wish to call it, as one joule of photons just released from the sun an instant before. Actually, let me restate that. If you are trying to put them to work at a temperature of absolute zero, in the coldest cold and blackest black of deepest space, then yes, indeed, they will still have the same useful energy (remember it all depends on the environment -- the Universe is apparently Post-Modern, it all depends on your context and point of view). But in an environment like, say, the surface of the Earth, where temperatures are significantly higher than absolute zero, the usefulness of these photons declines with every light-microsecond they travel away from the sun.

Hence, I don't see that it makes sense to use 5500K as the "flame temperature" to apply to sunlight on the earth. One idea of temperature is that heat flows from hot to cold, and given time objects in contact will equilibrate at the same temperature. This is one manifestation of entropy -- hot next to cold is a more ordered state than is uniformly lukewarm, so as entropy increases the temperature gradient decreases. You can leave an object in full raw unfocused unfiltered sunlight 93 million miles from the sun forever and its temperature will never approach 5500K. It occurred to me that the answer to the "flame temperature of sunlight" probably ought to be the answer to this question:

When you put a "blackbody" surface in the sunlight in a vacuum, what temperature will it equilibrate at?

This will be the temperature where the amount of energy entering the surface from absorbed sunlight is equal to the amount leaving the surface from its own blackbody emissions. This is not hard to calculate. A little algebra from well-known formulae gives you:

T⁴ = P/(A*s)

Where T is absolute temperature, P is power of incoming = outgoing radiation, A is the area of the surface, and s is the Stefan-Boltzman constant. Note that T is to the fourth power, so you need to take a "tesseract root" in this formula to get the final answer. Using 1400 W/m² as the power of sunlight at the top of the earth's atmosphere, you get a temperature of 397K, which is 124C. This is the temperature that an ideal surface will equilibrate at if it is facing directly at the sun above the atmosphere. This makes more sense to me as a "flame temperature" for sunlight than the temperature of the surface of the sun. If you want higher temperatures than this, you will need to do work to concentrate the energy.

So using the heat engine formula, this means that at "room temperature" of 298K, your maximum efficiency of conversion of sunlight into work would be (1 - 298/397), or only about 25%. This indicates that the entropy of incoming sunlight is about 14 times larger than the entropy of the ideal "photon gas" at 5500K used in Staniford's calculation. A 25% maximum theoretical efficiency is a pretty lousy number for a fuel source, far below that of fossil fuels. Given that in the real world it is hard to get much better than half of these theoretical efficiencies, it would change the picture of Our Solar Future rather dramatically.

So what is it, 5500K or 397K, 94% or 25%? The 397K number makes more sense to me, for the reasons I discused above. But, we all know that you can heat things up far hotter than this by focusing and concentrating the sunlight; won't that make the heat engine run more efficiently and give you a higher yield of work from the same energy input? Perhaps, but perhaps only if you believe in perpetual motion machines.

IF (and this is a big IF), my method of calculating the effective flame temperature of sunlight and therefore the entropy of sunlight at the earth is correct, then the answer to that question is "no." IF my calculation is correct, then the 25% limit on the useable energy in terrestrial sunlight is the limit imposed by the Second Law of Thermodynamics; need I reiterate that this Law cannot be broken under any circumstances by any means (so long as you are bigger than a quark, at least)? This really is God's Law, and there is no need for heaven and hell to enforce it because there is no way to violate it. Any efficiency gains you might think you have gotten above this number by concentrating, focusing, or otherwise manipulating the sunlight will be illusory; there will HAVE to be a hidden cost or additional energy input you have forgotten about.

Imagine that this Second-Law-limited efficiency is correct, and let's see what happens if we try to up the efficiency beyond this limit. Suppose you create a heat engine using raw unconcentrated 100% intensity sunlight. Your perfect absorber will operate at 397K; you can use no more than 25% of this heat energy to create useful work without violating the Inviable Second Law. Now build a system that concentrates this energy to produce a higher temperature on a smaller area, where you might think you are getting more useful work since you have made a bigger temperature difference. However, if the work you put in to concentrating the sunlight were to be less than the additional work you got out of the sunlight, you would STILL be in violation of the Second Law. The laws of thermodynamics don't care how your contraption is built; limits are limits and if you think you are getting something for nothing you are wrong. You have just overlooked another "something" that is actually providing the first "something," which cannot really come from "nothing." So if you focus the sunlight to make it appear that you can extract more than the theoretically maximum possible work from it, you are mistaken. It might be in the work that is done in bending the light to focus it, the depreciation of the focusing apparatus this causes, and the work necessary to build and maintain the apparatus. It might be in energy losses you have forgotten to include. There might be strange particle-wave phenomena happening in the focused beam that disperse energy in ways you did not anticipate. Most likely you have forgotten to include externalities that are necessary to keep your system running. Whatever, it has to be somewhere. Otherwise you would be creating energy or destroying entropy, neither of which is possible.

This 25% versus 94% number does not make solar unusable, but it does suggest that you can expect it to be about four times as expensive as what boosters are predicting now, even when all the inefficiencies are worked out. Note that even though I used the theoretical ideal heat engine in these calculations, it really does not matter how you attempt to get work from sunlight. The thermodynamic limits are the same. Also a side note: Using sunlight to heat something is not "work." These efficiency limits do not apply there. Sunlight can heat your water or your house, and cook your food, at an efficiency limited only by the cleverness of your design. In theory 100% of the solar calories can be converted to heat -- that is just the free flow of heat from hot to cold, with no thermodynamic work involved. But the second you try to turn it into something that can drive a motor, cool a refrigerator, compress a gas, etc. then the questions of entropy, useable energy, and the Second Law come in to play.

Which leaves me at the final question that I cannot answer: Did I do this right? The blackbody temperature of sunlight and the entropy of the confined photon gas as used by Staniford do not seem like the right answer at all. But is the alternative approach I used valid? I specifically want critiques and comments from people who know substantially more about physics than I do. Have I made a freshman mistake here, and if so what is it, why is it wrong, and what is the correct approach? On the other hand, if I do have the physics correct here, then there are some intriguing implications back at my initial, long-standing interest in this topic -- the significance of all this to the ecosphere as an "entropy machine."