Does the Greenhouse Effect Violate the Second Law of Thermodynamics?

(c) 2020 by Barton Paul Levenson

A number of global warming deniers deny that the greenhouse effect even exists, preferring to explain Earth's surface temperature--33 K warmer than the radiating temperature you would expect from sunlight alone--by some other mechanism. By which I mean, some other crackpot pseudoscience mumbo jumbo.

Many of these would-be Einsteins declare that the greenhouse effect can't possibly work, because you have the cooler atmosphere warming the warmer ground, and "the second law of thermodynamics says heat can only flow from a warmer source to a colder source, not the other way around!"

Of course, that isn't what the second law of thermodynamics says at all. It says that, absent extra energy pumped in from outside the system, net heat cannot spontaneously flow from colder to warmer. Net heat, not any heat at all.

Think about how absurd the denier claim is for a minute. You have a source, let's say the atmosphere at perhaps 255 K, radiating energy. What happens when that energy strikes the ground? Does it refuse to be absorbed because the ground is warmer? Does the atmosphere "know" somehow not to radiate to a warmer source? What do the deniers think happens when infrared photons from the air strike the ground? What happens to the energy they carry? Does it disappear? That would violate the first law of thermodynamics, which says that energy can neither be created nor destroyed.

But far be it for us to depend on logic alone for our rebuttal. Let's do the math.

For an isothermal reaction--one at a steady temperature--the second law of thermodynamics is:

where S is entropy in joules per kelvin (J K^-1), Q is the quantity of heat exchanged (J), and T the absolute temperature (K).

So let's examine the idea of back-radiation from the atmosphere to the Earth. This is how the greenhouse effect works--the ground, warmed by sunlight, radiates infrared light upward, which is absorbed by the air. The air radiates infrared light of its own. Some of this goes back to the ground, raising the temperature of the ground from what it would be otherwise.

Let's assume the atmosphere is at a temperature of 255 K and the ground is at 288 K. We assume both are blackbodies with perfect emissivity (ε = 1), so the Stefan-Boltzmann law:

tells us that the atmosphere radiates about 240 watts per square meter and the ground radiates about 390. In the equation above, F is the "flux density" coming out of a radiating object, in units of watts per square meter (W m^-2); ε is the emissivity, or radiating efficiency, which can vary from 0 to 1 depending on the substance doing the radiating; σ is the Stefan-Boltzmann constant (5.670373 x 10^-8 W m^-2 K^-4); and T is the absolute temperature (K).

We examine one second of radiation exchange, so we can talk about joules of energy instead of watts of power (since 1 W = 1 J s^-1). Here's a table, taken from an Excel spreadsheet I did, showing the resulting values of entropy. Note that the entropy decreases when a body loses energy (for example, by radiating it out), and gains entropy when it gains energy (for example, by absorbing radiation from outside).

Ignore the extra digits, which aren't really significant. Excel shows you all the digits of the precision it uses.

The air, at 255 K, radiates 240 W m^-2, while the ground, at 288 K, radiates 390 W m^-2. By radiating out, the air decreases 0.94 J K^-1 in entropy. By receiving radiation from the ground, it gains 1.53 J K^-1. Net entropy: A gain of 0.59 J K^-1.

Meanwhile, the ground loses 1.35 J K^-1 and gains 0.83 J K^-1, for a net decrease of -0.52 J K^-1. An apparent violation of the second law of thermodynamics.

But the system as a whole has increased in entropy by 0.07 J K^-1, and it's the system as a whole that counts. So there is no violation of the second law, and the greenhouse effect works just fine.

But wait! I hear you cry. Does that mean that an object, freely radiating away energy in space, can spontaneously decrease in entropy? Have I set up an example only to sabotage myself by a misunderstanding?

Nope. There is no universe consisting only of one radiating object, because an object that was its own universe would have nowhere to radiate to. An object freely radiating to space decreases in entropy, but the space it's radiating to gains more entropy. Try doing the math with space at a temperature of 2.7 K and you'll see what I mean. The second law of thermodynamics applies universally and is never violated. Pseudoscientists love to tell you something or other--evolution, the greenhouse effect--violates the second law of thermodynamics, but when you do the math, it invariably doesn't. Always do the math. It helps.