Thermodynamics is Essential for Understanding Effect of CO2 on Temperature
Written by Dr Pierre R Latour, Chemical Engineer
Can climate experts truly understand Earth’s climate without factoring in the role of thermodynamics? Experts in the laws of thermodyamics are increasingly saying that they can’t, as all predictions of human-caused catastrophic climate change fail.
Summary. Climate scientists promoting greenhouse gas theories usually omit or dismiss consideration of thermodynamics and rely on empirical models and observed data to assess the effect of anthropogenic CO2 (carbon dioxide) from combustion of ‘fossil fuels’ on the global and surface temperatures of the Earth.
This article shows the deep foundation thermodynamics provides for the way the atmosphere behaves and quantifies why, how, and how much CO2 affects temperature. This cannot be done without thermodynamics.
Article identifies two conservation equations , eight rate laws and two physical properties affected by CO2 that constitute a nonlinear algebraic model of the steady-state effect of CO2 on T. The first six relations come directly from thermodynamics.
Turns out there are several affects, one positive and at least two negative. The climate sensitivity, CS = change in temperature for doubling of atmospheric CO2 from 400 ppmv in 2014 to 800 is not much, vanishingly small, probably between -1C < CS < 0.8C. Replacing US coal fired power plants with natural gas probably changes Earth’s temperature after 50 years between -0.000001C < T50 – T0 < +0.0000008C.
Introduction. The science of thermodynamics is central to the practice of engineering; mechanical, electrical, aeronautical and particularly chemical. We hold thermo in reverence because we know we must obey the law and we earn our livings applying it.
“If your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.” — Arthur Eddington, The Nature of the Physical World (1928) 
“The fascination of a growing science lies in the work of the pioneers at the very borderland of the unknown, but to reach this frontier one must pass over well-traveled roads; of these one of the safest and surest is the broad highway of thermodynamics.” — Gilbert Lewis and Merle Randall, Thermodynamics and the Free Energy of Chemical Substances (1923) 
“A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.” — Albert Einstein, Autobiographical Notes (c. 1940s) 
“A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.” — Max Planck, on how Boltzmann‘s statistical thermodynamics and atomic hypothesis triumphed over those as Ernst Mach and others of the energetics school (c. 1947) 
“In whatever system where the weight attached to the wheel should be the cause of motion of the wheel, without any doubt the center of the gravity of the weight will stop beneath the center of its axle. No instrument devised by human ingenuity, which turns with its wheel, can remedy this effect. Oh, speculators about perpetual motion, how many vain chimeras have you created in the like quest. Go and take you place with the seekers after gold.” — Leonardo da Vinci (1494) 
“The future belongs to those who can manipulate entropy; those who understand but energy will be only accountants.” — Frederic Keffer 
Many participants in the greenhouse gas theory debate say things like laws of thermodynamics cannot refute GHGT, or thermodynamics doesn’t apply, or “Due to the interminable misunderstandings of Laws of Thermodynamics among climate scientists, I deliberately avoided the LoT.”
Laws of Thermodynamics, There are four, with nicknames ZLoT, FLoT, SLoT and TLoT. SLoT is the most interesting.
- If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.
- The increase in internal energy of a closed system is equal to the difference of the heat supplied to the system and the work done by the system. Or system energy accumulation rate plus energy output rate equals energy input rate. More generally, energy can be neither created not destroyed, only conserved and managed.
- 2. Heat cannot spontaneously flow from a colder location to a hotter location. Differences in temperature, pressure, and chemical potential tend to even out in a physical system that is isolated from the outside world. Entropy is a measure of how much this process has progressed. The entropy of an isolated system that is not in equilibrium tends to increase over time, approaching a maximum value at equilibrium. There are many versions of the second law, but they all have the same effect, which is to explain the phenomenon of irreversibility in nature. Derived from statistical mechanics.
- 3. As a system approaches absolute zero the entropy of the system approaches a minimum value. It is impossible to reach the absolute zero of temperature by any finite number of processes.
Do not dismiss SLoT so quickly. Those who dismiss SLoT in GHGT make a fatal error. Many famous luke-warmer skeptics merely claim it doesn‘t apply, undoubtedly because they don’t know how to apply it. So I will try to clear up the misunderstandings, knowing full well most climate scientists won’t understand or won’t like it. My audience is UN IPCC.
GHGT Violates SLoT. I proved GHGT hypothesis of enormous back-radiation heat transfer from cold atmospheric CO2 down to and absorbed by warmer surface, 333 w/m2 in famous K-T Global Energy Flows diagram, violates SLoT and hence FLoT. http://www.principia-scientific.org/no-virginia-cooler-objects-cannot-make-warmer-objects-even-warmer-still.html
I showed the infinite sequence of GHGT violating SLoT does not go to infinity, it converges to Es. The change in radiation energy rate to surroundings, Es, is the final rate minus initial rate minus amount initially diverted from surroundings to the injected cold plate. Es = [(1 + k)*K*F0 + (1 + K)*k*f0]/(1 – kK) – K*F0, where K is the fraction of radiation from the first warmer bar absorbed by the second colder bar, 0 < K <=1, and k is the fraction of re-radiation from the second colder bar absorbed by the first hotter bar, 0 <= k <=1. Subscript 0 denotes starting values > 0.
For any K > 0 and k > 0, Es > 0. This would constitute creation of energy, a violation of the first law of thermodynamics. Since K may be > 0, Es = 0 if and only if k = 0. Since this is the only possible solution, k must be identically zero, so no cold back-radiation is absorbed by warmer surface and its T remains 150 C. Quod Erat Demonstrandum, QED.
Rate Laws. Second Law of Thermodynamics is the Law of nature behind all mass and energy transfer rate laws, which rarely appears in astrophysics, meteorology and climate science but dominates chemical engineering.
SLoT determines which way stuff goes; downhill. Matter and energy flow, move, transfer forward at a rate proportional to a potential driving force (gradient) in the opposite direction to push it. Matter and energy cannot flow backwards; uphill. Why? It is just the way nature works, the SLoT, derivable from statistical mechanics.
Bodies move according to Newton 2nd Law of mechanics motion, F = mA, first expression of SLoT. F & A are vectors, point in opposite directions.
Bodies move in gravitational force fields proportional force intensity by universal law of gravitation intensity: F = g Mm/r2
Fluids flow in pressure fields proportional to pressure intensity drop: F = k(P1 – P0)0.5 >= 0
Electrons flow in electric fields proportional to voltage intensity drop: I = (E1 – E0)/R >= 0
Chemicals diffuse/flow in liquids and gases mass fields proportional to chemical potential (almost composition or concentration) intensity drop: D/A = k(C1 – C0) >= 0
Heat flows through solids, liquids and gases by conduction & convection at rate proportional to temperature (kinetic energy intensity) drop: Q/A = k(T1 – T0) >= 0
The rigorous expression for Fourier’s Law of heat transfer by conduction in harmony with SLoT is q = – k A dt/dx, heat conduction rate in x-direction. k is thermal conductivity of substance.
Even chemical reaction rates obey SLoT. And stars and human beings that die.
Every rate law has an associated resistance constant that depends on the situation, system physical properties.
Radiant energy flows through EMR field at rate proportional to energy intensity difference at every point1: Q = A(I1 – I0) >= 0
Where did all those little provisos >= 0 come from? SLoT!
Radiant Energy transfer. Let’s look closely at the last one: Q/A = I1 – I0 >= 0
Fortunately S-B Law gives us relationship for energy intensity/irradiance/power/strength from a radiator, I, depending on only two things, its temperature, T, and emissivity, ε.
I = σ εT4, w/m2, σ = 5.67, a universal constant of nature, like g for gravity. T = deg K/100. ε = 1 for theoretical, perfect black radiators, ε < 1 for real radiators.
Note if ε goes up for some chemical reason, either I goes up or T goes down. That is S-B Law of nature.
Energy transfer rate between real radiators gets a bit tricky because they only absorb part, α, of incident I, transmit & reflect the rest1.
Absorbed at “cold” surface 0 is I1 = σ α0 ε1 T14
Absorbed at “hot” surface 1 is I0 = σ α1 ε0 T04
I put “cold” in quotation marks to designate the lower intensity radiator, not lower temperature.
Of course α and ε are fractions. The subscripts are assigned such that Q >= 0. This is where SLoT creeps in.
I have not accounted for distance between radiators; intensity decreases with distance according to inverse square law of light. I do so for simplification, to show what I want to show.
According to SLoT, rate of radiant energy transfer from 1 to 0 is
Q/A = I1 – I0 = σ [α0 ε1 T14 – α1 ε0 T04] >= 0 (1)
This is the radiation rate law from Earth’s surface to atmosphere1. Each radiator emits and absorbs according to its emission & absorption spectrum; intensity vs wave length. Remember α and ε are both wavelength dependent, so I am using effective values here.
We consider atmosphere as lumped to a thin shell surrounding Earth at atmosphere’s mass centroid altitude 5 km, with properties the same as the effective bulk properties of our real, complex atmosphere. To do the job right, integration of two nonlinear partial differential equations in three dimensions is required. I am unable to go there with my own resources.
GHGT. GHGT promoters look at eqn (1) and say increasing CO2 increases atmosphere ε0, so to keep Q fixed at fixed T0, T1 at surface must increase, a greenhouse effect!
Not so fast. This logic is correct under the assumptions, which are not correct. So the conclusion is incorrect. We must look at the whole system to remove those assumptions and account for changes in Q and T0 with CO2. Rearrange the equation for the variable of interest:
T14 = [α1 ε0 T04 + σ Q/A]/ α0 ε1 (2)
These are the only variables and parameters that affect Earth’s surface T1.
Turns out CO2 has absorption bands in the solar spectrum, absorbing some and emitting it back to space. GHGT neglects this surface cooling effect. So Q drops, α0 increases and T0 decreases with increasing CO2; all of which decrease T1. In addition, increasing CO2 and T1 increases rate of photosynthesis chemical reaction, consuming CO2 and light. This reduces Q and ε0, which reduces T1. Neglecting these two cooling effects is an unforgivable intellectual error. Cherry picking increasing ε0 only to conclude CO2 increases T1 is not playing fair. And it is incorrect.
First, some say at the long wavelengths involved from cold CO2, α1 = 0. If so ε0 has no effect on T1 and Q no matter what; case closed. CS = 0. α1 has as much effect on T1 as ε0 does, and the smaller α1, the smaller the effect of changes in ε0 have on T1. I believe astrophysicists, meteorologists and GHGT promoters are unaware of that SLoT physics. Further if surface ε1 should change, say from greening, ice coverage or cities, those would change ε1 and T1. Neglecting these is clearly negligence.
Space. As an important aside, energy transfer rate from globe to space is
Q/A = Ig – I0s = σ [α0s εg Tg4 – αg ε0s T0s4] >= 0, but Ios and corresponding T0s = 3.7K are vanishingly small and may be assumed 0. And α0s = 1. Which leaves
Q/A = Ig = σ εg Tg4 > 0 (3)
Since fossil fuel combustion exchanges non-radiating O2 for radiating CO2 molecules, ε of atmosphere and hence εg increase. Since output Q depends on inputs Qi which are independent of CO2, if CO2 decreases εg, Earth’s global radiating Tg must go down.
This disproves GHGT predicting Tg goes up with CO2. In other words CO2 really causes global cooling. (I just proved it on the back of an envelope or a bar napkin. From well-known chemical engineering and no research.) All you have to do to quantify effect of CO2 on Tg is quantify effect of CO2 on εg. and use eqn (3).
Kirchhoff’s Law. Another aside is Kirchhoff’s Law is often invoked for both radiators to simplify things: α1 = ε1 and α0 = ε0. This makes α1 ε0 / α0 ε1 = 1.0 and eqns (1 & 2) simplify to
Q/A = I1 – I0 = σ α0 ε1 [T14 – T04] >= 0 (4)
T14 = T04 + σ Q/ A α0 ε1 (5)
In other words when Kirchhoff’s Law is valid, the driving force for radiant energy transfer simplifies to T14 – T04.
So when is Kirchhoff’s Law invalid? Almost everywhere, most of the time. Whenever there are other energy transfer mechanisms affecting the radiator, like electrical, mechanical, thermal or chemical, the assumptions of this law do not apply and it conclusions are inaccurate.
Kirchhoff’s Law does not apply for Earth’s atmosphere or surface because many energy transfer processes besides radiation are involved. Assuming this law just makes GHGT promoters’ work easier and incorrect.
Since it only provides a minor algebraic simplification, there is no reason to invoke it, so I shall not assume it and remain rigorous with eqn (2).
System. Equation (2) has two unknown variables with CO2: Q, T0, two physical properties that vary with CO2, α0 and ε0, and two properties which do not in short run vary with CO2, α1 and ε1.
So we need four more relationships to solve global system for effect of CO2 on T1.
- Effect of CO2 on property ε0
- Effect of CO2 on property α0
- Energy balance on atmosphere for T0, FLoT: input = output
- Energy balance on surface for Q, FLoT: input = output
These will include
- Rate law between solar input and atmosphere
- Rate law between solar input and surface
- Rate law between surface and space
- Rate law between atmosphere and space
- Rate laws between surface and atmosphere: radiation, conduction & convection
Two coupled energy balances (FLoT), eight rate laws (SLoT), and two physical property relations for CO2.
Two equations for two unknowns: T0 and T1. So the system is solvable!
Solution. Specify any input, like CO2, solar, crust, photosynthesis and these two algebraic equations can be solved simultaneously with numerical Newton-Raphson iteration to any degree of closure accuracy.
These two algebraic equations provide a pretty rigorous solution for lumped or effective variables commonly used. The accuracy of change in T1 for change in CO2 only depends on the accuracy of the rate constants and ε and α of the system.
Results. I am working to specify all these equations with inputs. I am sure many others can do it better than me. This eliminates any need for climate data and statistical computer models.
I wrote this to illustrate the role played by SLoT throughout is central. Neglect it and you are negligent, doomed to error. In fact the whole global warming system is described completely with thermodynamics: a little bit of easy FLoT and a whole lot of more interesting SLoT used by chemical engineers. Neglecting thermodynamics is preposterous chemical engineering nonsense.
My current assessment is the climate sensitivity result, CS = change in temperature for doubling of atmospheric CO2 from 400 ppmv in 2014 to 800 is not much, vanishingly small, probably between -1C < CS < 0.8C. Replacing US coal fired power plants with natural gas probably changes Earth’s temperature after 50 years between -0.000001C < T50 – T0 < +0.0000008C. Undoubtedly not worth whatever it cost.
The world it running riot with ignorant people practicing chemical engineering without a registered professional engineer license or degree from reputable college, reporting to UN IPCC and EPA. It is not running out of coal, oil and gas. It is running out of money and unbiased scientists.
- Martin Herztberg, “Earth’s Radiative Equilibrium in the Solar Irradiance”, Energy & Environment, v20, n1, 2009, p 83-93.
Dr Pierre R Latour is former Vice Chairman of Principia Scientific International (PSI). Among his many career achievements was his role at NASA as engineer on the Apollo space mission; Chemical Process Control Systems Engineer.