Category Archives: Global Warming

Scientific Folly of Averages

by James R. Barrante, Ph.D.


There is a misunderstanding among many scientists that using average values to do science is acceptable under any condition.  This is actually not the case.  To understand this, we must look at Error Theory.  Scientists are generally interested in two types of errors:  random error and systematic error in their measurements.  Random error is related to inherent errors in measuring devices, particularly when these measuring devices are being used near their limitations.  This type of error is associated with the precision of the measurement, how close individual measurements of the same parameter are to each other.  Systematic error is related to experimental design.  This type of error is associated with the accuracy of the measurement; that is, how close individual measurements of the same parameter are to the “true” value.   The precision of a measuring device can be increased by making the measurement more than once and taking an average only if the following is true:  the measurement must be made using the same measuring device on exactly the same sample over and again thousands of times.  Thus, the precision of the average or mean value will be greater than the precision of a single measured value.

No statistical analysis will affect the systematic error.  The average will be no more accurate than a single measurement.  What, then, is the advantage of measuring a sample several times and taking an average.  The reason is to insure that no systematic error was made in the measurement, a possibility if the measurement was made a single time.  For example, if you were determining the mass of an object, to insure that you did not misread the balance, you might take three consecutive readings, each one after zeroing the balance.  If they are all within the random error of the balance (generally supplied by the manufacturer), you can be sure that no error was made reading the scale of the balance.  Does this mean that the average of the three is accurate.  Not in the least.  If the balance had not been calibrated, the average could be precise (e.g., 24.55; 24.52; 24.54), but not accurate.  For example, the true weight could be 16.24.  Calibrating the balance is necessary to minimize systematic error.  It’s part of experimental design.

When do averages have no scientific meaning?  The study of climate change is a perfect example.  The idea of an average global temperature as it relates to an average global atmospheric CO2 level has about as much scientific meaning as the average diameter of a football has to its shape.  Knowing that the average diameter of a standard football is 9.00 inches tells you nothing about its shape.  In fact, the average presented as a single number would suggest that the football is spherical.  The average value of some measured parameter of a system (like pressure or temperature) is scientifically significant only if 1) the system is very small; 2) its boundaries are well-defined; and 3) measurements are made with the same measuring device on various points over the system where these measuring points have the same environment.  The globe satisfies none of these requirements.

You can certainly determine the temperature of the globe at various points around the globe and obtain an average that is valid as an average temperature.  But you cannot do anything scientifically meaningful with that number.  For example, suppose we wish to determine how atmospheric CO2 level affects that average.  For this to be scientifically meaningful, it would have to be true that only CO2 level could affect the temperature at every single measuring point over the entire system.  When the Soviet Union fell, a number of temperature measuring stations in Siberia were closed.  The average global temperature suddenly increased.  Climate change theory would have you believe that a sudden increase in CO2 level caused the sudden increase in average global temperature.  Another example, the average atmospheric CO2 level is found to suddenly increase.  It is assumed that this sudden increase in CO2 level was caused by an increase in burning fossil fuels.  In actuality, the opening of a coal-burning power plant occurred at one measuring point, while a volcanic eruption occurred at another measuring point.  Different causes, same effect.

Where cause and effect is meaningful is when the system is small with well-defined boundaries.  A small cylinder holding a gas is heated from 300 K to 400 K.  A pressure gauge on the cylinder measuring average pressure increases from 1.0 bar to 1.3 bar.  The average pressure change is consistent with the average temperature change, because the cylinder is a system that satisfies the three conditions mentioned above.  This is the nature of the natural sciences and it cannot be changed within the realm of the scientific method.  Unfortunately, many climate scientists do not seem to understand this.


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Author of best selling supplement, Applied Mathematics for Physical Chemistry, 3rd Edition, James R. Barrante has a new textbook, Physical Chemistry for the Biological Sciences, an iBook for iPads, iPhones, or Macs with OS X Maverick.  To download a sample chapter, go to the iBookstore.  At $9.99 it makes a great supplement for any physical chemistry course.


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August 19, 2014 · 1:15 am

Carbon Dioxide and Henry’s Law

by James R. Barrante, Ph.D.


There have been a number of unsuccessful attempts to relate the level of carbon dioxide in the atmosphere to the concentration of dissolved CO2 in the oceans using Henry’s Law.  The Henry’s Law equation is a simple linear relationship between the partial pressure of CO2 in the atmosphere to the activity of dissolved CO2 gas


Dissolved carbon dioxide reacts with water to form a weak acid, carbonic acid.  Consequently, the Henry’s Law constant k is not affected significantly by the high ionic strength of seawater.  The activity coefficient of undissociated carbonic acid is close to unity, so the activity of dissolved CO2 can be replaced by its concentration in moles per liter.  This, of course, is not true for the dissociation constants of carbonic acid.  (See Ocean Acidification I and II).   Like all equilibrium constants, the Henry’s Law constant is temperature dependent.  Therefore, an average partial pressure of atmospheric CO2 must be carefully calculated.  That is, it is not acceptable to determine the partial pressure of atmospheric CO2 at various points around the globe and average them together.

One must first recognize that the temperatures of the oceans are not randomly distributed around the globe, but are banded in zones following closely to the latitude lines.  Moreover, the latitude zonal surface seawater areas differ at least two ways.  First, from the equator to each pole the total surface area around the globe is different, and second, each area contains a different amount of land mass.  Consequently, before any calculations can be done, the total relative surface area of each zone must be determined.  This author found that a zonal width of 5˚ was adequate.  Since we are concerned with relative surface area, it is not necessary to use actual values.  The surface area of a zone is easily calculated using the equation

S  =  2 π r h

where is the radius of the globe and h is the height between successive latitude lines.  Table 1. shows the surface area of each zone on a globe of radius 50.0 units.  Notice that the difference in surface area between the first three zones is for all practical purposed zero.  We find that is it not necessary to go much beyond 80˚ north and south.

Table 1.

Latitude Range      Difference in Latitude          Surface Area
        (degrees)                                       Lines                          (square units)                                                    
0˚ to 5˚                                           4.2                                               1319
5˚ to 10˚                                          4.2                                               1319
10˚ to 15˚                                         4.2                                                1319
15˚ to 20˚                                         4.1                                                1288
20˚ to 25˚                                         4.1                                                1288
25˚ to 30˚                                         3.8                                                1194
30˚ to 35˚                                         3.8                                                1194
  35˚ to 40˚                                         3.5                                                1100
  40˚ to 45˚                                         3.4                                               1068 
  45˚ to 50˚                                         3.0                                                 942  
 50˚ to 55˚                                          2.5                                                  785
 55˚ to 60˚                                         2.4                                                  754
  60˚ to 65˚                                         2.1                                                  660
  65˚ to 70˚                                          1.7                                                  534
   70˚ to 75˚                                          1.4                                                  440
     75˚ to 80˚                                          0.9                                                  283   

Once the surface area of each zone is found, it is necessary to determine the fraction of that area that is water and the average temperature of that water.  Figure 1. shows a grid of the globe designed to do this.   The average temperature of

SS Area(2)

Figure 1.  Planar grid of globe.

each zone was found by layering the grid shown in Figure 1 on top of a SST map such as those produced by the National Weather Service Environmental Modeling Center for 2014.  Table 2. shows the fraction of each zone that is water and the average temperature of each zone.

Table 2.

Knowing the average sea surface temperature of each zone and using the equation describing the variation of the Henry’s Law constant with respect to temperature, we can determine the partial pressure of CO2 associated with each zone.  To do this, we must assume a level of dissolved CO2.  From the post Ocean Acidification II, a reasonable average value is 1.36 x 10-5M. By multiplying the fraction of ocean by the surface area of a particular zone, we can determine the zone’s contribution of partial pressure CO2 to the average total pressure of CO2.    The results of these calculations are shown in Table 3.

Table 3.

Table 3

We see that the total weighted average pressure of CO2 is 347.2 ppm, not too bad, considering the uncertainty in the concentration of dissolved CO2.  Keep in mind that this is an equilibrium value that may take a couple hundred years to be established. If the instantaneous level of CO2 is measured to be around 400 ppm, this would indicate that the contribution of CO2 from all other sources, including burning fossil fuels, is about 50 ppm.

It is important to note the silliness in measuring CO2 levels around the globe and assuming that the average has some statistical meaning.  Look at the data in Table 3.  Simply based on ocean temperature around the equator, CO2 levels are well over 400 ppm, while levels in Antarctica are similar to those found during ice ages.  This, obviously, has to be the case to get an average somewhere between these two extremes.  This really puts into question the validity of ice-core data as a proxy for “global” CO2 levels.  It is difficult to understand how samples of ice taken from Antarctica could represent global CO2 levels in violation of Henry’s Law.

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The Whole Truth

by James R. Barrante, Ph.D.

We have all seen the famous ice-core graphs from data taken from Vostok Station, Antarctica.  The CO2 graph going back some 400,000 years is particularly interesting.  The graph clearly shows atmospheric levels of CO2 rising and falling on a periodic basis.  The maxima, occurring approximately every 100,000 years at a level of about 280 ppm, represents the four interglacial periods, like the period we have been in for the last 10,000 years.  By contrast, there are four minima at approximately 180 ppm corresponding to the periodic ice ages experienced by the globe over the past 400,000 years.  We must keep in mind that on a scale of 400,000 years, a 200-year time period is about the width of the ink line.  Usually attached to the end of of the graph, representing a time period of about the last 150 years, is a vertical line shooting up to over 350 ppm, and then extrapolated into the future.  The caption found on many of these graphs is, “for the past 400,000 years CO2 levels in the atmosphere have never exceeded 280 ppm until now.”  It is perhaps one of the most dishonest interpretations of data I have seen in my long scientific career.

Now, before going on, let me say that there is nothing dishonest about the ice-core data itself.  It represents a beautiful piece of research done under miserably cold conditions by a group of scientists with the best intentions in mind.  What is dishonest is the idea that the graphs represent global temperature and global CO2 levels.  For example, ice core data came from samples of ice taken from the deep in the snowpack of Antarctica.  The ice came from snow that fell through the atmosphere of Antarctica (not New Jersey) and CO2 levels were determined from air trapped in those bubbles.  It is unlikely that those bubbles of air represent anything but the air over Antarctica.

The last part of the graph showing the last 150 years where “global CO2” shoots up to over 350 ppm was not constructed from Vostok ice-core data.  It was constructed from data obtained from measurements taken on Mauna Loa, an active volcano.  It would seem logical that the water temperature around Hawaii is a little warmer than the waters surrounding Antarctica, and since we know that atmospheric CO2 levels are controlled by water temperature, it would make sense that (volcanic action aside) CO2 levels around the Hawaiian Islands should be higher than around the South Pole.

So, when we say that CO2 levels never have exceeded 280 ppm for the last 400,000 years, that has only been verified over Antarctica.  We have no research suggesting that this is true for any other part of the globe.  You see, in science, an average of a specific property such as temperature, taken at different points with different measuring devices is just a number with no specific meaning.  The number describes something that does not exist.  For example, if you did not know the shape of an NFL football and I told you it had an average diameter of 6.64 inches, what shape would you expect it to have?  A sphere?  Obviously, the average diameter of a football doesn’t exist.  The same thing is true for average global temperature, average CO2 level, average sea level, or average global anything.  To suggest that it does is scientifically dishonest.

The only time that a an average is scientifically significant because it increases the precision of a measurement is when one measures the exact same thing with the exact same measuring device, under the exact same conditions hundreds of times.


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The Bogus Greenhouse Gas Demo

by James R. Barrante, Ph.D.

I am sure many of you out there have seen the famous experiment that attempts to show the greenhouse gas behavior of CO2.  It’s a relatively simple and inexpensive experiment, and so it is quite popular as a demonstration in elementary and secondary schools.  The results are convincing, but erroneous.  The experiment involves two identical glass jars with glass stoppers, similar to large cookie jars.  Each jar contains a very precise thermometer.  One jar is filled with air and the other jar is filled with high-purity CO2, which can be purchased from most gas-supply houses.  Two exactly the same infrared heat lamps are placed in the exactly same positions, so that the same amount of infrared light will strike each jar.  For the sake of argument, we will assume here that both setups are identical.

Before the lamps are turned on, the temperature in each jar is carefully recorded.  It is important to note that they do not have to be exactly the same, since the focus of the experiment is to measure a temperature change.  The lamps are turned on at the same time and infrared radiation strikes both lamps equally.  At some point, the lamps are turned off, and the temperature in each jar is recorded (a good approach would be to have two individuals recording the temperatures at exactly the same time.)

The demonstrations that I’ve seen have always shown that the temperature of the gas in the CO2 jar always goes up faster and to a higher temperature than the temperature of the gas in the air jar.  The experimenter then announces that the results are evidence that CO2 is a greenhouse gas.

Here are the problems with this not-so-well-thought-out experiment.  First, we know that CO2 is a greenhouse gas, so it is always easy to prove something we know is true.  The ability of CO2 to be a greenhouse gas is that it absorbs a band of infrared light at a wavelength around 15 microns (1 micron = 0.000000001 meters.  It turns out that this band of infrared light cannot pass through glass.  A better way to look at it is to say that the glass absorbs all the radiation from this 15 micron band.  Consequently, we know that we cannot be looking at the greenhouse gas effect.  So what is changing the temperature in the jars?  The simple answer is that when you heat up a container, any gas inside the container also will heat up by simple convection.  It’s a transfer of heat to the CO2, not light.  The major question is why did the CO2 gas heat up faster and to a higher temperature than the air?  It should not have. Since both jars had the same volume, each contained the same number of moles of gas.  But CO2 has a higher heat capacity than air (see Thermal Behavior of CO2).  The appropriate equation describing the absorption of heat by a substance is

q = n CvΔT

where n is the number of moles of gas, Cv is its heat capacity at constant volume and ΔT is the temperature change.  Assuming both jars received the same amount of heat

( Cv T)­­CO2  =  (Cv T)air

Since Cv for CO2 is greater than Cv for air, ∆T for air must be bigger than ∆T for CO2.  Any experiment showing just the opposite effect has either been rigged or was not performed carefully.  Note that even if sunlight is used in place of heat lamps (visible light from sunlight will pass through the glass and directly heat the interior of the jar), the results still would be questionable due to the higher density of the CO2 gas.  Before drawing any conclusions, it would be useful to replace the CO2 with argon gas, which we know is not a greenhouse gas, but is heavy like CO2.  My guess is that one would get the same results.


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Does CO2 Sensitivity Have Any Real Meaning?

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Thermal Behavior of CO2

Carbon dioxide is a triatomic molecule, but because it is linear, it behaves in many respects like a diatomic molecule.  In fact, the similarity between CO2 and other diatomic molecules is so striking that the molecule can be described as a Hooke’s Law harmonic oscillator, a model that generally works well only for diatomic molecules.

It is well-known that the heat capacity ratio,CP /CV, for most small molecule gases at room temperature or below, calculated using classical equipartition theory, does not agree well with the ratios determined experimentally.  For example, the calculated heat capacity ratio of N2 gas at 25˚C is 1.28, whereas the experimentally determined value is 1.39.  Contrast this to a monatomic gas like argon.  The calculated heat capacity ratio of Ar at 25˚C is 1.67, while the experimentally determined value is 1.62, very good agreement.  The general explanation for the difference between calculated and experimental heat capacity ratios in molecular systems is justified by the difference between the classical and quantum mechanical descriptions of the systems.  This is supported by the fact that as the temperature of N2 gas, for example, is increased, the capacity ratio begins to approach the calculated value of 1.28.

To understand this, consider the following:  according to classical equipartition theory, at any temperature molecules absorb thermal energy through molecular collisions and distribute this energy equally into their various molecular degrees of freedom.  In order for this to work, molecules must be able to absorb energy continuously.  The important thermal molecular degrees of freedom for gases are translational degrees, the motion of the molecule from one point in space to another as specified by its center of mass, rotations of atoms about various axes in the molecule, as specified by moments of inertia, and vibrations of atoms in molecules along certain allowed normal coordinate axes.  Additionally, one must consider that translational motion and rotational motion only involve kinetic energy, whereas vibrational motion involves both kinetic energy and potential energy.  Classically, all these modes of motion absorb energy continuously, and, according to the equipartition theory, are assigned an amount of energy equal to 1/2 kT for each degree, where k is the Boltzmann constant (1.381 x 10-23 J/K) and is the absolute temperature.  As a reminder, remember that monatomic gases possess only translational degrees, and solids possess (for the most part) only vibrational degrees.

To assign the position of the center of mass of a molecule, we need three coordinates.  Consequently, there are three translational degrees of freedom.  Assigning a value of 1/2 kT for each gives us a total of 3/2 kT energy for the translational degrees of freedom.  If a molecule is linear, there is no moment of inertia about the bond axis (assumed to be the x-axis).  Therefore, rotational degrees only involve the y– and z-axes, giving 2/2 kT energy for the rotational degrees of freedom (linear molecule).  Of course, for a non-linear molecule there are three rotational degrees of freedom, giving 3/2 kT energy for the rotational degrees of freedom (non-linear molecule).  We generally determine the vibrational degrees by difference.  Originally, equipartition theory was described in terms of a single point mass, giving a total of 3 degrees of freedom.  To extend this to a molecule, thought of as a collection of point masses, one needs to assign an x-, y, and z-coordinate for each atom in the molecule.  Consequently, a molecule has a total of 3N degrees of freedom, where N represents the total number of atoms in the molecule.

Let us apply this to N2 gas to see how all this works.  Nitrogen gas is a linear molecule and should have 3N  = 3(2) = 6 total degrees of freedom.  But 3 of those degrees are translational degrees and 2 of those are rotational degrees, so 6 – 5 =  1 must be a vibrational mode.  Generalizing these rules:

Translation:  3 degrees of freedom

Rotation:  (linear molecule) 2 degrees of freedom; (non-linear molecule) 3 degrees of freedom

Vibration:  3N – 5  (linear);  3– 6  (non-linear)

We now can calculate the heat capacity ratio of N2:

Translational Energy:    3/2 kT

Rotational Energy:  2/2 kT

Vibrational Energy:  (1 mode)  1/2 kT for kinetic energy; 1/2 kT for potential energy

Total Energy:  7/2 kT

To find the heat capacity at constant volume:


If we assume that N2 is an ideal gas, then  CP  =  CV  +  k  =  9/2 k.  The heat capacity ratio then is

γ  =  CP /CV  =  9/7  =  1.28

Application of Quantum Theory

We know that for all practical purposes translational energy states for all gases are so closely spaced that we can consider them as being continuous.  The same can be said for the rotational energy of most gases, particularly at room temperature.  However, this is not the case for vibrational energy states.  For many diatomic gases, such as nitrogen, the first excited vibrational energy state is separated from the ground state by a value that is too large for the thermal energy around room temperature, approximately 2500 J/mol, to excite nitrogen molecules to higher vibrational states.  Consequently, in the case of nitrogen, thermal energy is partitioned only between translational and rotational degrees.

Translational energy:  3/2 kT

Rotational energy:  2/2 kT

Total energy:  5/2 kT

Therefore:       CV  =  5/2 kCP  =  7/2 k

Ratio  =  7/5  =  1.40, which is very close to the experimentally determined value.

Application of Quantum Statistical Mechanics to Carbon Dioxide

As previously described, carbon dioxide can be treated as a Hooke’s Law diatomic molecule, because it is linear. We find that CO2 has four vibrational modes:  a symmetric stretching mode at 1388 cm-1  , an asymmetric stretching mode at 2349 cm-1  , and a degenerate pair of bending modes at 667 cm-1  .  We can apply the Boltzmann equation to determine the fraction of CO2 molecules in the first excited state for each vibrational mode.  Keep in mind that we are dealing with thermal excitation here, not light excitation.  It does not matter in this analysis whether the modes are infrared active or not.  Recognizing that ∆E  =  hc/λ, where h is Planck’s constant, c is the speed of light, and 1/λ is the wave number (in reciprocal meters), we have at 15˚C (288K)

Symmetric Stretch:  fraction in first excited state  =   CodeCogsEqn-14

  =    CodeCogsEqn-15

=  CodeCogsEqn-16   =   9.94 x 10-4

Asymmetric Stretch:  fraction  =  8.22 x 10-6

Each Bending Mode:     fraction  =  3.60 x 10-2

This is a surprising result.  While the asymmetric stretching mode is similar to most diatomic molecules, indicating that thermal energy would not promote CO2 molecules to higher energy states in this mode, this is not the case for the other two modes.  In fact, for the bending modes (3.6 x 2) about 7% of the molecules are already in the first excited state due to thermal collisions, irrespective of the presence of infrared radiation.  To check these numbers for accuracy, let us calculate the heat capacity ratio of CO2 and compare it to the experimental value.  The contributions for translation and rotation are the same as they are in the classical case.  The contributions for vibration are not that straight forward.  Certainly, we can eliminate the contribution from asymmetric stretch.  The contributions from the other modes must be calculated using statistical mechanics.  The appropriate equation to do this is


where R is the gas constant 8.314 J/mol·K.  This will give the heat capacity for a vibrational mode in joules per mole rather than in joules per molecule.  The following contributions are:

Symmetric stretch:    CV  =  0.397 J/mol·K

Bending modes:    CV   =  3.57  J/mol·K

The contributions from translation and rotation (per mole) are 3/2 R + 2/2 R  =  20.79 J/mol·K.  We now add all contributions, keeping in mind that there are two bending modes:

CV(total)  =  20.79  +  0.397  +  3.57  +  3.57  =  28.32 J/mol·K

Assuming CO2 is an idea gas (not a great assumption):    C =  36.64 J/mol·K

Ratio  =  1.294 at 15˚C

The actual published heat capacity ratio of CO2 at 15˚C is 1.304, which is quite close, considering we assumed that the gas was ideal.


The band of radiation that is infrared active and important in the greenhouse gas scenario is the bending mode at 667 cm-1 (15 microns).  Symmetric stretch, while able to store a small amount of energy (about 229 J/mol) is not IR active.  The fact that about 7% of the CO2 molecules are already in the first excited state at 15˚C due to molecular collisions will affect their ability to absorb IR radiation.  In fact, even in the absence of a source of IR radiation at 15 microns, CO2 will be radiating a small amount of 15 micron radiation.

If we look at the variation of this effect with respect to temperature, we find that from approximately 250K to 300K the fraction of molecules of CO2 in the first excited vibrational bending mode states changes from about 4.4% to 8.0%, which is relatively small, and most likely would not have a large impact on the molecule’s ability to absorb 15 micron IR radiation.  Nevertheless, this does not justify ignoring it, and it should be included in any calculations dealing with the greenhouse gas behavior of CO2.

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