Physical Chemistry for the Biological Sciences


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September 11, 2014 · 12:19 pm


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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Filed under Global Warming, Henry's Law, James R. Barrante

The Taste of Carbon Dioxide

by James R. Barrante, Ph.D.

A few years back, the Environmental Protection Agency deemed that the gas carbon dioxide is a pollutant.  The dictionary defines a pollutant as a substance that contaminates or poisons.  It would be useful to look at how carbon dioxide stands up to this definition.

The composition of Earth’s early atmosphere was approximately 95% carbon dioxide and water vapor, very similar to the atmosphere of Venus today.  It contained no oxygen.  Any oxygen that might have formed, reacted chemically with metals in the hot crust (e.g., iron, copper, calcium, magnesium, aluminum) and metalloids (silicon) to form oxides.  As the planet cooled to below 100˚C, something that Venus was never able to do because of its nearness to the sun, the water vapor began to condense forming lakes and rivers, and because of its high solubility in liquid water, the atmospheric carbon dioxide began to dissolve in the Earth’s waters.  When carbon dioxide dissolves in water, it forms a weak acid, known as carbonic acid, that will react with any alkaline (basic) substances to form bicarbonates and carbonates.  Earth’s waters began to attack the alkaline oxide minerals, like calcium oxide, to form carbonates, and since most carbonates are not soluble in water, they settled out of lakes and oceans to form rock such as limestone and marble.  This left more room for more carbon dioxide to dissolve.  The level of atmospheric CO2 began to drop.  Moreover, certain chemical reactions in the atmosphere began to produce nitrogen gas, a relatively inert gas that reacts slowly with other elements.  The composition of the atmosphere began to drastically change.

The appearance of living organisms on the planet further changed the chemistry of the planet.  While the chemical reaction

CO2     +     H2O       →     sugars     +       O2   

is not energetically possible, certain plants learned to make a photosensitizer, known as chlorophyll, that allowed them to use the sun’s energy to force this reaction to occur.  Because chlorophyll absorbs in the red region of the spectrum, plants appeared to be green.  The formation of large quantities of green algae further reduced the level of atmospheric carbon dioxide, producing an atmospheric waste poison, oxygen gas.  Luckily, the level of oxygen gas in the atmosphere would remain low.  This is because any newly formed oxygen gas would quickly react with metals in the Earth’s crust to form the metal ores that are present today.  Likewise, the reaction of vast amounts of silicon, the second most abundant element with oxygen, formed silicates and large regions of sand.

Animals began to evolve in the waters, feeding on the green plants (and on each other), but were confined to the waters, because of the deadly ultraviolet radiation reaching the surface of the Earth from the sun.  One good effect, however, did result from the oxygen in the air.  Lightening storms supplied large amounts of energy that allowed another type of oxygen, O3, and known as ozone, to form and this gas began to build up in the upper atmosphere.  Ozone has the ability to filter the ultraviolet radiation from sunlight, and this eventually allowed animals to move from the oceans onto the land.  We must keep in mind, however, that the food chain remained in tact.  Carbon based plants obtain their carbon atoms solely from atmospheric carbon dioxide.  Animals get their carbon atoms by eating the plants or eating the animals that eat the plants.  At levels of atmospheric CO2 in the thousands of parts per million range, lush plant life covered the Earth’s surface, allowing animals that fed on these plants or fed on the animals that ate these plants to grow very large. This could never happen today.  At levels of 100 ppm (parts per million) CO2 plants begin to die.  At levels of 200 ppm plants struggle to survive.  At levels of 300 ppm (pre-industrial CO2 levels) there was barely enough food for pre-industrial populations to survive, at todays levels of 400 ppm, it is not likely that there will be enough food on the planet to feed the expected large increases in population in the 21st century.

We know that atmospheric CO2 levels are controlled by the temperature of the oceans. If we are lucky, oceans will not cool and cause carbon dioxide levels to plummet. It appears that declaring our source of all the food on this planet a pollutant was not a stroke of genius. It was a stroke of ignorant stupidity.

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Filed under Basic Science, Carbon Dioxide Properties

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.


Filed under Basic Science, Global Warming

Acid – Base Chemistry

by James R. Barrante, Ph.D.

There have been a lot of articles and blogs online covering the so-called “acidification of the oceans.”  (See OAI, OAII on this website).  Much of the information found in many of these articles and blogs, while very well written, is scientifically incorrect.  The primary reason is a complete misunderstanding of acid-base physical chemistry.  So this post hopefully will address some of these errors.

There are a number of ways to define an acid and a base.  In this post we will use what is referred to as the Arrhenius definition.  An aqueous solution is acidic if (H+) > (OH) and basic (alkaline) if (OH) < (H+), where () refers to molar concentration.  There is a notion out there that these are the only two states available, and the system is “dipolar,” like electrical charge.  That is, if a system is less basic, it must be more acidic.  This is not the case, because there is a third state that is neither acidic nor basic and this is the neutral state, (H+) = (OH).

Because hydrogen ion and hydroxide ion concentration can vary over a wide range of values (many powers of ten), a logarithmic scale, known as pH and pOH, has been designed to make concentrations more manageable.  The definition of pH and pOH are:

pH  = – log aH+     and     pH  = – log aOH-,

where aH+ and aOH- are the activities (effective concentrations) of the ions, respectively.  For example, the assumption is made that if the pH of an alkaline solution drops, it is becoming more acidic.  The problem with this idea has to do with the neutral state.  You see, you can lower the pH of an alkaline solution simply by adding water, a neutral substance.  The solution is simply becoming less basic approaching neutrality.  To be acidic, it would have to cross the pH 7 boundary, and it will never do that, no matter how much water you add.  It is akin to asking what is the pH of a 10-8 M HCl solution?  I know you want to say 8.  The pH of an acid solution can never be greater than 7.  It’s a problem we give our students in Analytical Chemistry to figure out.

So, having said all this, it is clear that with a pH around 8, our oceans are alkaline and dropping the pH toward 7 doesn’t make them more acidic, it simply makes them less basic.  A solution that is not acidic in the first place cannot become more acidic.  And it is scientifically dishonest to suggest that it could.  The question now centers around what happens to an aqueous solution when CO2 gas is bubbled through it.  We know that CO2 gas reacts with water to form carbonic acid H2CO3.  But carbonic acid is a very weak acid and that is important.  (By the way, the term weak and strong, when referring to acids and bases, has nothing to do with concentration.)  The acid dissociates according to the equation (we will consider only the first dissociation at this point):


The double arrow here means that the reaction is reversible.  At some the concentrations of these substances will reach thermodynamic equilibrium, at which point


where () designates molar concentration and K1 is the apparent equilibrium constant.  A true thermodynamic equilibrium constant would require the use of activities (effective concentrations) of the substances rather than the concentrations themselves.  In this case we will take K1  = 4.45 x 10-7.  Obviously, the excess of hydrogen ion in solution makes the solution acidic.  But look at the equilibrium constant.  It is telling us that most of the dissolved carbon dioxide is H2CO3.  We know that at a temperature of 298.2K, when the pressure of CO2(g) is 400 ppm (0.0004 atm), the concentration of dissolved CO2 is about 1.35 x 10-5  M.  Using Eq (1), we can find that the concentrations of H+ and HCO3  are both equal to 2.47 x 10-6 M,  giving a pH = 5.61, which is mildly acidic.

Wouldn’t it be nice if nature was so simple.  Unfortunately, bicarbonate ion also is a weak acid.  That is, it dissociates according to the equation



where K2 is the second dissociation constant of carbonic acid equal to 4.69 x 10-11 .  From the size of K2 you can see that very little hydrogen ion is produced by the dissociation of bicarbonate.  In fact, the pH of a saturated solution of sodium bicarbonate can be found by the equation

pH  =  1/2 (pK1 + pK2)

where pK1 = – log K1  and pK2 = – log K2.

pH  =  1/2 (6.35 + 10.33)  =  8.34

considerably alkaline.  Of course, this explains why bicarbonate of soda is a good antacid.  Moreover, it turns out that the solution of sodium bicarbonate does not have to be saturated.  Since a solution of NaHCO3 is the first equivalence point in the titration of carbonic acid, any solution of  NaHCO3 in water will have a pH of 8.34.

So let’s see what would happen to ocean water if we added a little Na2CO3 to it.  Not too much, just enough to make it equal the carbonic acid concentration.  Combining the two equilibrium constant expressions gives


Since  (CO3-2)  =  (H2CO3) ,  (H+)2  =  K1K2  and pH  =  ½ (pK1  +  pK2)  =  8.34.  Adding a small amount of carbonate ion to the system drastically increases the pH to make the system alkaline.  You will find that at this point also, it is very difficult to change the pH of the solution.  The solution here is said to be “buffered.”  In fact, our oceans are gigantic buffered systems.  It is important to note that since we are using concentrations here, we should not expect to get very good answers.  This is because seawater has a high ionic strength and as the ionic strength goes up, the difference between activities and concentrations increases.  Either we use activities here or we correct equilibrium constants for salinity.

Of all the gases, CO2 gas is one of the most soluble.  But that being said, gases in general are not very soluble in water.  Even at the low temperature 280K, the solubility of CO2 in water at a pressure of 4000 ppm or 0.004 atm, is only about 2.4 x 10-4 M.  The point is that it is not enough to simply look at the production of hydrogen ions to decide on the acidity or alkalinity of a solution.  When the source of the hydrogen ions is a weak acid, the presence of other ions will drastically effect the pH of the solution.

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Filed under Ocean Acidification, Ocean Chemistry

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.


Filed under Global Warming