Harebrain Corner, Essays by Curt Hare

Logarithms and pH

In this essay we explain the mathematical concept of exponential behavior and logarithms and apply them to a few properties of water.  Let’s take a common and important property of water (see the Harebrain topic Water). The vapor pressure of water increases as the temperature of the water rises (Table 1). As the temperature increases the water molecules gain kinetic energy and more and more molecules have sufficient energy to break away from the collective interactions of bulk liquid water. At the normal boiling point, 100 °C and 760 mm Hg pressure, many molecules have enough energy to break away and form gaseous water bubbles within the liquid—at that point the water begins to boil.

            Table 1.  Vapor Pressure (mm Hg) of water from 0 to 100 °C


Temperature, oC               mm Hg     

            0                                4.6

           10                               9.2

           20                              17.5

           25                              23.8

           30                              31.8

           40                              55.3

           50                              92.5

           60                             149.4

           70                             233.7

           80                             355.1

           90                             525.8

           95                             633.9

          100                            760.0


If we plot the vapor pressure against the temperature we obtain what you see in Figure 1.

                        Figure 1.   A plot of the vapor pressure of water in mm Hg vs the temperature of the water in °C.


This is not a straight-line, or linear, relationship.  If you look at a small range, say, from 20-50 °C, it appears to be approximately linear as shown in Figure 2, where a straight line is drawn through four points.  But even in this small range it is clear that there is curvature in the plot (notice that the two middle points lie below the line and the outer two lie above the line).

                        Figure 2.  A plot of the vapor pressure of water in mm Hg vs the temperature of the water in °C over the range 10 to 40 °C.

Before we discuss this behavior in more depth, let’s take a look at some made-up numbers for vapor pressure.  We constructed this made-up trend by taking the real vapor pressure of water at 10 °C and then adding 10 to that value to get the vapor pressure at 20 °C, then adding 10 to that value, and so on.  This data (Figure 3) is clearly linear: the points fall on a straight line and the line has a slope of 1 (this is the number in front of the x in the equation y = x – 0.8) and an intercept of -0.8 (this is where the line meets the y axis, that is, at 0 °C the vapor pressure is -0.8 mm Hg).


            Figure 3.  A plot of some artificial numbers for the vapor pressure of water vs temperature.

Now let’s go back to the real vapor pressure data.  If you take 10, 20, 30, and 40 °C, does the vapor pressure vary by a constant amount as you go from 10 to 20, from 20 to 30, and so on?  No, the vapor pressures at the first two temperatures are 9.2 and 17.5, which is a difference of 8.3.  Is the difference between the vapor pressures at 20 and 30 °C (notice that we are considering temperatures that have the same increment (10 °C)) 8.3?  No, in fact the difference is 6.3 °C.  This tells us that if we increment the temperature by 10 °C, the vapor pressure does not change by the same value no matter whether we go from 10 to 20 °C or from 70 to 80 °C.  [Look at the made-up data and notice that the vapor pressure increases by 10 mm Hg for any 10 °C jump in temperature.]

The real vapor pressures do not vary linearly with temperature; they vary exponentially with temperature.  So let’s see what Excel will give us if we insist that the trendline is an exponential one (Figure 4).  Figure 4 contains both our previous linear fit to these four points, as well as an exponential fit.  The exponential curve goes through each of the four points, whereas the linear fit provides a much less satisfactory description of the points.  The equation obtained for the exponential curve is y = 5.17e 0.0598x.   This equation contains e, the base of natural logarithms with an approximate value of 2.718.  The equation could also be written using base 10:  y = 5.17(100.0598x/2.303), where 2.303 is an approximate conversion factor to convert from base 10 to base e.  The real power (pun intended) of this equation lies in the fact that x is part of the exponent.  Let’s see what happens to y when we use a value for x of 10, 20, and 30 in the exponent.


                        Figure 4.  The vapor pressure of water at four temperatures with an exponential fit.

For x = 10,  y = 5.1754e0.0598(10) = 9.41

For x = 20,  y = 5.1754e0.0598(20) = 17.1

For x = 30,  y = 5.1754e0.0598(30) = 31.1


Because these exponential equations and very large or very small numbers like Avogadro’s number (6.023 x 1023) are a bit unwieldy it is often convenient to use logarithms.  Logarithms can be used with any base number, but those for base e and base 10 are most frequently used.  For base e the logarithm is designated ln, and for base 10 the designation is log (though many writers use log for both bases).  We will use mainly base 10.

Let’s take a number like 100.  We could express this number as 102.  Or, let’s  take 500 and express it as an exponent, which would be 102.70.  The log of 100 is the number to which 10 (the base) must be raised to obtain 100.  So,

                        Log(100) = 2,    that is,  102 = 100.

Likewise, the log of 500 is 2.70.  When the log has numbers to the right of the decimal point, this part of the logarithm is called the mantissa.  In fact we could write the log of 500 as 

log(100) + log(5) = 2 + 0.70 = 2.70. 

Notice that while we must multiply 5 x 100 to get 500, we can add log(100) to log(5) to obtain the log of 500.  It is important also to be aware of significant figures.  The antilog (sometimes called inverse log) of 1.0 is 10; that is, log(10) = 1  The antilog of 0.70 is 5.0119; the antilog of 0.71 is 5.1286 and the antilog of 0.69 is 4.8978.  So, the use of just two significant figures in the mantissa (0.70) of 2.70 gives us a number somewhere between 490 and 513.  If we want to express the number more precisely, we need more significant figures.  If we use 2.700, we imply that the mantissa is known to ± 0.001.  The antilog of 2.701 is 502 and for 2.699 is 500.  Thus, three significant figures in the mantissa provide an antilog that also has three significant figures.

Logarithms and powers of 10 are used frequently in chemistry. Logs have a reputation of being difficult to use but they are much easier to understand since modern scientific calculators have log10x and 10x functions. Logs are used in measurements with exponential behavior and to compress large and small numbers to a more manageable form. Because logs are based on powers they are dimensionless numbers; that is, they have no units.

Avogadro’s number is a very large, uncountable, number 6.023 X 1023 but its log is 23.78. The reciprocal of Avogadro’s number, 1/(6.023 X 1023) = 1.67 X 10-24, is the mass of one atomic mass unit and its log is -23.78. Oh yes, -23.78 and 23.78 have only two significant numbers, the log mantissa 0.78. The 23 is only the power of 10 and sets the decimal point.

Addition is a simpler operation than multiplication and for many years tables of logarithms were used for accurate multiplication by adding the logs of the numbers. Also slide rules (the precursor of digital calculators) had log scales, which allowed rapid addition (multiplication) or subtraction (division) of the logs of numbers to obtain answers to three significant figures.

The dissociation of water into hydrogen and hydroxide ions makes good use of logs. The pH of an acid or basic solution is measured with a volt meter. In order to keep the scale positive and readable the pH was defined as –log [H+] in which the square brackets mean molar concentration. The p in pH stands for power; that is, 10-pH and the powers are usually negative which makes 10-pH a positive power.

For the auto dissociation equilibrium of water

H2O  +  H2O  --> H+ +  OH-,

The dissociation constant at room temperature is Kw = 10-14 and is written as: 

                                    [ H+ ] [ OH- ] = Kw = 10-14.

Taking the negative log of both sides of this equation gives:

 -log { [ H+ ] [ OH- ] = Kw = 10-14}. 

Because -log[H+] = pH and -log[OH-] = pOH, 

pH + pOH = pKw =14,

which is very useful in working acid-base equilibrium problems.  





Water is the most ubiquitous molecule on Earth and probably the most abundant molecule in the Universe. Water is part of us, the air, the oceans, the soil, our food…most everything. We drink it, bath in it, sweat it, flush it down the toilet. Our bodies are about 60% water and that percentage decreases as we age.  The primordial “soup” from which life evolved is believed to have been an aqueous (water-based) solution. Water is so commonplace that we generally do not have an appreciation for the very unique properties of water. It is a mistake to assume that all liquids are like water. They are not.

The best example of the unique properties of water is the very familiar observation of what happens when an ice cube is added to a glass of water liquid. The ice cube floats on top of the liquid water.  About 92% of the ice cube sinks below the waterline and 8% is above the waterline. Believe it or not this is very profound. Water is the only known substance in which the solid phase (ice) floats on the surface of the liquid phase. This common observation has a number of implications. It means that the density of the ice is less than that of the liquid. This is the opposite of what is observed for “normal” liquids, where the solid phase is always more dense than the liquid. When normal liquids freeze they form the solid phase starting at the bottom while water freezes from the top down. This means that the ice formed on the surface of frozen lakes and the polar ice caps act as protective layers for the liquid body.

The lower density of ice also implies that the water molecules need more space to be arranged into the solid phase while a tighter packing of molecules takes place in the liquid phase. This, of course, is the opposite of the common observation that solids are more compact than liquids. The fact that the solid needs more space to form from the liquid means that there is an expansion of water when it freezes. Normal liquids contract on freezing, but water expands. The density of liquid water varies with temperature and has a maximum value of 1.000…0g/cm3 at 3.98 oC. That is, the liquid density decreases slightly from 4 oC to 0 oC and from 4 oC to 100 oC. When liquid water freezes to ice at 0 oC and 1 atm pressure there is a sharp and dramatic change of the density to that of ice -- 0.92 g/cm3. This means that there is an expansion of the volume of 8%. The density and structure of ice varies with temperature and pressure. The phases of water as a function of temperature and pressure will be discussed later.

The boiling point of water is another property that is abnormally high. Figure 1 is a graph of the normal boiling points of the Group 16 hydrides (the term “hydride” in this case refers to covalent hydrogen compounds such as the very odiferous and toxic H2S) as a function of their row in the periodic table. There is a nearly linear relationship for the boiling points of H2S, H2Se, and H2Te in rows 3, 4, and 5 of the periodic table. If this is extrapolated to row 2, one would estimate that the normal boiling of H2O would be about -75 oC. This estimate is way off (by 175 oC) since the normal boiling point of water is 100 oC or 373K.


            Figure 1.  The boiling points (°C) of the Group 16 hydrides


Related to this is the fact that the molar heat of vaporization and the entropy of vaporization of water at the boiling point and 1 atm are both high values. The heat of vaporization of a liquid is the amount of heat energy needed to convert one mole of liquid (6.023 x 1023 molecules) into gas molecules.  The standard heat of vaporization of water at the boiling point,100 oC (373K) and 1 atm pressure, is 40,660 Joules per mol. The entropy of vaporization is the heat of vaporization divided by the boiling point in Kelvin and is a measure of disorder. Normal liquids have an entropy of vaporization of 88 J/mol-K (Trouton’s Rule) at their normal boiling point, but the standard entropy for water is 108.95J /mol-K. Since all gases have about the same degree of disorder, the high entropy of vaporization for water indicates that liquid water is well ordered.

There is one more heat phenomenon that needs to be mentioned: the specific heat of water, which you might guess is also a high value. The amount of heat energy required to increase the temperature of one gram of a substance at 1 atmosphere of pressure by one degree C is the specific heat. The specific heat of water liquid is 1.000 calories per gram, per degree C.  (This is the definition of a calorie.), The specific heat of liquid water is more commonly expressed in Joules as 4.184 J/g-K.  The specific hear of ice at -2 oC is 2.10 J/g-K.  Thus, the liquid phase of water is better at taking up heat energy. Good conductors like mercury liquid or copper metal have specific heats of 0.140 J /g-K and 0.385 J /g-K, respectively, but they do not store heat energy on a weight basis as well as water. Water is almost 11 times better at storing heat energy on a unit weight basis than is copper and 30 times better than liquid mercury.  Let’s say it again:  one gram of water is able to store about 11 times more heat than one gram of copper. (note: a change of one degree Celsius is the same as one Kelvin)

Why does water have all these highly unusual properties and other molecular compounds do not? The answer lies in the high polarity of the H2O molecule, the high electronegativity of oxygen, the slight dissociation of water into hydrogen (protons) and hydroxide ions and a phenomenon called hydrogen bonds. The water molecule has a wide V shape with a bond angle of 105 degrees. The oxygen apex is the negative end and the hydrogen ends are slightly positive. If the molecule were linear, the bond moments of the polar H-O bond would be in opposite


Figure 2. Electron distribution in a polar A-B bond (atom B is more electronegative than atom A)

directions, cancel one another and the molecule would be non-polar like CO2. The oxygen apex of the V also has two non-bonding pairs of electrons directed in a tetrahedral direction which makes the oxygen end rich in electron density (Figure 3).



Figure 3.  The dipole in the water molecule.  The negative end is at the more electronegative oxygen atom and there are two non-bonding, lone pairs of electrons pointing in two different tetrahedral directions.


The hydrogen atoms of one molecule are lacking in electron density and may be attracted to the highly electron rich oxygen end of another molecule. This attraction is called a hydrogen bond and may be a weak interaction between two molecules or may be a strong collective interaction involving many molecules (Figure 4). Hydrogen bonds are a property of compounds that have a hydrogen atom bound to a highly electronegative atom of N, O, or F and the bonded hydrogens are attracted to the lone pair of electrons of N, O, or F of another near by molecule.


Figure 4.  Hydrogen bonding between the positive and negative ends of neighboring water molecles.  Because the lone pairs of electrons and the hydrogen atoms of each water molecule are directed toward the corners of a tetrahedron, the other molecules surrounding and hydrogen bonding with the central molecule are also at the corners of a tetrahedron.


What then would 18g (one mole, 6.02 x 1023 molecules) of water be like? Frank and Wen have proposed that bulk water is filled with “flickering” clusters of molecules forming, breaking, and reorganizing into new clusters about a billion times faster than you can blink your eye (Figure 5).


 Figure 5. Illustration of the difference between the ordered, hexagonal arrangement of Ice I on the right and a momentary cluster in liquid water on the left. Solutes and impurities also play a role in organizing molecules in water. Solutes like urea, which lower the viscosity of bulk water are called structure breakers. Molecules like carbohydrates and proteins that increase the viscosity of water are called structure makers.

There are also small clusters and free individual molecules present in bulk liquid water. The number of free molecules increases and the complexity of the clusters decreases, as the temperature is increased toward the boiling point. It is the high kinetic energy that frees molecules that have sufficient energy to break away from the liquid state and pass into the gaseous state. This escaping tendency gives rise to the vapor pressure, which is a function of temperature.

Water has a tendency to form complex structures which are held together by hydrogen bonds. The best example of this is a class of compounds called clathrates or hydrates in which water molecules “wrap” around a substrate molecule and are held together by hydrogen bonds. An example is the methane (CH4) hydrate depicted in Figure 6. Methane hydrate is abundant at the bottom of the oceans particularly around the continental shelves.


Figure 6.  The structure of methane hydrate, with a molecule of methane (CH4) trapped inside a specific aggregate of water molecules connected to one another by hydrogen bonding.  Notice that the aggregate consists of four rings of five water molecules, which form a dodecahedron (12 faces).


 Water does this wrapping to form these “inclusion compounds” with many gas molecules and organic compounds. The anesthetic, nitrous oxide (N2O, laughing gas), and chloroform (CHCl3) form clathrate inclusion compounds with water. Many other anesthetic agents are believed to function by forming intracellular water clathrates, which temporarily disrupt cell function. When the anesthetic gas or agent is removed, the cells revive back to normal function. Nitrogen gas breathed at high pressures by deep sea divers causes nitrogen narcosis by a similar mechanism.

Hydrogen bonding plays an important role in another property of water liquid. As stated above, bulk liquid water is characterized by multiple and complex interactions between all the molecules. Each molecule in the body of the liquid is being “tugged” in all directions by these forces, but because it is in all directions, the net tugging force is zero. The molecules on the surface of the water do not experience forces in all directions. They only have a downward force to the body of liquid because there is no water above them. This creates a force across the outermost layer which is called surface tension. Surface tension is a lateral force which is like a very thin membrane at the interface. Because this layer is like the rubber surface of a trampoline; it can give and it can take weight up to the critical point. There are many common examples of surface tension. Most everyone is familiar with water striders (Figure 7),

Figure 7.  A water strider (Gerridae) out for a stroll on water.  Taken from Wikipedia, http://en.wikipedia.org/wiki/Gerridae, accessed Feb 13, 2015


and floating steel objects on the surface of water.

Dirt and grime on hands and clothing are better removed with soap and water than with water alone. Soap and detergents are surface-active agents that lower the surface tension between the dirt and water. Surface-active agents are molecules that are able to disrupt the structure of water and organize into structures called micelles (Figure 8). These molecules characteristically have two parts. One part “loves” water (hydrophilic) and the other part “loves” oil and grease and hates water (lypophilic or hydrophobic). This love-hate relationship is a result of the polarity of one end while the other is non-polar. The polar end may be non-ionic, ionic (plus or minus) or amphoteric (both plus and minus). Soap bubbles and the lipid bilayer of living cells have similar structures.             

Figure 8.  An arrangement of surfactant molecules with their hydrophobic “tails” pointing toward the center of the micelle and their hydrophilic “heads” pointing toward the surrounding water molecules (not shown).


Figure 9 is a phase diagram for water over a wide range of pressure and temperature. The pressures vary by several orders of magnitude and are plotted as the log of the pressure. There are numerous solid phases (The word “phase” is often used to distinguish the different states of matter—solid, liquid, gas—but it can also refer to different structural variations of a compound in the same state.) of ice over the range of P and T, but there is only one liquid phase and one vapor (gas) phase. The many solid phases are different crystal structures of ice. Ice I is the common phase of ice that forms here on Earth. The diagram also shows that there is only one point where the solid, liquid, and vapor may coexist in equilibrium together and this is the triple point ( 273.16 K and 611.73 Pa, 0.01oC and 4 mm Hg) . The line of equilibrium between Ice I and liquid water is interesting because it slopes to the left. That is; as pressure increases the freezing point of liquid water goes to lower temperatures. Or, in reverse, Ice I melts to liquid water as the pressure increases. This is one more abnormal property of water and Ice I, and should not be surprising because the water molecules in liquid water are closer together than the molecules in solid water I, as shown in Figure 3.  Thus, an increase in pressure will convert ice to its more compact liquid form.  Another way to say this is that an increase in pressure will generally convert a phase of lower density to a phase with a higher density.  For example, an increase in the pressure to 209.9 MPa at 251.165K creates the solid phase Ice III (see Figure X). At these pressures, the densities of Ice I and Ice III are about 0.95 g/cm3 and 1.16 g/cm3, respectively.  Ice III is normal in that a further increase in pressure takes the melting point (or freezing point) to higher temperatures. This also applies to Ice V, VI, and VII. What this all means is that water is abnormal up to a pressure of 209.9 MPa (2072 atm) and freezes to an ice cube that floats. Above 209.9MPa pressure water behaves like a normal liquid and freezes from the bottom up and ice cubes sink in water liquid. Fortunately, there is no place on the surface of Earth where the natural pressure reaches  209.9MPa (2072 atm), not even the bottom of the Marianas Trench, the deepest area in the Pacific ocean where the pressure is about 1000 atm.

Figure 9.  Phase diagram for water plotted as the log of pressure in Pascal vs. Kelvin temperature in black; the red is log of pressure in bar vs degrees C .  From KickassFacts, http://www.kickassfacts.com/given-time-to-decompress-slowly-could-a-human-survive-in-a-martian-summer-with-just-a-oxygen-mask/  accessed September 30, 2016.

Figure 10.  The structure of hexagonal ice Ih.  Taken from Wikipedia  http://upload.wikimedia.org/wikipedia/commons/3/31/Cryst_struct_ice.png.  Accessed Feb 13, 2015

Figure 11.  The structure of cubic ice Ic.  Taken from Martin Chaplin’s Water Structure and Science Site by permission, http://www1.lsbu.ac.uk/water/cubic_ice.html.  Accessed Feb 13, 2015

Figure 12.  The structure of Ice III.  Taken from Martin Chaplin’s Water Structure and Science Site by permission, http://www.lsbu.ac.uk/water/ice_iii.html, accessed Feb. 13, 2015


Earth has had abundant water on the surface since it cooled from a primordial molecular cloud. Earth’s position in the solar system is just right for the temperature range of liquid water. Our atmosphere has a pressure of 101.3 kPa (1 atm) and the cold zones freeze to Ice I which floats on the surface.  Thus, not only are the conditions on Earth right for the existence of us and other organisms, but also perfect for ice-skating!