Self-Ionization of Water

The pH scale is based on a logarithmic scale, meaning that an increase or decrease of an integer value changes the concentration by a tenfold. For example, a pH of 3 is ten times more acidic than a pH of 4. Likewise, a pH of 3 is one hundred times more acidic than a pH of 5. Similarly a pH of 11 is ten times more basic than a pH of 10. pH is often measured in chemistry, biochemistry, and biology. Because of the amphoteric nature of water (water can act as both an acid or a base), water does not always remain as H₂O. In fact, two water molecules react to form hydronium and hydroxide ions. This is also called the self-ionization of water. The equation is shown below: 

2 H₂O (l) H₃O⁺ (aq) + OH⁻ (aq)

The concentration of H₃O⁺ and OH^- are equal in pure water because of the stoichiometric ratio. The molarity of H₃O⁺ and OH^- in water are also both 1.0 X 10^-7 M at 25° Celsius. Therefore, a constant of water (K_w) is created to show the equilibrium condition for the self-ionization of water. The product of the molarity of hydronium and hydroxide ion is always 1.0 X 10^-14.

K_w= [H₃O⁺][OH-] = 1.0 X 10^-14

This equations also applies to all aqueous solutions. However, K_w does change at different temperatures, which affects the pH range discussed below. Note: H⁺ and H₃O⁺ is often used interchangeably. 

The equation for water equilibrium is:

 H₂O H⁺ + OH^-

• If an acid (H⁺) is added to the water, the equilibrium shifts to the left and the OH^- ion concentration decreases
• If base ( OH^-) is added to water, the equilibrium shifts to left and the H⁺ concentration decreases. 

pH and pOH

Because the constant of water, K_w is always 1.0 X 10^-14, the pK_w is 14, the constant of water determines the range of the pH scale. To understand what the pK_w is, it is important to understand first what the "p" means in pOH, and pH. The danish biochemist Soren Sorenson proposed the term pH to refer to the "potential of hydrogen ion."  He defined the "p" as the negative of the logarithm, -log, of [H⁺]. Therefore the pH is the negative logarithm of the molarity of H. The pOH is the negative logarithm of the molarity of OH^- and the pK_w is the negative logarithm of the constant of water. These definitions give the following equations:

pH= -log [H⁺]
pOH= -log [OH^-]
pK_w= -log [K_w]

A Logarithm, used in the above equations, of a number is how much a power is raised to a particular base in order to produce that number. To simplify this, look at the equation: log_ba=x. This correlates to b^x=a. A simple example of this would be log₁₀100=2, or 10²=100. It is assumed that the base of Logarithms is ten if it is not stated. So for the sake of pH and pOH problems it will always be ten. When x is a negative number that means you are dividing it by the power. So, if log₁₀0.01=-2 which can be written 10^-2=0.01. 10^-2 also means 1/10². The log function can be found on your scientific calculator. Now if we apply this to pH and pOH we can better understand how we calculate the values.

The constant of water is always 1.0 X 10^-14. So pK_w=-log [1.0 X 10^-14]. Using what we know about Logarithms, we can write this as 10^-pK_w=10^-14. By substituting we see that pK_w is 14. The equation also shows that each increasing unit on the scale decreases by the factor of ten on the molarity. For example, a pH of 1 has a molarity ten times more concentrated than a solution of pH 2. Also, the pK_w of water is 14 and the addition of pH and pOH is always 14 at 25° Celsius.

pK_w= pH + pOH = 14

Range of pH and how to read it

                                                                                                           Figure 1: Solutions and the placement of them on pH scale

The pH scale is often referred to as ranging from 0-14 or perhaps 1-14. Neither is correct. The pH range does not have an upper nor lower bound, since as defined above, the pH is an indication of concentration of H⁺. For example, at a pH of zero the hydronium ion concentration is one molar, while at pH 14 the hydroxide ion concentration is one molar. Typically the concentrations of H⁺ in water in most solutions fall between a range of 1 M (pH=0) and 10^-14 M (pH=14). Hence a range of 0 to 14 provides sensible (but not absolute) "bookends" for the scale. However, in principle, one can go somewhat below zero and somewhat above 14 in water, because the concentrations of hydronium ions or hydroxide ions can exceed one molar. Figure 1 depicts the pH scale with common solutions and where they are on the scale. 

Figure 2: pH Scale and the concentrations of H⁺ and OH^-

• From the range 7-14, a solution is basic. The pOH should be looked in the perspective of OH^- instead lf H⁺. Whenever the value of pOH is greater than 7, then it is considered basic. And therefore there are more OH^- than H⁺ in the solution   

• At pH 7, the substance or solution is at neutral and means that the concentration of H⁺ and OH^- ion is the same.

• From the range 1-7, a solution is acidic. So, whenever the value of a pH is less than 7, it is considered acidic. There are more H⁺ than OH^- in an acidic solution.                                                                               

Proper Definition of pH

In 1909 S.P.L. Sorensen published a paper in Biochem Z in which he discussed the effect of H⁺ ions on the activity of enzymes. In the paper he invented the term pH to describe this effect and defined it as the -log[H⁺ ]. In 1924 Sorensen realized that the pH of a solution is a function of the "activity" of the H⁺ ion not the concentration and published a second paper on the subject. A better definition would be pH=-log[aH⁺ ], where aH⁺ denotes the activity of the H⁺ ion. The activity of an ion is a function of many variables of which concentration is one. It is unfortunate that chemistry texts use a definition for pH that has been obsolete for over 50 years.

Because of the difficulty in accurately measuring the activity of the H⁺ ion for most solutions the International Union of Pure and Applied Chemistry (IUPAC) and the National Bureau of Standards (NBS) has defined pH as the reading on a pH meter that has been standardized against standard buffers. The following equation is used to calculate the pH of all solutions:

The activity of the H⁺ ion is determined as accurately as possible for the standard solutions used. The identity of these solutions vary from one authority to another, but all give the same values of pH to ± 0.005 pH unit. The historical definition of pH is correct for those solutions that are so dilute and so pure the H⁺ ions are not influenced by anything but the solvent molecules (usually water). In most solutions the pH differs from the -log[H⁺ ] in the first decimal point.

Above, the pH was approximated as the measure of H⁺ concentration:

pH= -log [H⁺]

Note: concentration is abbreviated by using square brackets, thus  [H⁺] = hydrogen ion concentration.  When measuring pH, [H⁺] is in units of moles of H⁺ per liter of solution. This is a reasonably accurate definition at low concentrations (the dilute limit) of H⁺. At very high concentrations (10 M hydrochloric acid or sodium hydroxide, for example,) a significant fraction of the ions will be associated into neutral pairs such as H⁺Cl^–, thus reducing the concentration of “available” ions to a smaller value which we will call the effective concentration. It is the effective concentration of H⁺ and OH^– that determines the pH and pOH. The pH scale as shown above is called sometimes "concentration pH scale" as opposed to the "thermodynamic pH scale". The main difference between both scales is that in thermodynamic pH scale one is interested not in H⁺concentration, but in H⁺activity. What a person measures in the solution is just activity, not the concentration. Thus it is thermodynamic pH scale that describes real solutions, not the concentration one.

For solutions in which ion concentrations don't exceed 0.1 M, the formulas pH = –log [H⁺] and pOH = –log[OH^–] are generally reliable, but don't expect a 10.0 M solution of a strong acid to have a pH of exactly –1.00! However, this definition is only an approximation (albeit very good under most situations) of the proper definition of pH, which depends on the activity of the hydrogen ion:

pH= -log a{H+}

The activity is a measure of the "effective concentration" of a substance, is often related to the true concentration via an activity coefficient, γ:

a{H⁺}=γ[H⁺]

Calculating the activity coefficient requires detailed theories of how charged species interact in solution at high concentrations (e.g., the Debye-Hückel Theory). The following table gives experimentally determined pH values for a series of HCl solutions of increasing concentration at 25 °C. 

Table 3: HCl Solutions with Corresponding pH values

Christopher G. McCarty and Ed Vitz, Journal of Chemical Education, 83(5), 752 (2006) and G.N. Lewis, M. Randall, K. Pitzer, D.F. Brewer, Thermodynamics (McGraw-Hill: New York, 1961; pp. 233-34).

1) If the concentration of NaOH in a solution is 2.5 X 10^-4 M, what is the concentration of H₃O⁺?

Answer

1) Because 1.0 X 10^-14 = [H₃O⁺][OH^-],
    To find the concentration of H₃O⁺, solve for the [H₃O⁺].
    (1.0 X 10^-14)/[OH^-] = [H₃O⁺]
    (1.0 X 10^-14)/(2.5 X 10^-4) = [H₃O⁺] = 4.0 X 10^-11 M

1) In one solution, there is 0.002M of HCl. Find the pH.
2) If there is a solution of 0.00005M NaOH. Find the pH.

Answer:
1)
    The equation for pH is -log [H+]
    [H⁺]= 2.0 X 10^-3 M
    pH = -log [2.0 X 10^-3] = 2.70 

2)
    The equation for pOH is -log [OH^-]
    [OH^-]= 5.0 X 10^-5 M
    pOH = -log [5.0 X 10^-5] = 4.30
    pK_w = pH + pOH and pH = pK_w - pOH
    Then pH = 14 - 4.30 = 9.70.

If soil has a pH of 7.84, what is the H⁺ concentration of the soil solution?

Answer:

pH = -log [H⁺]
7.84 = -log [H⁺]
[H+] = 1.45 x 10^-8 M

(Hint: place -7.84 in your calculator and take the antilog (often inverse log or 10^x) = 1.45 x 10^-8M

Living Systems

Molecules that make up or are produced by living organisms usually function within a narrow pH range (near neutral) and a narrow temperature range (body temperature). Many biological solutions, such as blood, have a pH near neutral. pH influences the structure and the function of many enzymes (protein catalysts) in living systems. Many of these enzymes have narrow ranges of pH activity.

1. In a solution of 2.4 X 10^-3 M HI, find the concentration of OH^-.
2. Determine the pH of a solution that is 0.0035 M HCl.
3. Determine the [H₃O⁺] of a solution with a pH = 5.65
4. If the pOH of NH₃, ammonia, in water is 4.74. What is the pH?  
5. Pepsin, a digestive enzyme in our stomach, has a pH of 1.5. Find the concentration of OH^- in the stomach. 

Contributors

• Emmellin Tung (UCD), Sharon Tsao (UCD), Divya Singh (UCD), Patrick Gormley (Lapeer Community School District)

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