Chemistry 1020—Lecture 17—Notes

Last period we introduced the concept of molar concentration, abbreviated M, which stands for moles/L.

Now lets apply that to the discussion of the quantity of acid in a solution. We express the quantity of acid simply as the molar concentration of H⁺, abbreviated [H⁺], where the square brackets indicate molar concentration.

Recall that H⁺ is really a hydrated proton, and could also be represented as H₃O⁺. What is the maximum concentration we might expect to have of hydrogen ion in aqueous solution?

Clearly there can’t be more H₃O⁺ species than there are water molecules to begin with. So we might first ask, what is the molar concentration of water, i.e. [H₂O]?

1000 g/L x 1 mol/18.0 g = 55.6 mol/L or 55.6 M

[H₃O⁺] must be less than this because there must be some kind of anion in the solution to neutralize the positive charge. Therefore the maximum acid concentration is probably somewhere about 20 M.

What would be the minimum concentration of H⁺? At first thought, you might be tempted to say zero, but that is not the case because of the following reaction that water undergoes:

H₂O + H₂O  --->   H₃O⁺ + OH^-

This reaction represents the proton from one water molecule that is hydrogen bonded to a neighbor actually being transferred to the neighbor. This reaction is more easily explained using the Bronsted-Lowery definition of an acid mentioned earlier. One of the water molecules is acting as an acid (a proton donor), and the other as a base (a proton acceptor). After the transfer the conjugate base of the donor water is formed (OH^-), and the conjugate acid of the acceptor water is formed (H₃O⁺). The proton can be transferred back, so the reaction can actually occur in both directions, as illustrated in the following equation:

When the rate of the forward reaction is equal to the rate of the reverse reaction, there is no further increase in hydrogen ions and hydroxide ions, and the solution is in what we call a state of dynamic equilibrium.

Equilibrium is a term applied to a system when there is no apparent change in the composition of the system with time. There are two types of equilibrium: static and dynamic.

In a static equilibrium there is no change because nothing is happening. In a dynamic equilibrium, there is no net change because two things are happening in the opposite direction at equal rates. Imagine yourself paddling a kayak upstream, where you are just able to paddle as rapidly as the stream is flowing. Your position remains stationary relative to the bank of the stream, just next to a car parked on the bank. The car is not moving, and is in a state of static equilibrium. Your kayak is also stationary relative to the bank, but it is in a state of dynamic equilibrium. Another example would be running on a treadmill, where you have no net forward motion because your forward motion is counteracted by the motion of the tread.

Dynamic equilibria in chemical reactions can be expressed quantitatively by what is called an equilibrium constant. In the case of the reaction stated above, the position of the equilibrium is expressed by the relationship that;

[H₃O⁺][OH^-] = a constant, K_w, which = 10^-14.

That is, the product of the concentration of hydrogen ion and the concentration of hydroxide ion remains constant. If one goes up, the other must go down.

In a neutral solution, [H₃O⁺] = [OH^-]. Let’s call that value x.

Then from the relationship above:

x² = 10^-14

x = 10^-7

[H₃O⁺] can be decreased further by increasing the hydroxide concentration. But just as there is a maximum hydrogen ion concentration, limited by the number of water molecules, there is a maximum hydroxide concentration, also in the region of about 20 M.

If [OH^-] were 20 M, what would the [H₃O⁺] be?

[H₃O⁺] x 20 = 10^-14

[H₃O⁺] = 5 x 10^-16

In other words, hydrogen ion concentration can range from a high of near 20 M to a low of around 5 x 10^-16. This is a range of almost seventeen orders of magnitude (powers of ten).

When [H⁺] > [OH^-], the solution is acidic.

When [OH^-] > [H⁺], the solution is basic.

Following are some possible combinations:

So you are told a solution has a concentration of 3.5 x 10^-4 [H⁺].

Q: Is it acidic or basic?

A: Acidic, because [H⁺] > 10^-7

 

Q: What is its [OH^-]?

A: 10^-14/3.5 x 10^-4 = 2.9 x 10^-11

It is inconvenient to express such a broad concentration range with a linear scale. It is impossible to get more than three orders of magnitude on one graph, and even that is difficult. For this reason, chemists revert to a logarithmic scale. (Review the discussion in Lecture 15 about logarithms).

The logarithm of a number is the power of ten equivalent to that number. For example:

These numbers are for even powers of ten. What about other numbers? By entering the numbers on your calculator, and punching the logarithm button, you will find that:

log 1.0 = 0

log 2.0 = 0.30

log 3.0 = 0.48

log 4.0 = 0.60

log 5.0 = 0.70

log 6.0 = 0.78

log 7.0 = 0.85

log 8.0 = 0.90

log 9.0 = 0.95

Now, use your calculator for the following series of numbers:

log 3.0 = 0.48

log 30.0 = 1.48

log 300 = 2.48

log 0.3 = -0.52

log 0.03 = -1.52

Remember that you multiply powers of ten by adding exponents:

10^a x 10^b = 10^(a+b)

It follows that log (10^a x 10^b) = log (10^(a+b)) = a + b

So:

log 300 = log (3.0 x 10²) = log 3.0 + log 10² = 0.48 + 2.0 = 2.48

And:

log 0.03 = log(3.0x10^-2) = log 3.0 + log 10^-2 = 0.48 – 2.0 = -1.52

A logarithm consists of two parts, related to the two parts of a number expressed in exponential notation.:

The mantissa, which is the part of the logarithm after the decimal, refers to the numerical part of the exponential notation.

The characteristic, which is the number before the decimal, refers to the power of ten in the exponential notation.

In the logarithm 2.48, 0.48 is the mantissa, referring to 3.0, and 2 is the characteristic, referring to 10².

Back to hydrogen ion concentration

We define a logarithmic scale of hydrogen concentration as follows:

pH = -log [H⁺]

So, expressing [H⁺] as pH in the earlier table;

Notice that as acidic strength, and hydrogen ion concentration increase, pH decreases. (Therefore, the anecdote about a legislator trying to pass a law to reduce pH to zero because of the problem of acids shows a gross misunderstanding of the meaning of this term.)

Let’s review a couple of problems from the book.

Your turn 6.9 page 194

(b) Q: [H⁺] = 6 x 10^-4; what is the pH

     A: log 6 x 10^-4 = -3.22; pH = 3.22

 

(c) Q: [H⁺] = 5 x 10^-5; what is the pH

     A: log 5 x 10^-5 = -4.30; pH = 4.30

Number 11, page 216

Convert the following pH values to [H⁺]; indicate whether the solution is acidic or basic, and calculate the [OH^-].

(a) pH = 4.25 acidic

     [H⁺] = 10^-4.25 = 10^0.75 + 10^-5 = 5.6 x 10^-5

     [OH^-] = 10^-14/5.6 x 10^-5 = 1.8 x 10^-10

 

(b) pH = 6.78 acidic

     [H⁺] = 10^-6.78 = 10^0.22 + 10^-7 = 1.7 x 10^-7

     [OH^-] = 10^-14/1.7 x 10^-7 = 6.0 x 10^-8

 

(c) pH = 8.16 basic

     [H⁺] = 10^-8.16 = 10^0.84+ 10^-9 = 6.9 x 10^-9

     [OH^-] = 10^-14/6.9 x 10^-9 = 1.4 x 10^-6

 

(d) pH = 11.33 basic

     [H⁺] = 10^-11.33 = 10^0.67 + 10^-12 = 4.7 x 10^-12

     [OH^-] = 10^-14/4.7 x 10^-12 =

A word about significant figures:

The number of significant figures of a number is reflected in the significant figures in the mantissa of the logarithm. values of [H⁺] and pH are seldom expressed to more than two significant figures.

For practice interconverting pH, and [H⁺], try the interactive drill problems at http://proton.csudh.edu/lecture_help/phcalcs.html.

(A note about the web site: You are also asked there to calculate values for pOH, which is simply defined as –log[OH^-].  Since [H⁺][OH^-] = 10^-14, then pH + pOH = 14. )