### Video Transcript

In this video, we will learn how to
define and calculate the degree of the dissociation of a weak acid and use it to
derive Ostwald’s law of dilution.

Let’s consider the difference
between a strong and weak acid. When hydrochloric acid, a strong
acid, is dissolved in water, all of the molecules disassociate into hydrogen ions
and chloride ions. So if 50 molecules of hydrochloric
acid were dissolved in water, all 50 molecules would disassociate. In contrast, when hydrofluoric
acid, a weak acid, is dissolved in water, only a portion of the molecules will
disassociate and an equilibrium is established. If 50 molecules of hydrofluoric
acid are dissolved in water, on average, only one molecule would disassociate into
hydrogen ions and fluoride ions.

The extent to which an acid
dissociates can be represented by the degree of dissociation. The degree of dissociation can be
defined as the proportion of the substance that dissociated. It is often given the symbol 𝛼 and
can be calculated by dividing the amount of dissociated substance by the total
amount of substance, both dissociated and undissociated. We can use the number of molecules
or the number of moles for the amounts. Let’s go ahead and calculate the
degree of dissociation for each acid.

The total number of hydrochloric
acid molecules was 50, and all 50 molecules dissociated. So the degree of dissociation is
one. There were also 50 molecules of
hydrofluoric acid, but only one molecule dissociated. So the degree of dissociation is
0.02. Notice that when calculating the
degree of dissociation, the total number of ions produced is not used in the
equation. The degree of dissociation may also
be given as a percentage by multiplying the resulting decimal value by 100
percent.

Weak acids typically have very low
degrees of dissociation. In general, as the strength of the
acid increases, so does the degree of dissociation, with strong acids having a
degree of dissociation of one. So the degree of dissociation can
give us some insight into the strength of an acid. But the extent to which a weak acid
dissociates is dependent on the concentration of the acid. So to better quantify the strength
of an acid, we can calculate an acid dissociation constant. An acid dissociation, constant
symbol 𝐾 a is the equilibrium constant for the dissociation of an acid in
water. It is temperature dependent, but it
is not dependent on the acid concentration.

For the dissociation of a generic
mono product weak acid, represented here by HA, we can write this acid dissociation
constant expression. The brackets indicate that we
should use the concentration of each species. And as we are calculating in
equilibrium constant, we should use the concentrations at equilibrium. If the acid is very weak, very few
molecules will disassociate. So the concentrations of the ions
at equilibrium will be low, while the concentration of the acid is high. This will result in a low acid
dissociation constant. As the strength of the acid
increases, the concentration of the ions will increase, and the acid dissociation
constant will increase.

We now have two quantities degree
of dissociation and acid dissociation constants that can help us to understand the
strength of an acid. These two quantities can be related
to one another. Let’s consider the following
example. In this diagram, one HA picture
represents one mole of HA. So initially, 10 moles of HA, a
weak acid, are dissolved in water to produce 10 liters of solution. This means that the initial
concentration of HA is one molar and the initial concentration of H plus and A minus
is zero molar.

If the degree of dissociation of
one molar HA is 0.2, then at equilibrium 20 percent of the acid will have
dissociated. At equilibrium, the concentration
of HA will be 0.8 molar and the concentration of H plus and A minus will be 0.2
molar. This means that the concentration
of the acid decreased by 0.2 molar, while the concentration of the ions increased by
0.2 molar. Thus, the concentration’s changed
by a value equal to the degree of dissociation. If we replace the changing
concentration with the degree of dissociation, then the equilibrium concentration of
HA in moles per liter is one minus the degree of dissociation and the equilibrium
concentration for H plus and A minus in moles per liter is equal to the degree of
dissociation.

We can substitute these equilibrium
concentrations into the acid dissociation constant expression to determine that 𝐾 a
is equal to the degree of dissociation squared divided by one minus the degree of
dissociation. This expression allows us to relate
the acid dissociation constant and degree of dissociation but only when the initial
concentration of acid is one molar. We said earlier that the degree of
association is dependent on concentration. So we need an expression that
includes the initial concentration of the acid.

Let’s define the initial
concentration of the acid as 𝑐 naught. We know that initially the
concentrations of H plus and A minus will be zero molar. As the dissociation proceeds, the
concentration of HA will decrease by the initial concentration times the degree of
dissociation, while the concentration of H plus and A minus will increase by the
initial concentration of the acid times the degree of dissociation. Adding the initial concentration
and changing concentration gives us these equilibrium concentrations. We can substitute these equilibrium
concentrations into the 𝐾 a expression. We can simplify the expression by
factoring the initial concentration and by combining the degrees of
dissociation.

We are left with the following
expression, which allows us to relate the acid dissociation constant and degree of
dissociation for any initial concentration. This is known as Oswald’s dilution
law. And it can be applied to any
dissociation constant, not just acid dissociation constants. For many weak acids, we can
simplify Oswald’s dilution law even further. Many weak acids have a degree of
dissociation that is very small. If the degree of association is
very small, then one minus the degree of dissociation will be approximately equal to
one. This means that for weak acids with
very small degrees of dissociation, we can use the approximation 𝐾 a is
approximately equal to the degree of dissociation squared times the initial
concentration. Rearranged to solve for the degree
of dissociation, we find that the degree of dissociation is approximately equal to
the square root of the 𝐾 𝐚 divided by the initial concentration.

It’s important to recognize that
this approximation only works for acids which have very small degrees of
dissociation. And it’s also worth mentioning that
all of these relationships are temperature dependent. With that said, we can see from the
approximation that the degree of dissociation is inversely proportional to the
square root of the initial concentration. This means that as the initial
concentration of acid increases, the degree of dissociation will decrease, and vice
versa. So if we dilute a solution, that
is, we add more solvent, and decrease the concentration, the proportion of molecules
that disassociate will increase.

The relationship between degree of
dissociation and concentration is of particular importance when dealing with
electrolytes. In general, the electrical
conductivity of an electrolyte is proportional to the ion concentration. So we might expect that doubling
the concentration of a weak electrolyte will double the electrical conductivity.

Ethanoic acid is a weak
electrolyte. If the concentration of ethanoic
acid is doubled, then the degree of dissociation will decrease. This means that the proportion of
molecules that disassociate will decrease, and the concentration of hydrogen ions
and acetate ions in solution will be less than double the original amount. So doubling the electrolyte
concentration does not double the ion concentration. Therefore, the electrical
conductivity will be less than double the original amount.

Likewise, if we halve the
concentration, the degree of dissociation will increase, and the concentration of
the ions will be slightly greater than one-half their original value. So the electrical conductivity will
also be slightly greater than one-half its prior value.

Before we summarize what we’ve
learned about degree of dissociation, let’s take a look at some questions.

A 0.3 molar solution of benzoic
acid was found to be 1.47 percent dissociated. What is the value of 𝐾 𝐚 to two
decimal places for this acid? Assume one minus 𝛼 is
approximately equal to one.

Benzoic acid is a weak acid. When it is dissolved in water, only
a portion of the molecules will dissociate into benzoate ions and hydrogen ions. We are told that 1.47 percent of
the acid was found to be dissociated. This means that if we dissolved 100
moles of benzoic acid in water, at equilibrium 1.47 moles will have dissociated into
ions, while the remaining 98.53 moles are undissociated. The proportion of the substance
that is dissociated is called the degree of dissociation. It is usually represented by the
symbol 𝛼 and is typically given as a decimal. We were given the percentage of
dissociation. We can convert the percentage of
dissociation to a decimal by dividing by 100 percent. So the degree of dissociation of
the benzoic acid given in the question is 0.0147.

We want to use the degree of
dissociation to calculate the value of 𝐾 𝐚. 𝐾 𝐚 is the acid dissociation
constant, an equilibrium constant for the dissociation of an acid in water. The acid dissociation constant and
the degree of dissociation can be related using Oswald’s dilution law. In this expression, 𝑐 naught
represents the initial concentration of the acid. The question tells us that the
concentration is 0.3 molar. We know the degree of dissociation
and the concentration. So we could put those values into
the expression and solve for 𝐾 𝐚. But the question tells us to assume
that one minus 𝛼 is approximately equal to one.

If the denominator of the
expression is approximately equal to one, we can rewrite the expression as 𝐾 𝐚 is
approximately equal to 𝛼 squared times 𝑐 naught. We can substitute the degree of
dissociation and concentration into the expression and perform the calculation to
determine the acid dissociation constant. Rounding our answer to two decimal
places, we have determined that the value of 𝐾 𝐚 for benzoic acid is 6.48 times 10
to the negative fifth molar or moles per liter.

Which of the following factors
would increase the degree of dissociation of a weak acid? (A) Increasing dilution, (B)
decreasing volume, (C) decreasing dilution, (D) increasing acid concentration.

When a weak acid is mixed with
water, only a portion of the molecules dissociate into ions. We can represent the extent to
which the molecules dissociate with the degree of dissociation. Degree of dissociation, typically
given the symbol 𝛼, is the proportion of the substance that is dissociated. The degree of dissociation is
dependent on a number of factors, including the strength of the acid, temperature,
and concentration. The strength of an acid can be
quantified using an acid dissociation constant, or 𝐾 a. The degree of dissociation, 𝐾 𝐚,
and the initial concentration of the acid, represented by 𝑐 naught, can be related
to one another using Oswald’s dilution law.

For weak acids with very low
degrees of dissociation, one minus 𝛼 can be approximated to one. And Oswald’s dilution law can be
simplified to 𝐾 𝐚 is approximately equal to 𝛼 squared times 𝑐 naught. This can be rearranged to solve for
the degree of dissociation. Here, we can see that the degree of
dissociation is inversely proportional to the square root of the acid
concentration. Thus, increasing the concentration
will decrease the degree of dissociation, and decreasing the concentration will
increase the degree of dissociation. We want to determine which factor
would cause the degree of dissociation to increase. So we need to determine which
factor would cause the concentration to decrease.

Answer choice (A), increasing
dilution means that we are adding more solvent to the solution. Increasing the amount of solvent
will cause the concentration to decrease. A decrease in concentration
corresponds with an increase in the degree of dissociation. Let’s look at answer choice
(B). Decreasing the volume of the
solvent will increase the concentration, so the degree of dissociation will
decrease. Answer choice (C) is the opposite
of answer choice (A). Decreasing dilution will cause an
increase in the concentration, so the degree of dissociation will decrease. Lastly, we know that increasing the
acid concentration will cause a decrease in the degree of dissociation. So the factor that would increase
the degree of dissociation of a weak acid is answer choice (A), increasing
dilution.

Now let’s summarize what we’ve
learned. Degree of dissociation is the
proportion of a substance that has dissociated. It can be calculated by dividing
the amount of substance dissociated by the total amount of substance, where the
amount may be given as a number of molecules or a number of moles. The degree of dissociation can be
related to the acid dissociation constant via Ostwald’s dilution law, where 𝐾 𝐚 is
the acid dissociation constant, 𝛼 is the degree of dissociation, and 𝑐 naught is
the initial concentration of the acid.

For weak acids with very low
degrees of dissociation, one minus 𝛼 is approximately equal to one. So we can use the approximations 𝐾
𝐚 is approximately equal to 𝛼 squared times 𝑐 naught or 𝛼 is approximately equal
to the square root of 𝐾 𝐚 divided by 𝑐 naught. The degree of dissociation is
inversely proportional to the square root of the acid concentration. So increasing the concentration of
the acid will decrease the degree of dissociation, and vice versa.