### Video Transcript

According to Oswaldβs law of
dilution, which of the following is the correct relationship between the
dissociation constant πΎ π and the degree of dissociation πΌ? Assume that one minus πΌ is
approximately equal to one. (A) πΎ π equals the square root of
πΌ divided by πΆ π. (B) πΎ π equals πΌ multiplied by
πΆ π. (C) πΎ π equals the square root of
πΆ π divided by πΌ. (D) πΎ π equals πΌ multiplied by
πΆ π squared. (E) πΎ π equals πΌ squared
multiplied by πΆ π.

Some molecules when placed into
water dissociate into separate ions. Dissociation is the separation of a
substance into individual components. Degree of dissociation is a measure
of the proportion of the substance that actually dissociated. Mathematically, itβs given the
symbol πΌ, and it equates to the amount of dissociated substance divided by the
total amount of substance that is dissociated and undissociated. Degree of dissociation is usually
expressed as a decimal, and it can be expressed in terms of moles of dissociated
substance divided by total moles of substance dissociated and undissociated. We can apply it to all homogeneous
systems that have arrived in an equilibrium state.

If our starting molecule is that of
a weak acid, shown here as HA, weβll find that it only dissociates partly in
water. We can see that the dissociation of
HA is a reversible process, and eventually equilibrium will be established. For our weak acid, we can write out
an expression for the acid dissociation constant denoted by πΎ π. In our πΎ π expression, the
product ion concentrations are seen multiplied together on the top half of the πΎ π
fraction, whilst the reactant molecule concentration, in this case HA, is seen on
the bottom half of our fraction.

The size of the πΎ π value gives us
an indication of the strength of the acid. Smaller values of πΎ π mean weaker
acids. Larger values of πΎ π mean
stronger acids. Remember that for a weak acid, all
three species seen in the equation here will be present at equilibrium. In a weak acid, there will always
be some HA left in the solution as itβs not fully dissociated. The amount of H+ ions and Aβ ions
in this scenario depends on the strength of the acid. In the context of our weak acid HA,
the degree of dissociation or πΌ for HA can be viewed as the proportion of HA that
actually dissociates.

We can see how the acid
concentration changes from an initial concentration, denoted πΆ π, as equilibrium
is approached. πΆ π, which represents the
starting concentration of our weak acid, is seen in all the possible answers. So, for our weak acid, weβll take
the initial concentration before any dissociation occurs as being πΆ π. For this scenario, the
concentrations of H+ and Aβ would both be zero as no dissociation has taken place
yet. For the weak acid, the change in
concentration as equilibrium is attained would be negative πΆ π multiplied by
πΌ. Here, we are adjusting the original
concentration of HA by the proportion of HA that dissociates or πΌ to find the new
concentration at equilibrium.

The change in concentration is
negative here as HA is being used up. Itβs a reactant in this
process. The changes in the concentrations
of our product ions H+ and Aβ are both πΆ π times πΌ, respectively. These changes in concentrations
both take positive values as theyβre being formed from zero. The changes of these product ion
concentrations are identical to each other due to the stoichiometry of the reaction
occurring. For each species, by adjusting the
initial concentration by the change in concentration, we can get the equilibrium
concentrations. The equilibrium concentration for
HA becomes πΆ π subtract πΆ π multiplied by πΌ. Since the initial concentrations of
H+ and Aβ were both zero initially, the adjusted concentrations are both πΆ π
multiplied by πΌ.

Now letβs focus more closely on the
equilibrium concentrations for HA, H+, and Aβ. Weβve got them in terms of πΆ π
and πΌ, but we can simplify them a little further. When factorized, the equilibrium
concentration for HA simplifies to πΆ π multiplied by one minus πΌ. Since we now have our equilibrium
concentrations in terms of πΆ π, the initial concentration of the acid, and πΌ, the
degree of dissociation, weβll substitute these values into our πΎ π expression. Weβll substitute the concentration
of HA into the bottom of the πΎ π expression. And weβll substitute the
concentrations of H+ and Aβ into the top half of the πΎ π expression.

We can now see more clearly the
relationship between the acid dissociation constant πΎ π, the initial concentration
of the weak acid, and the degree of dissociation πΌ. To simplify this expression a
little further, we can cancel πΆ π on the top and bottom half of this fraction. This gives us πΎ π equals πΌ
squared multiplied by πΆ π all divided by one minus πΌ. But weβre also told in the question
that one minus πΌ is approximately equal to one. This means that for a weak acid,
the degree of dissociation or πΌ is a very small quantity indeed. If one minus πΌ is approximately
equal to one, our πΎ π expression can be further simplified.

Our πΎ π expression therefore
simplifies to πΌ squared multiplied by πΆ π divided by one. So πΎ π equals πΌ squared
multiplied by πΆ π, which agrees with answer (E). The correct answer is πΎ π equals
πΌ squared multiplied by πΆ π.