In this video I'm going to calculate the freezing point depression equation, namely derive the
equation that allows us to calculate what is the temperature decrease when we add a
solute to a pure solvent. So pure solvent will freeze at a temperature Tf, the freezing
temperature, and when we add a solute then the freezing temperature is decreased and
we'll refer to that as Td, now the solid that forms does not contain the solute, it's the
pure solvent now in solid form. Need a couple other terms to use in the derivation, so X1
is the mole fraction of the solvent X2 is the mole fraction of the solute, we're using
only two components so X2 is one minus X1. What we're interested in is at equilibrium,
at temperature Td. So I've written down the equilibrium equation, the solid is a pure
phase and so this is the fugacity of the pure solid, but at temperature Td, this is the
thing you have to keep in mind as you follow through this derivation, this is the mixture
fugacity for component 1, both cases component 1 being our solvent and so mole fraction
we defined, this is the activity coefficient for component 1, and then this is the pure
component fugacity, liquid phase at temperature Td. So something we want to keep in mind is
the fugacity of the liquid, pure component, does not equal the fugacity of the solid at
the temperature Td. At the freezing point of the pure solvent, Tf, these fugacities
would be equal. So we're going to take this equation relating fugacity to mole fraction,
so I'm going to take this equation, I'm going to rewrite it in terms of the ratio of fugacities,
and then I'm going to take the log of both sides of the equation, well the reason we're
doing this is because the log of the fugacity ratio we can relate to Gibbs free energies,
so this is now the Gibbs free energy of the solid, component 1, at temperature d, minus
the Gibbs free energy of the liquid, component 1, pure component at temperature d, divided
by R times the temperature d, well I can write this as delta G component one, at temperature
d of fusion, with a minus sign, over R Td, so delta G of fusion is when we take the solid
and we convert it to the liquid at the same temperature and so this is minus delta G of
fusion. And so what we're going to do is calculate the delta G of fusion in terms of delta H
and delta S, so I've written down, from the definition of delta G in terms of H, T and
S, where these are the changes, fusion, component one, at temperature d. So we can relate this
to delta H and delta S of fusion, at the freezing temperature of the pure solvent, Tf because
these are indeed values that we know. So what I've drawn is a diagram where we can take
advantage of the idea of state function, we're going from this point to this point, solid
at temperature Td to liquid, this is delta H of fusion, of the solvent, component one
at temperature d, what we're going to do is we're going to say first we'll take this
solid, raise its temperature to the freezing temperature, now convert it from solid to
liquid, where delta H of fusion, at the freezing temperature is a number that we know, and
then we're going to lower the temperature of the liquid back to the final Td, we can
imagine we'd do this, doesn't mean physically we're doing this, we just can imagine, because
now delta H of fusion at the temperature d is just the sum of one plus two plus three,
and so we can write the equation that describes that state function relationship. So what
I've written is that delta H of fusion, namely what we're trying to calculate, at temperature
d is equal to the term one plus two plus three where we assume constant heat capacities,
so pretty good assumption for a liquid particularly over relatively narrow temperature range.
Well we can simplify this equation, so I've just rearranged it and I've defined delta
Cp as this difference in heat capacity of the liquid and the solid just to make it easier
to write the equations down. Now I'm going to do the same thing for entropy change but
I won't go through the detailed calculations the same way, I'll just write down the answer.
So here's the same idea, state function, so I have in this case delta S for fusion of
the solvent, component 1, at the temperature d, is equal to delta S of fusion of the solvent
at its freezing temperature for a pure component and a delta Cp, a log term, and then I've
substituted for delta S of fusion, delta H of fusion over Tf, and so this is an equation
for all phase changes, delta S is delta H over the temperature where the phase change
takes place. So what I've done is just rewritten the equation from earlier, basically definition
of Gibbs free energy, and now I can substitute back this equation and this equation, and
just rearrange, and I'll just write down the final form of the equation after the rearrangement,
we're going to assume this term is small compared to the first two terms, and then we can, these
first two terms factor out fusion, and if I want to have a common denominator Tf, then
it becomes this where I've ignored these other terms. So delta G of fusion of the solvent
at the temperature d is related to delta H of fusion at the freezing point of the pure
solvent divided by that freezing point temperature and then this temperature difference. so I've
substituted in from the earlier equation and then I used the relation between delta G and
delta H and this temperature difference and so now we can make some simplifications, the
activity coefficient is approximately 1, which is a good approximation when X1 is close to
one and then we also will assume that in the denominator that Td is approximately Tf, the
freezing point depression is not a large number in general, when we're using this equation,
and so that's a reasonable assumption to make to simplify the equation, we'll make another
simplification, so the log of X1 is the same as the log of 1 minus X2, but this is approximately
equal to minus X2 when X2 is small, this is just a mathematical relationship. Now I can
write down this difference in temperature and then this equation says since delta H
of fusion, this is going from solid to liquid, so delta H of fusion is greater than zero,
so the right side is positive which means the left side is positive, which means that
the freezing point of the pure component is greater than the freezing point of the mixture,
so we have a freezing point depression. So this then is the equation that we can use
in calculating the freezing point depression.