In this case we're calculating chemical equilibrium, where we have a liquid, calcium, and a crystal
solid reacting to form two other crystals, two other solids, and we're given delta G
standard state conditions at 1000 degrees and 1200 degrees centigrade. We want to know
what is the maximum temperature where the calcium will actually reduce the thorium oxide,
to thorium metal, and a critical aspect is we're assuming all these phases are pure.
In other words the thorium solid does not mix with the calcium oxide solid, they're
in separate phases, and this dramatically effects the equilibrium. So we're going to
apply chemical equilibrium where equilibrium constant is the ratio of activities, so calcium
oxide squared, activity of thorium solid, activity of calcium squared, and then the
activity of thorium oxide. Because the activities are the ratio of fugacities and standard conditions
is one bar. Since they're pure components, pressure doesn't effect the activities very
much for solids or liquids. Each of the activities are equal to one, therefore the equilibrium
constant is one, and of course this is dimensionless. So the equilibrium constant is related to
the change in the Gibbs free energy at a given temperature which means one is equal to, and
so this means that delta G is zero at the conditions where we satisfy equilibrium. So
what we're going to do, delta G from it's definition is delta H minus T delta S. And
we're going to assume that we can write this equation at the two temperatures where we
know values for delta G and we assume they have the same delta H, namely delta H doesn't
change much with temperature, that's usually a pretty good assumption, that over smaller
temperature ranges heat of reaction doesn't change very much, we're also assuming here
that delta S is not changing very much since we're dealing with solids, so the delta S
for the reaction we'll assume is not changing very much. If we make those assumptions then
we can essentially write down two equations from our data here. We can write down at 1000
degrees C and I'll write this now in calories, is delta H minus temperature, in absolute
temperature and delta S, and then at 1200 degrees C, 2000 is the value that we're given,
same delta H, absolute temperature, same delta S, and we basically can solve these simultaneously
and we can calculate delta H as minus 12,320 calories per mole and delta S as minus 7 calories
per mole Kelvin. So we're looking for the temperature where delta G is zero. So zero
is equal to delta H minus T delta S and so therefore when we substitute in temperature
is 1760 Kelvin. So what this means is that at temperatures below this the reaction will
be in a forward direction and will reduce, so for T less than 1760 Kelvin we reduce the
thorium oxide, at higher temperatures the equilibrium constant decreases, the reaction
will go to the left because the activities are one for the pure components, reaction
will proceed to the left until there's no thorium left. So to get reduction we have
to be below 1760 Kelvin.