In this example we are going to calculate the equilibrium conversion for a gas phase
reaction of methanol and water reacting to carbon dioxide and hydrogen. The reaction
is at 2 bar, and the water to methanol ratio is 1.5 coming into the reactor. The question
is, what is the equilibrium conversion at 80 degrees C if we assume these are ideal gases?
This means you are going to have to calculate first the equilibrium constant, and so it's a
multi-step problem. The first step is to calculate the Gibbs free energy change at 25 degrees
C for this reaction using the delta G of formations for each of the components, and we look these
up in a table. The second step is to use the delta G of reaction at 25 degrees C to determine
the equilibrium constant at 25. So I will call this K1 to represent the value at 25
degrees C. Next we want to calculate the equilibrium constant at 80 degrees C. So first we are going
to assume that the heat of reaction is independent of temperature. This is a pretty good approximation
because the temperature range is pretty small here. So we are going to calculate the heat
of reaction at 25 from the heats of formation at 25 degree C, and again these are values
we look up in a table, and then we are going to use the Van't Hoff equation to get the equilibrium constant
at 80 degrees C from the equilibrium constant at 25 degrees C and the heat of reaction, and
then finally we are going to use that equilibrium constant in an expression that relates the
activities of each of the species raised to their stoichiometric coefficient. The product
of the activities to the equilibrium constant. So what I will first do is look up the values
for the delta Gs and delta Hs of formation for each of the species. So I have listed the
values for delta H and delta G, and notice that for hydrogen delta G and delta H are
zero, because remember when we say heats of formation we mean what is the energy to form this species
from the elements, and since hydrogen is elementary and is at standard state then delta H and delta
G of formation are zero for hydrogen. So the first step is to calculate delta G of the
reaction at 25 from delta G of CO2, and this is all at 25, plus 3 times delta G of hydrogen
minus delta G of methanol minus delta G of water. So the stoichiometric coefficients
are used where they're negative for reactants and positive for products. These are all at 25
degrees C. So these are all delta Gs of formation. So I will substitute the values from the table.
So I have made the substitution and calculated delta G to minus 3.6 kJ/mol. I am now going
to use this delta G of reaction at 25 degrees C to calculate the equilibrium constant at
25 degrees C, and there is a relationship, the equilibrium constant is the exponential of
the change in the Gibbs free energy at 25 over RT, Gibbs free energy change at standard conditions.
I will substitute in the numbers, and I want to be careful with the units. So first thing that's important
is the minus sign here. It is minus delta G over RT. So we have minus, minus 3.6, which I am
going to convert to joules per mole, and then I can use 8.314 joules per mole per Kelvin, the gas constant, and
then the temperature is absolute temperature 273 plus 25 to make it Kelvin. And all this is in the
exponential. When I do that I get the equilibrium constant 4.28 for an equilibrium constant
at 25, and I am going to call that K1 to make the next calculations a little easier. So
as I indicated earlier, in order to get to the equilibrium constant at another temperature
I first have to calculate the heat of reaction, and the format is the same as calculating
the delta G of reaction at 25 degrees C, and I am going to calculate the heat of reaction
at 25 degrees C. So I have written down the heat of reaction at 25 in terms of
heat of formations of products minus heat of formation of reactants, and I have substituted
in the values from the table. So we get the heat of reaction 49.2 kJ/mol, and this is
positive, so it is endothermic. We are going to assume constant over a narrow temperature
change, and we can then apply the Van't Hoff equation to relate the equilibrium constant at
two temperatures. So the Van't Hoff equation says the natural log equilibrium constant
at temperature 2 over the equilibrium constant at temperature 1 is related to the negative
heat of reaction, which we're assuming is constant, over the gas constant 1 over temperature 2 minus
1 over temperature 1, where these must be absolute temperatures. So I can substitute in the values.
So when I have substituted in the values note that I converted the kJ to J, put in the correct
gas constant and units. Put in the temperatures, temperature T2 is 80 degrees C or 353 Kelvin, and
temperature T1 is 25 degrees C, 298 Kelvin. I know K1, that is 4.28. So I can solve this
equation for K2. So this is equilibrium constant at 80 degrees C. Remember the equilibrium constant
is dimensionless, and then I can substitute in the equilibrium constant in terms of the
activities raised to their stoichiometric coefficients, and then in the next screencast
part 2 we'll apply this equation to determine the equilibrium conversion.