In this screencast I am going to use heats of formation at 25 degrees C and then heat
capacities that are temperature dependent to calculate the heat of reaction at 250 degrees
C for the reaction that is indicated. The first step in this calculation is to use the
heats of formation at 25 degrees C to determine the heat of reaction at 25 degrees C. So what
we're going to do is determine the heat of reaction. This heat of reaction I am going
to switch now to kelvin since we are going to want to work in kelvin for the rest of
the calculations. The heat of reaction will be 2 times the heat of formation of HCN at
298 minus the heat of formation of nitrogen at 298 and minus the heat of formation of acetylene
also at 298. So it's products minus reactants and using the stoichiometric coefficient.
Now this is 0 because it's an element and heats of formation are the enthalpy change when
we go from elements to the particular species. So to make HCN it's reacting the elements: hydrogen
is H_2, carbon is graphite and nitrogen is N_2 to make HCN. Then we can look up in a table
the remaining values and we substitute in the heats of formation values. We ended with
the heat of reaction of 43.5 kJ/mol. This is at 298 K. Next we are going to take
advantage of the idea of state function to calculate the heat of reaction at 250 degrees
C. So I am going to represent this in a diagram. The idea is we are starting out with a mol
of nitrogen and a mol of acetylene. We are at 523 K. We are going to form 2 mol
of HCN at 523 K. We want to know what is the heat of reaction at 523 K.
So what we are going to do is let's cool the reactants down to 298. We can calculate that energy
change from heat capacities. Carry out the reaction to 298 and then heat them back up
to 523. So what I have drawn is an alternate pathway then in black where first we cool
down our reactants. There is going to be some enthalpy change. We are then going to carry
out the reaction at 298. This is the heat of reaction at 298 which we calculated. Then
we are going to heat the product back to 523. Call that delta H_3. The sum of the three delta
H's, the black pathway, is going to be the same as delta H for the blue pathway for the
reaction at 523. That is the value we wanted to determine. We have this value. We need
to calculate the changes from the heat capacities and the temperature change. Here are the heat
capacities as a function of temperature where the temperature is in Kelvin in these equations.
We will first calculate delta H_1. Delta H_1 is going to be integral from our starting
temperature, 523, to our final temperature, 298. The heat capacity of nitrogen plus the
heat capacity of acetylene. We are going to integrate over this temperature range. I am
going to substitute in the values for the heat capacities and I am going to combine terms.
I will combine this term plus this term for the first term in the integration. Likewise
for the other terms. So we have 4 terms that we are going to integrate. Then we've substituted
in the limits for each of the terms. Now we will do the math to calculate each of these
terms. This gives us a value for delta H_1 of -18,362 J which is -18.4 kJ. Now we are
going to do the same calculation for delta H_3. Now we have two mol of HCN and we are
now heating from 298 to 523. We only have 1 species which is HCN. We are going to substitute
in the value for the heat capacity. Here is the heat capacity for HCN. We are going to
now do the same kind of integration as we did before and substitute in the limits. The
value of delta H_3 is 17,580 J or 17.6, if I round off, kJ. Delta H of reaction at 523 K is delta
H_1 plus delta H of reaction at 298 plus delta H_3. I can substitute these three values in.
Delta H reaction at 523 is 43.0 kJ. This is per mol of acetylene reacting. Notice it
is not very different from the 43.8 at 298. This is fairly common that these two heat
capacity terms almost cancel out. It is often that we assume the heat of reaction is not
a function of temperature. For this temperature range of 225 degrees C there is only a 1.8%
difference in the heat of reaction assuming delta H of reaction is constant is not a bad
assumption in many cases.