How can I find the heat that will be released in kJ per gram for the reaction of Al with Fe2O3?

Question

Find the heat that will be released in kJ per gram when aluminium ($\ce{Al}$) reacts with $\ce{Fe2O3}$ as follows: $$\ce{2Al(s) + Fe2O3(s) → Al2O3(s) + 2Fe(l)}$$

The information given is:

• $\Delta H = 12.40 \space \rm{kJ/mol}$ (for $\ce{Fe}$)
• $\Delta H = -822.2 \space \rm{kJ/mol}$ (for $\ce{Fe2O3}$)
• $\Delta H = -1669.8 \space \rm{kJ/mol}$ (for $\ce{Al2O3}$)

If it asked me to find the heat released in kJ/mol that would be easy:

$$ \Delta H^\circ = \sum \nu_{i}\Delta H^\circ_f (\rm{products}) - \sum \nu_{i}\Delta H^\circ_f (\rm{reactants}) $$

I assumed that there is one mole and I found the mass in grams by multiplying the molar mass of each compound by one mole.

Then I divided the $H^\circ_f$'s by those amounts to get the new values "expressed in kJ per gram"

Then I used the previous equation and I got an answer around 10 but the correct answer is 15.24 kJ/g.

The question is: is the way I used wrong, or is there something else I didn't pay attention to?

Answer 1

When you converted the heats to kJ/g, you inadvertently "broke" the Hess's law equation that you plugged them into.

Each of the enthalpies of formation in that equation is multiplied by the number of moles of the matching species, and so the units of the enthalpy of formation are kJ/mol.

If you assume one mole of product and convert that to grams, you then would have to convert all of the other species to grams (two moles of Al, two moles of Fe, etc).

It's easier to just do the whole calculation in moles, then divide the final answer by grams. However, you need to specify which species is the gram basis. In other words, you are asking for the heat per gram, but per gram of what?

Whatever the answer is, then find the total heat output in terms of kJ/mol of that substance, then divide it by the molar mass of the substance (in g/mol).

One way to avoid these kinds of problems is to always explicitly include the units in any calculation that you do. In this case, you would have quickly seen that the Hess's Law equation would not give you the correct units.

Answer 2

I'm guessing that you're looking for the heat released per gram of aluminum.

You're correct to here

ΔH° = Σ(ν × ΔHf°) (products) - Σ(ν × ΔHf°) (reactants)

Since there are 2 moles of aluminum, you need to divide this ΔH° by 2, then you have heat released in kJ/mol of aluminum.

To convert this to kJ/gram you need to now divide by the atomic weight of aluminum (26.98 gm/mol).

When I did this I got an answer of 15.25 kJ/gm aluminum heat released.