[Win,odd Dhamnekar]

How many cubic feet of air at a pressure of 760 torr and 0 °C is required per ton of Fe₂O₃ to convert that Fe₂O₃ into iron in a blast furnace? For this exercise, assume air is 19% oxygen by volume.

My answer:
In a blast furnace, iron(III) oxide (Fe₂O₃) is reduced to iron by carbon monoxide (CO) according to the following balanced chemical equation:

[tex]Fe_2O_3 + 3CO \to 2Fe + 3CO_2[/tex]

This means that for every mole of Fe₂O₃ converted to iron, 3 moles of CO are required.

The molar mass of Fe₂O₃ is 159.69 g/mol. One ton of Fe₂O₃ is equivalent to 907185 g, so the number of moles of Fe₂O₃ in one ton is 907185 g / 159.69 g/mol = 5680.9 mol.

Since 3 moles of CO are required for every mole of Fe₂O₃, the number of moles of CO required to convert one ton of Fe₂O₃ into iron is 5680.9 mol * 3 = 20000 mol.

At a pressure of 760 torr and a temperature of 0 °C (273.15 K), the volume occupied by one mole of an ideal gas is given by the ideal gas law: V = nRT/P, where n is the number of moles, R is the ideal gas constant (62.3637 L·torr/K·mol), T is the temperature in kelvins, and P is the pressure in torr.

Substituting the values for n, R, T, and P into the ideal gas law gives the volume occupied by 20000 mol of CO at a pressure of 760 torr and a temperature of 0 °C: V = (20000 mol)(62.3637 L·torr/K·mol)(273.15 K) / (760 torr) = 400000 L.

Since air is assumed to be 19% oxygen by volume and oxygen is converted to CO in the blast furnace, the volume of air required to provide this amount of CO is 400000 L / 0.19 = 2000000 L.

There are approximately 28.317 liters in one cubic foot, so this volume in cubic feet is approximately 70000 cubic feet.

In my opinion, this answer seems to look correct. Do you agree with me?

[Hunter2]

Its missing the reaction 2 C + O2 => 2 CO. Its not asked how much CO is used, Its asked the air used where the oxygen was obtained.

[Borek]

Short ton, long ton, or metric ton?

For this exercise, assume air is 19% oxygen by volume.

Rather surprising.

5680.9 mol * 3 = 20000 mol.

Really?

And as Hunter stated, CO/O₂ is not taken into account.

[Win,odd Dhamnekar]

5680.9 mol × 3 = 20000 mol using significant figures.

How can I include the reaction 2 C + O₂ 2CO into my answer  and how would the inclusion of that reaction  change the final answer?

[Borek]

5680.9 mol × 3 = 20000 mol using significant figures

No, you apply significant figures only to the final result, not to intermediate values.

[Win,odd Dhamnekar]

My corrected answer:

To solve this problem, we need to first determine the amount of air required to convert one ton of [itex]Fe_2O_3[/itex] into iron. From the first reaction, we see that for every 1 mole of [itex]Fe_2O_3[/itex] reacted, we need 3 moles of CO. From the second reaction, we see that for every 1 mole of CO, we need 1/2 mole of O₂. Therefore, for every mole of [itex]Fe_2O_3[/itex] reacted, we need 3/2 moles of O₂.

One ton of [itex]Fe_2O_3[/itex] is equal to 2000 pounds. The molar mass of [itex]Fe_2O_3[/itex] is 159.69 g/mol, so one ton of [itex]Fe_2O_3[/itex] is equal to 5680.9 moles. Therefore, we need 3/2 * 5680.9 = 8521.35 moles of O₂.

Since air is 19% oxygen by volume, we know that 1 volume of air contains 0.19 volumes of oxygen. Therefore, we need 8521.35 / 0.19 = 44849.21 volumes of air.

At standard temperature and pressure (STP), 1 mole of gas occupies 22.4 L. Therefore, 44849.21 volumes of air will occupy 44849.21 * 22.4 = 1004622.31 L of air.



Therefore, we need 1004622.31/28.317=35478 cubic feet of air at a pressure of 760 torr and 0°C per ton of [itex]Fe_2O_3[/itex] to convert that [itex]Fe_2O_3[/itex] into iron in a blast furnace.

Now, in my opinion, this answer seems to look correct. Isn't it? what would you say about this answer?

[Enthalpy]

Much unburnt CO exits the blast furnace. Whoever asked that question ignores it.

You use more digits in the intermediate steps, good, but in reasonable amounts please. 9 digits don't fit a blast furnace. 1kg=2lbs and 5.7*3=20 aren't correct neither. You can compute with 1 or 2 places more than the final result.

[Borek]

You can compute with 1 or 2 places more than the final result.

Typically when using calculator you just use whatever it displays, if it is nine digits - so be it.

Copying them all here is a waste of time though, on that we can agree.