Question

For the thermochemical equation \(\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{~s})+3 \mathrm{SO}_{3}(\mathrm{~g}) \rightarrow \mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) $$ \text { (s) } \Delta H=-570.2 \mathrm{~kJ} $$ what mass of \(\mathrm{SO}_{3}\) is needed to generate \(1,566 \mathrm{~kJ} ?\)

Short answer

You need approximately 659.77 grams of SO3 to generate 1566 kJ.

Step-by-step solution

Understand the given thermochemical equation

The given thermochemical equation is \( \mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{~s})+3 \mathrm{SO}_{3}(\mathrm{~g}) \rightarrow \mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}(s) \). The enthalpy change \( \Delta H \) for this reaction is \(-570.2 \mathrm{~kJ} \). This means that 570.2 kJ of energy is released when 1 mole of \( \mathrm{Fe}_{2} \mathrm{O}_{3} \) reacts with 3 moles of \( \mathrm{SO}_{3} \). We need to find the mass of \( \mathrm{SO}_{3} \) required to release 1566 kJ.

Calculate moles of SO3 needed for given energy

First, determine how many moles of \( \mathrm{SO}_{3} \) are needed to release 1566 kJ of energy. According to the given equation, 3 moles of \( \mathrm{SO}_{3} \) release 570.2 kJ. The relationship can be set up as: \( 3 \text{ moles of } \mathrm{SO}_{3} \rightarrow 570.2 \mathrm{~kJ} \), thus, find the moles for 1566 kJ: \(\text{moles of } \mathrm{SO}_{3} = \frac{3 \times 1566}{570.2} \approx 8.24 \text{ moles} \).

Calculate mass of SO3 needed

Now convert the moles of \( \mathrm{SO}_{3} \) to mass. The molar mass of \( \mathrm{SO}_{3} \) is the sum of the atomic masses: Sulfur (S) has a molar mass of 32.07 g/mol and Oxygen (O) has a molar mass of 16.00 g/mol. Thus,\( \text{Molar mass of } \mathrm{SO}_{3} = 32.07 + (3 \times 16.00) = 80.07 \text{ g/mol} \).Therefore, the mass needed is: \(\text{Mass of } \mathrm{SO}_{3} = 8.24 \text{ moles} \times 80.07 \text{ g/mol} \approx 659.77 \text{ grams} \).

Key concepts

Molar Mass Calculation

To find the mass of a chemical compound in a reaction, we first determine the molar mass. The molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). It's calculated by adding the atomic masses of all elements in the compound, based on the chemical formula.
The atomic mass of each element can be found on the periodic table. For instance, in our exercise involving \(\mathrm{SO}_{3}\), sulfur (S) has an atomic mass of 32.07 g/mol, and oxygen (O) has an atomic mass of 16.00 g/mol. Since \(\mathrm{SO}_{3}\) contains one sulfur atom and three oxygen atoms, its molar mass is:

• \(\text{Molar mass of } \mathrm{SO}_{3} = 32.07 + 3 \times 16.00 = 80.07 \text{ g/mol}\)

Now you have the molar mass, which is crucial for converting between the mass of a substance and the amount in moles during chemical reactions. This conversion is essential for accurate stoichiometric calculations.

Enthalpy Change

Enzymes play a key role in chemistry by describing the heat exchange during reactions. Specifically, it's the heat absorbed or released under constant pressure. In thermochemical equations, enthalpy change (\(\Delta H\)) indicates whether a reaction releases or absorbs energy.
For the equation \(\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})+3 \mathrm{SO}_{3}(\mathrm{g}) \rightarrow \mathrm{Fe}_{2}(\mathrm{SO}_{4})_{3}(s)\), \(\Delta H = -570.2 \text{ kJ}\), meaning the reaction exothermically releases 570.2 kJ when 3 moles of \(\mathrm{SO}_{3}\) react. The negative sign signifies energy release, making it an exothermic reaction.
If you need to find the energy required or released for a different quantity in the reaction, you adjust using the given \(\Delta H\) value, scaling appropriately. For example, to produce -1566 kJ of energy, you calculate the proportion from the specified \(\Delta H\). This is often followed by adjusting reactant masses using stoichiometry.

Stoichiometry

Stoichiometry is an essential concept that revolves around the quantitative relationships in chemical equations. It allows us to predict the amounts of substances consumed and produced in a reaction.
In our exercise, stoichiometry tells us that 3 moles of \(\mathrm{SO}_{3}\) produce 570.2 kJ, a vital stoichiometric relationship. When determining how much \(\mathrm{SO}_{3}\) is needed to release 1566 kJ, the stoichiometric ratio helps guide this conversion:

• \(3 \text{ moles of } \mathrm{SO}_{3} \rightarrow 570.2 \text{ kJ}\)
• For 1566 kJ: \(\text{moles of } \mathrm{SO}_{3} = \frac{3 \times 1566}{570.2} \approx 8.24 \text{ moles}\)

This calculation provides the moles of \(\mathrm{SO}_{3}\) needed, linking the energy output to physical quantities used in the experiment. Stoichiometry takes calculated moles to mass conversion using the molar mass to find practical quantities of reactants needed for given energy change, completing the task.