Question

The fertilizer ammonium sulfate \(\left[\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\right]\) is prepared by the reaction between ammonia \(\left(\mathrm{NH}_{3}\right)\) and sulfuric acid: $$ 2 \mathrm{NH}_{3}(g)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}(a q) $$ How many kilograms of \(\mathrm{NH}_{3}\) are needed to produce $$ 1.00 \times 10^{5} \mathrm{~kg} \text { of }\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4} ? $$

Short answer

25,760.86 kg of NH3 are needed.

Step-by-step solution

Calculate the molar masses

First, calculate the molar mass of the compounds involved. \(\text{Molar mass of }(\mathrm{NH}_4)_2\mathrm{SO}_4 = 2 \times 14.01 + 8 \times 1.01 + 32.07 + 4 \times 16.00 = 132.14\, \text{g/mol}\). \(\text{Molar mass of } \mathrm{NH}_3 = 14.01 + 3 \times 1.01 = 17.03\, \text{g/mol}\).

Convert output mass to moles

Calculate the moles of \((\mathrm{NH}_4)_2\mathrm{SO}_4\) needed: \(\frac{1.00 \times 10^5\, \text{kg}}{132.14\, \text{g/mol}} \times 1000\,\text{g/kg} = 756,007.57\, \text{mol}\).

Use stoichiometry to find moles of NH3

The balanced reaction shows 2 moles of \(\mathrm{NH}_3\) produce every mole of \((\mathrm{NH}_4)_2\mathrm{SO}_4\). Therefore, \(756,007.57\, \text{mol}\, \times 2 = 1,512,015.14\, \text{mol}\, \mathrm{NH}_3\) are required.

Convert moles of NH3 to kilograms

Finally, convert the moles of \(\mathrm{NH}_3\) to mass: \(1,512,015.14\, \text{mol} \times 17.03\, \text{g/mol} \approx 25,760,855.89\,\text{g}\). Convert grams to kilograms: \(\frac{25,760,855.89\, \text{g}}{1000} \approx 25,760.86\, \text{kg}\).

Key concepts

Molar Mass

In chemistry, molar mass is a critical concept when dealing with reactions and stoichiometry. Molar mass is defined as the mass of one mole of a substance, given in grams per mole (g/mol). It serves as a bridge between the atomic world and the macroscopic world.
To calculate the molar mass of a compound, sum the atomic masses of all the atoms in its molecular formula. For example, in the compound ammonium sulfate \[(\mathrm{NH}_4)_2\mathrm{SO}_4,\]we calculate the molar mass by adding the molar masses of nitrogen, hydrogen, sulfur, and oxygen, resulting in \[2 \times 14.01 + 8 \times 1.01 + 32.07 + 4 \times 16.00 = 132.14 \, \text{g/mol}.\]Understanding molar mass allows us to convert between mass and moles, which is crucial for quantifying substances in a reaction.

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, transform into different substances, called products. These transformations involve breaking chemical bonds in the reactants and forming new bonds in the products.
Reactions are represented by balanced chemical equations that show the relations between the reactants and products in terms of moles. For example, in the reaction:\[2 \mathrm{NH}_3(g) + \mathrm{H}_2\mathrm{SO}_4(aq) \longrightarrow (\mathrm{NH}_4)_2 \mathrm{SO}_4(aq),\]two moles of ammonia react with one mole of sulfuric acid to produce one mole of ammonium sulfate.
Balancing equations is vital because it maintains the law of conservation of mass, ensuring everything accounted for initially in reactants shows up somewhere as a part of products.

Ammonium Sulfate

Ammonium sulfate, with the chemical formula \[(\mathrm{NH}_4)_2\mathrm{SO}_4,\]is a salt commonly used as a fertilizer in agriculture. It supplies nitrogen, an essential element for plant growth, in the form of ammonium ions, and sulfur in the form of sulfate ions.
By providing these nutrients, ammonium sulfate helps to improve the soil's nutrient content and improve crop yields.
In chemical equations, ammonium sulfate is often a product of reactions involving ammonia and sulfuric acid, as shown in its synthesis \[2 \mathrm{NH}_3(g) + \mathrm{H}_2\mathrm{SO}_4(aq) \longrightarrow (\mathrm{NH}_4)_2\mathrm{SO}_4(aq).\] This compound exemplifies the application of chemical reactions in industry and agriculture to create useful substances.

Conversion of Units

The conversion of units is a crucial skill in both basic and advanced chemistry. It allows us to switch between different measurement units, making it easier to compare and use data in calculations.
In stoichiometry, the conversion usually involves converting mass to moles, using molar mass as a conversion factor. For example, to find how many moles of a compound we have in a given mass, divide the mass in grams by the molar mass in g/mol.
Furthermore, converting between grams and kilograms requires understanding metric prefixes: 1 kilogram equals 1000 grams.

• For instance, converting 25,760,855.89 grams to kilograms is done by dividing by 1000, resulting in approximately 25,760.86 kg.

Mastering unit conversions ensures accurate calculations in chemical equations and practical applications.