Question

Potassium nitrate decomposes on heating, producing potassium oxide and gaseous nitrogen and oxygen: $$ 4 \mathrm{KNO}_{3}(s) \longrightarrow 2 \mathrm{~K}_{2} \mathrm{O}(s)+2 \mathrm{~N}_{2}(g)+5 \mathrm{O}_{2}(g) $$ To produce \(56.6 \mathrm{~kg}\) of oxygen, how many (a) moles of \(\mathrm{KNO}_{3}\) and (b) grams of \(\mathrm{KNO}_{3}\) must be heated?

Short answer

(a) 1415 mol KNO\textsubscript{3}, (b) 143.05 kg

Step-by-step solution

Balanced Chemical Equation

Identify the balanced chemical equation for the decomposition of potassium nitrate: \[4 \text{KNO}_{3} \rightarrow 2 \text{K}_{2} \text{O} + 2 \text{N}_{2} + 5 \text{O}_{2} \]

Molar Mass of Oxygen

Find the molar mass of oxygen (O\textsubscript{2}). The molar mass of one oxygen atom (O) is approximately 16.00 g/mol. Thus, the molar mass of O\textsubscript{2} is:\[2 \times 16.00 = 32.00 \text{ g/mol} \]

Convert Mass of Oxygen to Moles

Convert the given mass of oxygen produced (56.6 kg) to moles using the molar mass of O\textsubscript{2}:\[56.6 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} \times \frac{1 \text{ mol}}{32.00 \text{ g}} = 1768.75 \text{ mol O}_{2} \]

Mole Ratio from Balanced Equation

Use the stoichiometric coefficients from the balanced equation to find the mole ratio between KNO\textsubscript{3} and O\textsubscript{2}. From the equation: \[4 \text{KNO}_{3} \rightarrow 2 \text{K}_{2} \text{O} + 2 \text{N}_{2} + 5 \text{O}_{2} \]4 moles of KNO\textsubscript{3} produce 5 moles of O\textsubscript{2}. Therefore, the ratio is:\[\frac{4 \text{ mol KNO}_{3}}{5 \text{ mol O}_{2}} \]

Calculate Moles of KNO\textsubscript{3}

Calculate the moles of KNO\textsubscript{3} needed to produce 1768.75 mol of O\textsubscript{2}.\[1768.75 \text{ mol O}_{2} \times \frac{4 \text{ mol KNO}_{3}}{5 \text{ mol O}_{2}} = 1415.00 \text{ mol KNO}_{3} \]

Molar Mass of KNO\textsubscript{3}

Determine the molar mass of KNO\textsubscript{3}. The molar mass is calculated as follows:\[(39.10 \text{ g/mol K}) + (14.01 \text{ g/mol N}) + (3 \times 16.00 \text{ g/mol O}) = 101.11 \text{ g/mol KNO}_{3} \]

Convert Moles of KNO\textsubscript{3} to Grams

Convert the moles of KNO\textsubscript{3} to grams using its molar mass:\[1415.00 \text{ mol KNO}_{3} \times 101.11 \text{ g/mol KNO}_{3} = 143049.65 \text{ g} \approx 143.05 \text{ kg} \]

Key concepts

Balanced Chemical Equation

Understanding a balanced chemical equation is crucial in any chemical reaction. A balanced equation has the same number of each type of atom on both sides of the reaction arrow. This means the law of conservation of mass is upheld, where matter is neither created nor destroyed. In our example, the equation for the decomposition of potassium nitrate is already balanced:
o $$ 4 \text{KNO}_{3} \rightarrow 2 \text{K}_{2}\text{O} + 2 \text{N}_{2} + 5 \text{O}_{2} $$
In this reaction, we have:

• 4 KNO₃ molecules producing 2 K₂O molecules
• 2 N₂ molecules
• 5 O₂ molecules

The balanced equation provides the basis for mole ratios, which are essential in stoichiometric calculations.

Mole Conversion

Mole conversion involves converting between mass, moles, and number of particles. Here, we need to convert the mass of oxygen (O₂) into moles. The molar mass of a molecule is the sum of the atomic masses of its atoms. Oxygen (O) has an atomic mass of approximately 16.00 g/mol. Since an O₂ molecule contains two oxygen atoms, its molar mass is:
$$ \text{Molar mass of } O_{2} = 2 \times 16.00 = 32.00 \text{ g/mol} $$
To convert 56.6 kg of O₂ to moles, we use:
o $$ 56.6 \text{ kg} \times \frac{1000 \text{ g}}{1 \text{ kg}} \times \frac{1 \text{ mol}}{32.00 \text{ g}} = 1768.75 \text{ mol O}_{2} $$
This conversion gives us the moles of O₂ we need to move to the next calculation.

Stoichiometry

Stoichiometry is the study of the quantitative relationships in a chemical reaction. It uses mole ratios from the balanced equation to relate the amounts of reactants and products. In our balanced equation, the mole ratio between KNO₃ and O₂ is:
$$ \frac{4 \text{ mol KNO}_{3}}{5 \text{ mol O}_{2}} $$
o For every 5 moles of O₂ produced, 4 moles of KNO2, we calculate:
o $$ 1768.75 \text{ mol O}_{2} \times \frac{4 \text{ mol KNO}_{3}}{5 \text{ mol O}_{2}} = 1415.00 \text{ mol KNO}_{3} $$ This step ensures that we have the correct amount of reactants for the desired amount of products.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance, calculated by summing the atomic masses of its elements. For KNO

Potassium (K): 39.10 g/mol

Nitrogen (N): 14.01 g/mol

Oxygen (O): 16.00 g/mol (each molecule has 3 atoms of Oxygen)

Adding these values:o $$ (39.10 \text{ g/mol K}) + (14.01 \text{ g/mol N}) + (3 \times 16.00 \text{ g/mol O}) = 101.11 \text{ g/mol KNO}_{3} $$
By knowing the molar mass, we can convert the moles of KNO_{3Thus, 1415.00 moles of KNO3}