Problem 64 - Chemistry: Principles and Reactions

Question

Phosphine gas reacts with oxygen according to the following equation: $$ 4 \mathrm{PH}_{3}(g)+8 \mathrm{O}_{2}(g) \longrightarrow \mathrm{P}_{4} \mathrm{O}_{10}(s)+6 \mathrm{H}_{2} \mathrm{O}(g) $$ Calculate (a) the mass of tetraphosphorus decaoxide produced from \(12.43 \mathrm{~mol}\) of phosphine. (b) the mass of \(\mathrm{PH}_{3}\) required to form \(0.739 \mathrm{~mol}\) of steam. (c) the mass of oxygen gas that yields \(1.000 \mathrm{~g}\) of steam. (d) the mass of oxygen required to react with \(20.50 \mathrm{~g}\) of phosphine.

Step-by-Step Solution

Answer

Answer: The mass of tetraphosphorus decaoxide produced is 883.28 g.

Step 1: Write down the balanced chemical equation

The balanced chemical equation is given by: $$ 4 \mathrm{PH}_{3}(g)+8 \mathrm{O}_{2}(g) \longrightarrow \mathrm{P}_{4} \mathrm{O}_{10}(s)+6 \mathrm{H}_{2} \mathrm{O}(g) $$

Step 2: Calculate molar masses of the molecules involved

Note the molar masses for each molecule needed for the stoichiometry calculations. - PH3: 1P + 3H = (1 x 30.97) + (3 x 1.01) = 34.00 g/mol - P4O10: 4P + 10O = (4 x 30.97) + (10 x 16.00) = 283.88 g/mol - O2: 2O = (2 x 16.00) = 32.00 g/mol - H2O: 2H + 1O = (2 x 1.01) + (1 x 16.00) = 18.02 g/mol (a) Calculate the mass of tetraphosphorus decaoxide produced from 12.43 mol of phosphine

Step 3a: Use mole ratios to calculate the amount of P4O10 produced

The mole ratios from the balanced chemical equation are: 4 PH3 : 1 P4O10 Using the 12.43 mol of PH3: 12.43 mol PH3 x (1 mol P4O10 / 4 mol PH3) = 3.1075 mol P4O10

Step 4a: Calculate the mass of P4O10 produced

To find the mass of P4O10, multiply the moles of P4O10 by its molar mass: 3.1075 mol P4O10 x (283.88 g P4O10 / 1 mol P4O10) = 883.28 g P4O10 (b) Calculate the mass of PH3 required to form 0.739 mol of steam (H2O)

Step 3b: Use mole ratios to calculate the amount of PH3 needed

The mole ratios from the balanced chemical equation are: 6 H2O : 4 PH3 Using the 0.739 mol of H2O: 0.739 mol H2O x (4 mol PH3 / 6 mol H2O) = 0.4927 mol PH3

Step 4b: Calculate the mass of PH3 needed

To find the mass of PH3, multiply the moles of PH3 by its molar mass: 0.4927 mol PH3 x (34.00 g PH3 / 1 mol PH3) = 16.75 g PH3 (c) Calculate the mass of oxygen gas that yields 1.000 g of steam (H2O)

Step 5c: Calculate the moles of H2O from the mass given

To find the moles of H2O, divide 1.000 g H2O by its molar mass: 1.000 g H2O x (1 mol H2O / 18.02 g H2O) = 0.0555 mol H2O

Step 6c: Use mole ratios to calculate the amount of O2 needed

The mole ratios from the balanced chemical equation are: 6 H2O : 8 O2 Using the 0.0555 mol of H2O: 0.0555 mol H2O x (8 mol O2 / 6 mol H2O) = 0.0740 mol O2

Step 7c: Calculate the mass of O2 needed

To find the mass of O2, multiply the moles of O2 by its molar mass: 0.0740 mol O2 x (32.00 g O2 / 1 mol O2) = 2.368 g O2 (d) Calculate the mass of oxygen gas required to react with 20.50 g of phosphine

Step 5d: Calculate the moles of PH3 from the mass given

To find the moles of PH3, divide 20.50 g PH3 by its molar mass: 20.50 g PH3 x (1 mol PH3 / 34.00 g PH3) = 0.6029 mol PH3

Step 6d: Use mole ratios to calculate the amount of O2 needed

The mole ratios from the balanced chemical equation are: 4 PH3 : 8 O2 Using the 0.6029 mol of PH3: 0.6029 mol PH3 x (8 mol O2 / 4 mol PH3) = 1.2058 mol O2

Step 7d: Calculate the mass of O2 needed

To find the mass of O2, multiply the moles of O2 by its molar mass: 1.2058 mol O2 x (32.00 g O2 / 1 mol O2) = 38.59 g O2

Key Concepts

Molar Mass Calculation

Finally, molar mass calculation is a fundamental concept in stoichiometry. It's the mass of one mole of a substance, typically expressed in grams per mole (g/mol). To find the molar mass, we sum the atomic masses of all the atoms in the molecule, as seen in the step-by-step solution provided for each reactant and product of the reaction equation (1).

Calculating Molar Mass

For instance, the molar mass of phosphine (PH3) is calculated by adding the atomic mass of one phosphorus atom and three hydrogen atoms, resulting in 34.00 g/mol. Similarly, the molar mass of tetraphosphorus decaoxide (P4O10) is found by adding the atomic mass of four phosphorus atoms and ten oxygen atoms, which totals 283.88 g/mol.

In the given exercise, after determining the number of moles of the substance in question (phosphine, water, or oxygen, for example), the student would use the respective molar mass to convert this to grams.

By mastering the skill of molar mass calculation, students can efficiently switch between the mass of a substance and the amount in moles, which is instrumental in solving stoichiometry problems. With this understanding, students can apply the concept to a variety of chemical reactions to determine the mass of reactants needed or predict the mass of products formed.