Question

Elemental phosphorus occurs as tetratomic molecules, \(\mathrm{P}_{4}\). What mass (g) of chlorine gas is needed to react completely with \(455 \mathrm{~g}\) of phosphorus to form phosphorus pentachloride?

Short answer

2610 g of Cl_2

Step-by-step solution

Write the Balanced Chemical Equation

First, write the balanced chemical equation for the reaction between elemental phosphorus (P\textsubscript{4}) and chlorine gas (Cl\textsubscript{2}) to form phosphorus pentachloride (PCl\textsubscript{5}): \[ \text{P}_4 + 10 \text{Cl}_2 \rightarrow 4 \text{PCl}_5 \]

Convert Mass of Phosphorus to Moles

Calculate the number of moles of phosphorus \textsubscript{4} using its molar mass. The molar mass of \textsubscript{4} is calculated as follows: \[ 4 \times 30.97 \text{ g/mol} = 123.88 \text{ g/mol} \] Therefore, convert the mass of phosphorus to moles: \[ \frac{455 \text{ g}}{123.88 \text{ g/mol}} = 3.673 \text{ mol} \text{ P}_4 \]

Determine the Moles of Chlorine Needed

Use the stoichiometry from the balanced equation to determine the moles of chlorine gas needed. From the balanced equation, 1 mole of \textsubscript{4} reacts with 10 moles of Cl\textsubscript{2}: \[ 3.673 \text{ mol} \text{ P}_4 \times 10 \frac{\text{mol Cl}_2}{\text{mol P}_4} = 36.73 \text{ mol} \text{ Cl}_2 \]

Convert Moles of Chlorine to Mass

Calculate the mass of chlorine gas needed using its molar mass (70.90 g/mol): \[ 36.73 \text{ mol} \text{ Cl}_2 \times 70.90 \text{ g/mol} = 2605.97 \text{ g} \]

Finalize the Answer

Round the final answer to reflect appropriate significant figures based on input data. Given input mass has three significant figures, the final answer should thus be reported as: \[ 2610 \text{ g} \]

Key concepts

balanced chemical equation

Before you can start any stoichiometric calculations, you need a balanced chemical equation. This step ensures that the law of conservation of mass is followed in your reaction.
A balanced chemical equation has equal numbers of each type of atom on both sides of the reaction.

For instance, in the reaction between phosphorus (P\textsubscript{4}) and chlorine gas (Cl\textsubscript{2}) to form phosphorus pentachloride (PCl\textsubscript{5}), the balanced equation is:
\[ \text{P}_4 + 10 \text{Cl}_2 \rightarrow 4 \text{PCl}_5 \]

This equation tells us that 1 molecule of P\textsubscript{4} reacts with 10 molecules of Cl\textsubscript{2} to form 4 molecules of PCl\textsubscript{5}. Balancing the equation is crucial for accurate calculations.

molar mass

The molar mass is the mass of one mole of a substance, and it is usually expressed in grams per mole (g/mol). Knowing the molar mass allows you to convert between mass and moles, which is a key step in stoichiometric calculations.

You can find the molar mass of a molecule by adding up the atomic masses of each element in the molecule.

• For example, the molar mass of phosphorus (P\textsubscript{4}) is \[4 \times 30.97 \text{ g/mol} = 123.88 \text{ g/mol}\]


• Similarly, the molar mass of Cl\textsubscript{2} is \[2 \times 35.45 \text{ g/mol} = 70.90 \text{ g/mol}\]

With these molar masses, you can move on to calculating the moles involved in the reaction.

mole conversions

Mole conversions are necessary when you need to switch between the mass of a substance and the number of moles, or vice versa. This step is fundamental in reaction stoichiometry.

• First, convert the mass of phosphorus to moles: \[ \frac{455 \text{ g}}{123.88 \text{ g/mol}} = 3.673 \text{ mol} \text{ P}_4 \]


• Next, use the stoichiometric relationship from the balanced equation to find the moles of chlorine needed: \[3.673 \text{ mol} \text{ P}_4 \times 10 \frac{\text{mol Cl}_2}{\text{mol P}_4} = 36.73 \text{ mol} \text{ Cl}_2 \]

These conversions ensure you're correctly applying the balanced equation to real-world scenarios.

stoichiometric calculations

Stoichiometric calculations use the balanced chemical equation to find out how much of each reactant is necessary or how much of each product will be formed. They follow a series of straightforward steps that you can use to solve many types of chemical problems.
Here’s how we would proceed with our example:

• We calculated the moles of P\textsubscript{4} and Cl\textsubscript{2} needed.
  
• Finally, we converted the moles of Cl\textsubscript{2} to grams: \[36.73 \text{ mol} \text{ Cl}_2 \times 70.90 \text{ g/mol} = 2605.97 \text{ g} \]

The result is 2610 grams of Cl\textsubscript{2} (rounded to three significant figures).
By mastering these calculations, you can predict the outcomes of reactions and determine the quantities of reactants and products accurately.