Question

A compound was analyzed and was found to contain the following percentages by mass: phosphorus, 90.10\%; hydrogen 8.90\%. Determine the empirical formula of the compound.

Short answer

The empirical formula of the compound is \(PH_3\).

Step-by-step solution

Convert mass percentages into grams

Assume that we have 100 grams of the compound, which makes it easier to work with the percentages given. With this assumption, we have 90.10 grams of phosphorus and 8.90 grams of hydrogen.

Convert grams into moles

Next, we need to convert the grams of each element into moles using their molar masses. The molar mass of phosphorus (P) is 30.97 g/mol, and the molar mass of hydrogen (H) is 1.01 g/mol. Using their givens gram values: Moles of phosphorus (P) = 90.10 g / 30.97 g/mol ≈ 2.91 moles Moles of hydrogen (H) = 8.90 g / 1.01 g/mol ≈ 8.81 moles

Find the mole ratio

To find the empirical formula, we need to find the simplest whole-number ratio of moles of each element. We can do this by dividing both mole values by the smallest value: Mole ratio of P:H = (2.91 moles / 2.91 moles) : (8.81 moles / 2.91 moles) ≈ 1 : 3

Determine the empirical formula

Now that we have the simplest whole-number ratio of the moles of each element (1:3), we can write the empirical formula as \(PH_3\). So, the empirical formula of the compound is \(PH_3\).

Key concepts

Understanding Mass Percentage

To determine the empirical formula of a compound, one of the first steps is to understand the concept of mass percentage. It represents the percent by mass of each element in a compound and is crucial for empirical formula calculations. By understanding mass percentage, you're essentially identifying how much of each element is present in a compound by weight. This is expressed as:

• The mass percentage of each element = (mass of the element in 1 mole of the compound / molar mass of the compound) × 100%.

For example, in the compound described, phosphorus represents 90.10% and hydrogen represents 8.90% of the compound. Assuming a 100-gram sample makes it simpler to convert these percentages directly into grams, thus leading to 90.10 grams of phosphorus and 8.90 grams of hydrogen. This conversion from percentage to mass is the precursor to further calculations like finding moles, which eventually help in deriving the empirical formula.

Calculating Molar Mass

Once you have converted the mass percentages into grams, the next key step is to understand molar mass, a critical concept in chemistry. Molar mass is the mass of one mole of a substance (usually expressed in g/mol). It represents the weight of Avogadro’s number of atoms of an element, or molecules of a compound, in grams. In our example,

• The molar mass of phosphorus (P) is approximately 30.97 g/mol.
• The molar mass of hydrogen (H) is approximately 1.01 g/mol.

To calculate the number of moles, you divide the mass of the element by its molar mass, which gives you the amount of substance in moles:

• Moles of phosphorus = 90.10 g / 30.97 g/mol ≈ 2.91 moles
• Moles of hydrogen = 8.90 g / 1.01 g/mol ≈ 8.81 moles

These values are essential as they are used in determining the mole ratio of the elements, which is crucial in finding the empirical formula.

Finding the Mole Ratio

The next step, after computing the moles of each element, is to find the mole ratio which helps in deducing the empirical formula of a compound. The mole ratio is the simplest whole-number ratio of moles of each element in a compound. Due to its simplicity, this ratio gives direct insight into the subscripts used in the empirical formula. To find it, follow these steps:

• Identify the smallest value among the moles calculated. Here, it is 2.91 for phosphorus.
• Divide each of the mole values by this smallest number, to simplify the ratio to the smallest possible whole numbers.

In this example, the steps are:

• Mole ratio of Phosphorus:Hydrogen = (2.91 moles / 2.91) : (8.81 moles / 2.91) ≈ 1:3

This ratio directly translates to the subscripts in the empirical formula, thus providing the empirical formula as \(PH_3\). The mole ratio is key to writing correct chemical formulas in chemistry.