Question

A large family of boron-hydrogen compounds has the general formula \(\mathrm{B}_{x} \mathrm{H}_{y}\). One member of this family contains \(88.5 \%\) B; the remainder is hydrogen. Which of the following is its empirical formula: \(\mathrm{BH}_{2}, \mathrm{BH}_{3}, \mathrm{B}_{2} \mathrm{H}_{5}, \mathrm{B}_{5} \mathrm{H}_{7}\) or \(\mathrm{B}_{5} \mathrm{H}_{11} ?\)

Short answer

The empirical formula is \( \mathrm{B}_{5} \mathrm{H}_{7} \).

Step-by-step solution

Calculate Mass of Hydrogen in 100g

If 88.5% of the compound is boron, then 11.5% is hydrogen. In 100g of the compound, this equates to 11.5g of hydrogen.

Calculate the Moles of Boron

The molar mass of boron (B) is approximately 10.8 g/mol. Therefore, the moles of boron in 88.5g of boron is given by \( \frac{88.5 \text{ g}}{10.8 \text{ g/mol}} = 8.194 \text{ mol} \).

Calculate the Moles of Hydrogen

The molar mass of hydrogen (H) is approximately 1 g/mol. Therefore, the moles of hydrogen in 11.5g of hydrogen is given by \( \frac{11.5 \text{ g}}{1 \text{ g/mol}} = 11.5 \text{ mol} \).

Determine Mole Ratio of Boron to Hydrogen

Calculate the mole ratio of boron to hydrogen by dividing each by the smallest number of moles:\[ \text{Boron: } \frac{8.194 \text{ mol B}}{8.194} = 1 \] \[ \text{Hydrogen: } \frac{11.5 \text{ mol H}}{8.194} \approx 1.4 \]

Find Approximate Whole Number Ratio

The approximate ratio is 1:1.4, which can be multiplied by 5 to give the smallest whole number ratio. This yields B: 5 and H: 7, suggesting an empirical formula of \( \mathrm{B}_{5} \mathrm{H}_{7} \).

Verify against Given Empirical Formulas

The empirical formula \( \mathrm{B}_{5} \mathrm{H}_{7} \) matches one of the given options. So, the compound's empirical formula is \( \mathrm{B}_{5} \mathrm{H}_{7} \).

Key concepts

Empirical Formula Calculation

Calculating the empirical formula of a compound involves determining the simplest ratio of atoms in that compound. For this, it's crucial to know the percentage composition of each element present. In the boron-hydrogen compound mentioned, it's given that 88.5% of the compound is boron. The rest, 11.5%, must be hydrogen. This information allows us to convert these percentages directly into grams, assuming a 100g sample:

• 88.5g of Boron (B)
• 11.5g of Hydrogen (H)

Once we have these masses, the next step is to convert them into moles. This is achieved by dividing the mass of each element by its respective molar mass. With a molar mass of approximately 10.8 g/mol for boron, and 1 g/mol for hydrogen, the conversion becomes straightforward. Thus, we calculate the number of moles as follows:

• Moles of Boron = \( \frac{88.5 \text{ g}}{10.8 \text{ g/mol}} \approx 8.194 \text{ mol} \)
• Moles of Hydrogen = \( \frac{11.5 \text{ g}}{1 \text{ g/mol}} = 11.5 \text{ mol} \)

This provides the initial data needed for determining the empirical formula.

Mole Ratio Determination

Determining the mole ratio is crucial for finding the empirical formula. After calculating the moles of each element, the next objective is to find the simplest whole number ratio between them. This involves dividing the number of moles of each element by the smallest number of moles obtained from the elements involved.

• Boron: Divide 8.194 mol by 8.194: \( \frac{8.194}{8.194} = 1 \)
• Hydrogen: Divide 11.5 mol by 8.194: \( \frac{11.5}{8.194} \approx 1.4 \)

This yields the ratio 1:1.4. To obtain whole numbers, we can multiply both values by a factor that will remove the decimal. In this case, multiplying by 5 yields 5:7, indicating that the empirical formula is \( \mathrm{B}_{5} \mathrm{H}_{7} \). This step is critical as it establishes the simplest set of whole numbers representing the proportions of each element.

Chemical Composition Analysis

Chemical composition analysis helps verify the empirical formula by comparing experimental data with known possibilities. Here, in confirming that the empirical formula \( \mathrm{B}_{5} \mathrm{H}_{7} \) matches one of the given options (\( \mathrm{BH}_{2}, \mathrm{BH}_{3}, \mathrm{B}_{2} \mathrm{H}_{5}, \mathrm{B}_{5} \mathrm{H}_{7}, \mathrm{B}_{5} \mathrm{H}_{11} \)), the calculated result must align with chemical principles and experimental results.Understanding chemical composition involves:

• Evaluating the mass percentages of known elements.
• Ensuring the calculated empirical formula is among the feasible options.
• Applying logical steps based on mole ratios to deduce correct formulas.

This analysis is vital to substantiate that the empirical formula selected not only fits our calculated data but also aligns with the given theoretical possibilities. Confirming \( \mathrm{B}_{5} \mathrm{H}_{7} \) validates the result within the practical context of the exercise, ensuring both numerical and theoretical consistency.