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. What is its empirical formula?

Short answer

The empirical formula is \( \mathrm{B}_5\mathrm{H}_7 \).

Step-by-step solution

Assume 100g of Compound

Since the problem provides percentages, we begin by assuming that the total mass of the compound is 100 grams. This means there are 88.5 grams of boron (B) and 11.5 grams of hydrogen (H).

Calculate Moles of Boron

The atomic mass of boron (B) is approximately 10.81 g/mol. To find the moles of boron, use the formula:\[\text{Moles of } B = \frac{88.5 \, g}{10.81 \, g/mol} \approx 8.186 \, mol\]

Calculate Moles of Hydrogen

The atomic mass of hydrogen (H) is approximately 1.01 g/mol. To find the moles of hydrogen, use the formula:\[\text{Moles of } H = \frac{11.5 \, g}{1.01 \, g/mol} \approx 11.386 \, mol\]

Determine the Mole Ratio

To find the simplest whole number ratio, divide each mole value by the smallest number of moles:\[\text{Ratio of B} = \frac{8.186}{8.186} = 1\]\[\text{Ratio of H} = \frac{11.386}{8.186} \approx 1.39\]

Adjust to Whole Numbers

Since 1.39 is close to 1.4, multiply both ratios by 5 to convert them to whole numbers:\[\text{Whole ratio of B} = 1 \times 5 = 5\]\[\text{Whole ratio of H} = 1.39 \times 5 = 7\]Thus, the empirical formula is \(\mathrm{B}_5\mathrm{H}_7\).

Key concepts

Boron-Hydrogen Compounds

Boron-hydrogen compounds are fascinating substances due to their unique bonding and structural properties. These compounds are known as boranes, and they are a group of organoboron chemical substances that are formed between boron and hydrogen. General formulas for these compounds generally look like \( \mathrm{B}_x \mathrm{H}_y \). Boranes are interesting because they showcase unusual electron deficiency, which refers to the fact that traditional electron-sharing does not fully describe these structures.
Boron atoms in these compounds often bond in multi-centered structures where electrons are shared over more than two atoms, unlike traditional two-center, two-electron bonds. Because of this electron-sharing anomaly, boranes have complex three-dimensional structures involving triangular, pentagonal, or other polyhedral formats. Understanding boron-hydrogen compounds helps scientists design materials with specific properties useful in various industries, including rocket fuels and cryogenics.
These unique bonds and structures also make boron-hydrogen compounds a significant area of study in coordination chemistry, leading to innovations and diverse applications.

Mole Ratio

Mole ratio plays a crucial role in determining the empirical formula of a compound. It is essentially the ratio between the amounts of moles of elements in a compound. To find this ratio, you calculate how many moles of each element are present in a given sample. Then, you divide each mole value by the smallest mole value among them. This process helps in determining the simplest, whole number ratio of elements in a compound.
This ratio is crucial for the understanding of chemical formulas at the empirical level, which represents the simplest form of a compound, rather than its actual molecular structure. Let's consider the example in question, where we calculated the mole ratio between boron and hydrogen. By dividing the moles of boron and hydrogen by the smallest common mole value, we obtained a simplified ratio. This simple comparison (mole ratio) then guided us to adjust into whole numbers, helpful in revealing the empirical formula \( \mathrm{B}_5\mathrm{H}_7 \).
It's vital to remember that empirical formulas offer the most reduced ratio, not necessarily the actual ratio of atoms within a given molecular compound. Understanding how to determine mole ratios empowers better insight into these fundamental components of chemistry.

Percentage Composition Analysis

Percentage composition analysis is a valuable tool that determines what percentage of each element contributes to the whole compound. It provides insights into the structure and proportions of elements within compounds. The process begins by assuming a total mass (often, conveniently, 100 grams) for easy calculations. This simulates the real-world scenarios where these compounds might be weighed and analyzed.

• The amounts of each element are converted to grams based on the given percentage, which is translated into moles by dividing by the respective atomic masses of the elements.
• This conversion to moles is key because chemical reactions and formulas are mole-based, highlighting the interconnectedness of these concepts.

Once mole quantities are determined, these values are used to compute the simplest mole ratio, revealing the empirical formula of the compound.
In the context of our example with boron and hydrogen, percentage composition analysis told us that 88.5% of the compound was boron, while the rest was hydrogen. Knowing the atomic masses of boron (10.81 g/mol) and hydrogen (1.01 g/mol), this information then laid the groundwork to find the empirical formula \( \mathrm{B}_5\mathrm{H}_7 \).
Mastering percentage composition analysis is essential for anyone deeply invested in chemistry and contributes significantly to analytical procedures needed across scientific research and industrial applications.