Question

Diborane, \(\mathrm{B}_{2} \mathrm{H}_{6}\), can be prepared by the following reaction: $$ 3 \mathrm{NaBH}_{4}(s)+4 \mathrm{BF}_{3}(\mathrm{~g}) \longrightarrow 2 \mathrm{~B}_{2} \mathrm{H}_{6}(g)+3 \mathrm{NaBF}_{4}(s) $$ (a) How many moles of \(\mathrm{NaBH}_{4}\) react with \(1.299 \mathrm{~mol}\) of \(\mathrm{BF}_{3} ?\) (b) How many moles of \(\mathrm{B}_{2} \mathrm{H}_{6}\) can be obtained from \(0.893 \mathrm{~mol}\) of \(\mathrm{NaBH}_{4} ?\) (c) If \(1.987 \mathrm{~mol}\) of \(\mathrm{B}_{2} \mathrm{H}_{6}\) is obtained, how many moles of \(\mathrm{NaBF}_{4}\) are produced? (d) How many moles of \(\mathrm{BF}_{3}\) are required to produce \(4.992 \mathrm{~mol}\) of \(\mathrm{NaBF}_{4} ?\)

Short answer

Short answer: (a) 0.974 moles of \(\mathrm{NaBH}_{4}\) react with 1.299 moles of \(\mathrm{BF}_{3}\). (b) 0.595 moles of \(\mathrm{B}_{2} \mathrm{H}_{6}\) can be obtained from 0.893 moles of \(\mathrm{NaBH}_{4}\). (c) 2.9805 moles of \(\mathrm{NaBF}_{4}\) are produced when 1.987 moles of \(\mathrm{B}_{2} \mathrm{H}_{6}\) are obtained. (d) 6.656 moles of \(\mathrm{BF}_{3}\) are required to produce 4.992 moles of \(\mathrm{NaBF}_{4}\).

Step-by-step solution

Identify the relevant information from the equation

The balanced equation tells us that 3 moles of \(\mathrm{NaBH}_{4}\) react with 4 moles of \(\mathrm{BF}_{3}\). So, the molar ratio is \(\frac{3}{4}\).

Calculate moles of \(\mathrm{NaBH}_{4}\) that reacts with \(1.299 \mathrm{~mol}\) \(\mathrm{BF}_{3}\)

Using the molar ratio, we can calculate the moles of \(\mathrm{NaBH}_{4}\) that react with \(1.299 \mathrm{~mol}\) of \(\mathrm{BF}_{3}\): Moles of \(\mathrm{NaBH}_{4} =\) moles of \(\mathrm{BF}_{3}\times \frac{3}{4} = 1.299\times \frac{3}{4} =0.974\) moles. (b) How many moles of \(\mathrm{B}_{2} \mathrm{H}_{6}\) can be obtained from \(0.893 \mathrm{~mol}\) of \(\mathrm{NaBH}_{4}?\)

Identify the relevant information from the equation

The balanced equation tells us that 3 moles of \(\mathrm{NaBH}_{4}\) produce 2 moles of \(\mathrm{B}_{2}\mathrm{H}_{6}\). So, the molar ratio is \(\frac{2}{3}\).

Calculate moles of \(\mathrm{B}_{2} \mathrm{H}_{6}\) produced from \(0.893 \mathrm{~mol}\) of \(\mathrm{NaBH}_{4}\)

Using the molar ratio, we can calculate the moles of \(\mathrm{B}_{2} \mathrm{H}_{6}\) produced: Moles of \(\mathrm{B}_{2} \mathrm{H}_{6} =\) moles of \(\mathrm{NaBH}_{4} \times \frac{2}{3} = 0.893\times \frac{2}{3} =0.595\) moles. (c) If \(1.987 \mathrm{~mol}\) of \(\mathrm{B}_{2} \mathrm{H}_{6}\) is obtained, how many moles of \(\mathrm{NaBF}_{4}\) are produced?

Identify the relevant information from the equation

The balanced equation tells us that 2 moles of \(\mathrm{B}_{2}\mathrm{H}_{6}\) produce 3 moles of \(\mathrm{NaBF}_{4}\). So, the molar ratio is \(\frac{3}{2}\).

Calculate moles of \(\mathrm{NaBF}_{4}\) produced from \(1.987 \mathrm{~mol}\) of \(\mathrm{B}_{2} \mathrm{H}_{6}\)

Using the molar ratio, we can calculate the moles of \(\mathrm{NaBF}_{4}\) produced: Moles of \(\mathrm{NaBF}_{4} =\) moles of \(\mathrm{B}_{2} \mathrm{H}_{6} \times \frac{3}{2} = 1.987\times \frac{3}{2} =2.9805\) moles. (d) How many moles of \(\mathrm{BF}_{3}\) are required to produce \(4.992 \mathrm{~mol}\) of \(\mathrm{NaBF}_{4}?\)

Identify the relevant information from the equation

The balanced equation tells us that 4 moles of \(\mathrm{BF}_{3}\) produce 3 moles of \(\mathrm{NaBF}_{4}\). So, the molar ratio is \(\frac{4}{3}\).

Calculate moles of \(\mathrm{BF}_{3}\) required to produce \(4.992 \mathrm{~mol}\) of \(\mathrm{NaBF}_{4}\)

Using the molar ratio, we can calculate the moles of \(\mathrm{BF}_{3}\) required: Moles of \(\mathrm{BF}_{3} =\) moles of \(\mathrm{NaBF}_{4} \times \frac{4}{3} = 4.992\times \frac{4}{3} =6.656\) moles.

Key concepts

Stoichiometry Calculations

Stoichiometry calculations are the mathematical calculations that allow chemists to predict the quantities of reactants and products involved in a chemical reaction. These calculations are based on the principle of conservation of mass and the mole concept, ensuring that the atoms involved in the reactions are accounted for.

For instance, in the synthesis of diborane, we use stoichiometry to determine how many moles of sodium borohydride (\textbf{NaBH}\(_4\)) react with a given amount of boron trifluoride (BF\(_3\)), and vice versa. Understanding the stoichiometry of a reaction is crucial for calculating precise amounts of reactants needed for a desired product yield, thus helping to minimize waste and maximize efficiency in chemical synthesis.

Molar Ratio

The molar ratio is a critical component of stoichiometry and is derived from the coefficients of a balanced chemical equation. It tells us the proportional relationship between the amounts (in moles) of reactants and products. In the diborane reaction, the balanced chemical equation provides the molar ratios, such as 3 moles of \textbf{NaBH}\(_4\) for every 4 moles of BF\(_3\).

These ratios are used directly in stoichiometric calculations to convert between moles of different substances. For example, if you know the amount of \textbf{NaBH}\(_4\), you can use the molar ratio to find out how many moles of BF\(_3\) are required or how many moles of diborane (B\(_2\)H\(_6\)) will be produced.

Chemical Reaction Equations

Chemical reaction equations are representations of chemical reactions wherein the reactants are written on the left side and the products on the right. They reflect the changes occurring during a chemical reaction. It's essential that these equations are balanced, meaning that the number of atoms of each element is conserved across the reaction.

In the equation for diborane synthesis, careful balancing accounts for the six hydrogen atoms and the two boron atoms on both sides of the equation. The balanced equation not only serves as a recipe for chemists but also as a basis for all stoichiometry calculations, as it determines the mole ratio for the reactants and products.

Mole Concept

The mole concept is a foundational principle of chemistry, defining the quantity of substance. One mole of any substance contains Avogadro's number (approximately \(6.022 \times 10^{23}\)) of entities, whether they are atoms, molecules, ions, or electrons.

In the context of the diborane synthesis, understanding the mole concept allows one to relate the mass of a substance to the number of moles using molar mass. This concept enables the calculation of the amount of substances consumed and produced in a chemical reaction, as seen in the synthesis reactions where moles of reactants and moles of products are converted based on their molar ratio.