Question

What volume of hydrogen gas at \(273 \mathrm{~K}\) and 1 atm pressure will be consumed in obtaining \(21.6 \mathrm{~g}\) of elemental boron (atomic mass \(=10.8\) ) from the reduction of boron trichloride by hydrogen? (a) \(89.6 \mathrm{~L}\) (b) \(67.2 \mathrm{~L}\) (c) \(44.8 \mathrm{~L}\) (d) \(22.4 \mathrm{~L}\)

Short answer

The volume of hydrogen gas consumed is 67.2 L.

Step-by-step solution

Write the Chemical Equation

Write the balanced chemical equation for the reaction.The reduction of boron trichloride ( \( ext{BCl}_3 \) ) by hydrogen ( \( ext{H}_2\) ) is represented by:\[2 ext{BCl}_3 + 3 ext{H}_2 \rightarrow 2 ext{B} + 6 ext{HCl}\]

Calculate Moles of Boron

Calculate the moles of boron required.Given mass of boron = \(21.6\, \text{g}\) Atomic mass of boron = \(10.8\, \text{g/mol}\) Moles of boron = \(\frac{21.6}{10.8} = 2.0\, \text{mol}\)

Relate Moles of Boron to Moles of Hydrogen

Use the balanced chemical equation to relate moles of boron to moles of hydrogen.From the equation, \(2 \text{ mol B}\) is produced by \(3 \text{ mol } \text{H}_2\).Therefore, \(2 \, \text{mol B}\) requires \(3 \, \text{mol } \text{H}_2\).

Calculate Volume of Hydrogen

Calculate the volume of hydrogen gas using the ideal gas law (under STP conditions where 1 mole of gas occupies 22.4 L).Using the relation from Step 3, \(3\) moles of \(\text{H}_2\) correspond to a volume of:\[3 \text{ mol } \text{H}_2 \times 22.4 \text{ L/mol} = 67.2 \text{ L}\]

Key concepts

Boron Trichloride Reduction

In the chemical world, reduction is when a substance loses oxygen or gains hydrogen. Boron trichloride (\(\text{BCl}_3\)) is a compound where boron is bonded with chlorine. To reduce \(\text{BCl}_3\), hydrogen (\(\text{H}_2\)) is used as the reducing agent. This means hydrogen will react with \(\text{BCl}_3\) to produce elemental boron (\(\text{B}\)) and hydrogen chloride (\(\text{HCl}\)).

When this reaction occurs, it looks like this: \[2\text{BCl}_3 + 3\text{H}_2 \rightarrow 2\text{B} + 6\text{HCl}\]. When balanced, this equation shows how the atoms are rearranged during the reaction. Just imagine the hydrogen gas swooping in to pull the chlorine away, leaving behind pure boron.

Understanding the reduction of \(\text{BCl}_3\) by hydrogen is essential for grasping more complex chemical processes. By visualizing these reactions step by step, students can better appreciate how elements interact in chemical equations.

Chemical Equation Balancing

Balancing chemical equations is like solving a puzzle where all pieces must fit just right. It ensures that the number of atoms of each element is the same on both sides of the equation. In the equation \[2\text{BCl}_3 + 3\text{H}_2 \rightarrow 2\text{B} + 6\text{HCl}\], each type of atom is balanced.

Think of it this way: on the left side, 2 molecules of \(\text{BCl}_3\) and 3 molecules of \(\text{H}_2\) are required. The result? 2 atoms of boron (\(\text{B}\)), and 6 molecules of \(\text{HCl}\). This process ensures the law of conservation of mass is respected, meaning no atoms are lost in the reaction.

To balance a chemical equation, adjust the coefficients (the numbers in front of molecules) without changing the actual chemical species. Once balanced, the equation tells us the exact proportions needed for a complete reaction, which prevents waste and maximizes efficiency.

Moles Calculation

The concept of moles ties directly into the world of chemistry as a way to count particles. Since atoms and molecules are super tiny, we use moles to understand quantities on a human scale. One mole equals Avogadro's number, \(6.022 \times 10^{23}\) particles, just like a dozen means 12.

To find out how many moles of boron are involved, one would take the given mass, 21.6 grams, and divide by the atomic mass of boron, which is 10.8 g/mol: \[\frac{21.6 \text{ g}}{10.8 \text{ g/mol}} = 2.0 \text{ mol}\]This tells us that there are 2 moles of boron.

Understanding how to calculate moles helps students determine how much of a chemical they have or need. It’s like counting individual grains of sand using buckets.

Ideal Gas Law

The ideal gas law describes how gases behave under normal conditions. Its formula \[PV = nRT\] combines several gas laws, which describes the behavior in terms of pressure (\(P\)), volume (\(V\)), number of moles (\(n\)), the gas constant (\(R\)), and temperature (\(T\)).

For standard temperature and pressure (STP) conditions, 1 mole of any ideal gas occupies 22.4 L. This is crucial for our problem because once we know we need 3 moles of hydrogen to produce 2 moles of boron, we can easily calculate for the gas volume. At STP, 3 moles of hydrogen take up: \[3 \text{ mol} \times 22.4 \text{ L/mol} = 67.2 \text{ L}\]This volume calculation shows us how much space the hydrogen gas will occupy when the reduction process is finished.

Using the ideal gas law helps chemists predict the behavior of gases under different conditions, making it easier to control reactions and understand molecular properties.