Question

Disilane, \(\mathrm{Si}_{2} \mathrm{H}_{x},\) contains \(90.28 \%\) silicon by mass. Calculate the value of \(x\) in this compound.

Short answer

The value of \(x\) is 6.

Step-by-step solution

Understand the Problem

We need to find the number of hydrogen atoms in disilane, noted as \(x\), using the given percentage of silicon by mass \((90.28\%).\) This means we need to calculate the mass of hydrogen because the total mass percentage of silicon and hydrogen must sum to 100%.

Define Known Quantities

The molar mass of silicon (\(\text{Si}\)) is approximately \(28.09 \, \text{g/mol}.\) Therefore, the molar mass for two silicon atoms (\(\text{Si}_2\)) is \(2 \times 28.09 \, \text{g/mol} = 56.18 \, \text{g/mol}.\)

Express Total Mass of Compound

The total mass of \(\text{Si}_2\text{H}_x\) can be expressed as \(56.18 \, \text{g/mol} + x \times 1.01 \, \text{g/mol},\) where \(1.01 \ ext{g/mol}\) is the approximate molar mass of hydrogen (\(\text{H}\)).

Use Percentage by Mass to Set Up Equation

Since silicon makes up 90.28% of the mass, we set up the equation: \[\frac{56.18}{56.18 + x \times 1.01} = 0.9028\]

Solve the Equation for x

Rearrange and solve for \(x\): - Multiply both sides by \(56.18 + x \times 1.01\) to clear the fraction: \[0.9028 \times (56.18 + x \times 1.01) = 56.18\]- Expand the left side: \[50.6992 + 0.9028x \times 1.01 = 56.18\]- Subtract \(50.6992\) from both sides: \[0.9028x \times 1.01 = 5.4808\]- Divide by \(0.9028 \times 1.01\): \[x = \frac{5.4808}{0.912828} \approx 6\]

Conclude the Calculation

Thus, the number of hydrogen atoms \(x\) in disilane is 6.

Key concepts

Molar Mass Calculation

Molar mass is the mass of one mole of a chemical compound or element. It is usually expressed in grams per mole (g/mol). Calculating the molar mass of a compound helps in understanding how much of each element is present in a mole of that compound. Let's go through how this was done with the example of disilane, which contains two silicon (Si) atoms and an undetermined number of hydrogen (H) atoms.

To calculate the molar mass of silicon in disilane, you start by knowing the molar mass of a single silicon atom, which is about 28.09 g/mol. Since disilane has two silicon atoms, you multiply this value by two. Hence, the molar mass of Si in disilane is 56.18 g/mol.

When adding hydrogen, each hydrogen has a molar mass of approximately 1.01 g/mol. If you were to calculate the total molar mass of disilane, you would add these together. Thus, molar mass calculation is crucial in determining the exact composition and behavior of chemical compounds.

Percentage by Mass

Percentage by mass, also known as weight percent, is a way to express the concentration of an element in a compound. It shows how much of the total mass of a compound is made up by a particular element.

For disilane, it's given that silicon composes 90.28% of the compound by mass. This information is crucial when determining how much hydrogen is in the compound. The sum of the percentage by mass of all elements in a compound must equal 100%.

Using the provided 90.28% for silicon, you can deduce that the hydrogen in disilane makes up the remaining percentage of the mass, which is 9.72%. This understanding allows you to set up equations involving the masses of individual elements in the compound, making it possible for us to solve for unknowns, like the number of hydrogen atoms.

Stoichiometry

Stoichiometry is the area of chemistry that involves calculating the quantities of reactants and products in chemical reactions. It uses relationships derived from balanced chemical equations and molecular formulas to determine these quantities.

In the disilane problem, stoichiometry plays a role when setting up and solving equations involving molar masses and percentage compositions to find the unknown number of hydrogen atoms, represented by x. By using the given percentage by mass of silicon and the total calculated molar mass of the compound, we derived an equation that helped solve for x.

The stoichiometry involved includes setting up the equation: \[ \frac{56.18}{56.18 + x \times 1.01} = 0.9028 \]
Solving this equation via stoichiometric principles provided the insight needed to determine that x, the number of hydrogen atoms, is 6. Stoichiometry helps chemists understand and anticipate the outcomes of chemical processes by working through these calculations.