Question

The molar mass of \(\mathrm{HCl}\) is \(36.5 \mathrm{~g} / \mathrm{mol}\), and the average mass per \(\mathrm{HCl}\) molecule is \(36.5 \mathrm{u}\). Use the fact that \(1 \mathrm{u}=1.6605 \times 10^{-24} \mathrm{~g}\) to calculate Avogadro's number.

Short answer

Avogadro's number is approximately \(6.022 \times 10^{23}\).

Step-by-step solution

Understand the given values

We are given the molar mass of \(\mathrm{HCl}\) as \(36.5\, \mathrm{g/mol}\), the average mass per \(\mathrm{HCl}\) molecule as \(36.5\, \mathrm{u}\), and \(1\, \mathrm{u} \approx 1.6605 \times 10^{-24}\, \mathrm{g}\). Our goal is to determine Avogadro's number, which is the number of molecules in one mole of a substance.

Convert the average molecular mass into grams

To find out how many grams are in one molecule, we need to convert the mass from atomic units (u) to grams using the conversion factor. Multiply the average mass per molecule by the conversion factor: \[36.5\, \mathrm{u} \times 1.6605 \times 10^{-24}\, \mathrm{g/u} = 6.055825 \times 10^{-23}\, \mathrm{g}\]Each molecule of \(\mathrm{HCl}\) has a mass of approximately \(6.055825 \times 10^{-23}\, \mathrm{g}\).

Calculate Avogadro's number

Avogadro's number can be calculated by dividing the molar mass in grams by the mass of a single molecule in grams:\[N_A = \frac{36.5\, \mathrm{g/mol}}{6.055825 \times 10^{-23}\, \mathrm{g/molecule}}\]Performing the division gives:\[N_A \approx 6.022 \times 10^{23}\, \mathrm{molecules/mol}\]This is Avogadro's number, which is approximately \(6.022 \times 10^{23}\).

Key concepts

Molar Mass

Molar mass is a key concept in chemistry that represents the mass of one mole of a substance. When we talk about \({\mathrm{HCl}}\), for instance, its molar mass is given as \({36.5\, \mathrm{g/mol}}\).

• "Mole" is a standard unit in chemistry, similar to how a "dozen" represents twelve items.
• Molar mass allows us to convert between the mass of a substance and the number of particles, like molecules or atoms, present in a given sample.

When you measure out \({36.5\, \mathrm{g}}\) of \({\mathrm{HCl}}\), you possess one mole of \({\mathrm{HCl}}\). This is true because one mole of any substance contains Avogadro's number of particles, \({6.022 \times 10^{23}}\) of them, to be precise.Understanding molar mass provides a basis for moving between the macroscope world of grams and the microscopic realm of atoms and molecules. For practical lab work, molar mass helps bridge the observable quantities we can measure with scales and equipment to the chemical equations and reactions occurring at the molecular level.

Atomic Mass Unit

Atomic mass unit, often abbreviated as u, is a unit designed to express atomic and molecular masses.

• It provides a way to convey mass values that are otherwise exceedingly tiny when measured in grams.
• The definition is based on a carbon atom: one atomic mass unit is 1/12th the mass of a carbon-12 atom.

For \({\mathrm{HCl}}\), we know the average mass per molecule to be \({36.5\, \mathrm{u}}\), which simplifies the comparison among molecules by providing a uniform scale.To convert \({\mathrm{u}}\) to grams, a conversion factor of \({1\, \mathrm{u} = 1.6605 \times 10^{-24}\, \mathrm{g}}\) is utilized. This allows scientists to relate the minuscule mass of molecules to a palpable scale, facilitating calculations and comparisons.

Conversion Factor

A conversion factor is a fundamental tool in chemistry that helps transition between units of measurement. In this case, converting the mass of a molecule from atomic mass units (u) to grams is achieved through the conversion factor \({1\, \mathrm{u} = 1.6605 \times 10^{-24}\, \mathrm{g}}\).

• This step is crucial when determining Avogadro's number.
• It allows for meaningful comparisons and calculations, particularly with direct measurements possible only in grams.

By applying the conversion factor, we determine the mass of one \({\mathrm{HCl}}\) molecule in grams, bridging the gap to apply molar mass concepts effectively.Through this conversion, we unlock the ability to measure on an atomic level while performing calculations that require results in the gram-based standard units, bringing precision to chemical measurements and enhancing understanding.