Question

You are given \(15 \mathrm{g}\) each of yterium, boron, and copper. Which sample represents the largest number of atoms?

Short answer

Boron has the largest number of atoms.

Step-by-step solution

Determine Molar Mass

First, calculate the molar mass of each element. The molar mass of Yttrium (Y) is 88.906 g/mol, Boron (B) is 10.811 g/mol, and Copper (Cu) is 63.546 g/mol.

Calculate Moles of Each Element

Using the formula \( n = \frac{m}{M} \) where \(n\) is the number of moles, \(m\) is the mass, and \(M\) is the molar mass, calculate the moles of each element.\[n_{Y} = \frac{15}{88.906} \approx 0.169 \text{ mol}\] \[n_{B} = \frac{15}{10.811} \approx 1.387 \text{ mol}\] \[n_{Cu} = \frac{15}{63.546} \approx 0.236 \text{ mol}\]

Compare Number of Moles

Look at the number of calculated moles to determine which element has the highest value. Boron has the largest number of moles at approximately 1.387 mol.

Determine Number of Atoms

To find the number of atoms, use Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol). Multiply the number of moles by Avogadro's number to find the total number of atoms for each element.\[\text{Atoms of } Y = 0.169 \times 6.022 \times 10^{23} \approx 1.018 \times 10^{23} \]\[\text{Atoms of } B = 1.387 \times 6.022 \times 10^{23} \approx 8.352 \times 10^{23} \]\[\text{Atoms of } Cu = 0.236 \times 6.022 \times 10^{23} \approx 1.421 \times 10^{23} \]

Conclusion

Compare the number of atoms calculated for each element. Boron has approximately \(8.352 \times 10^{23}\) atoms, which is the largest number among the three elements. Therefore, the sample of Boron represents the largest number of atoms.

Key concepts

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry, known to be approximately 6.022 x 10^{23}. This number represents the quantity of atoms, molecules, or particles in one mole of a substance. It provides a bridge between the atomic scale and the macroscopic scale that we can observe and measures quantities, enabling precise chemical calculations.

• Avogadro's Number is important because it allows scientists and students to convert between the number of particles and the amount of substance in moles, facilitating the understanding of reactions and compositions.
• When given the number of moles of a substance, multiplying by Avogadro's Number enables calculation of the total number of atoms or molecules present.
• In this exercise, for example, Avogadro's Number was used to determine the different amounts of atoms in yttrium, boron, and copper from their respective number of moles.

Understanding Avogadro's Number is crucial for various applications in chemistry as it foundationally helps determine the scale at which reactions occur, compositions are balanced, and quantities are measured in laboratory settings.

Molar Mass

Molar Mass is the mass of one mole of a given substance, generally expressed in grams per mole (g/mol). It is a vital component in chemical calculations as it serves as a conversion factor between mass and moles.

• Each element has a unique molar mass determined by its atomic weight. For instance, yttrium, boron, and copper have molar masses of 88.906 g/mol, 10.811 g/mol, and 63.546 g/mol respectively.
• Molar mass is used to find the number of moles in a given sample by using the formula: \( n = \frac{m}{M} \), where \( n \) is the number of moles, \( m \) is the mass of the sample, and \( M \) is the molar mass.
• This exercise demonstrates the calculation of moles using molar mass, which is a necessary step before finding out the number of atoms with Avogadro's Number.

Understanding molar mass is fundamental in chemistry as it aids in stoichiometric calculations, balancing chemical equations, and making precise predictions about reactions.

Chemical Calculations

Chemical Calculations often involve converting between masses, moles, and the number of particles. They are critical in predicting the outcomes of reactions, understanding compositions, and engaging in various analytical procedures.

• Such calculations typically follow a sequence: determining molar mass, calculating the number of moles, and then using Avogadro's Number to find the number of atoms or molecules.
• In the step-by-step solution for this exercise, each element's moles were calculated, compared, and then converted to the number of atoms using Avogadro's Number.
• Comprehending chemical calculations ensures better insight into the quantitative aspects of chemical equations and reactions. It aligns lab measurements with theoretical predictions and helps in resource management during experiments.

Mastering chemical calculations enhances one's ability to efficiently and accurately execute experiments and solve complex chemical problems.