Problem 113 - Chemistry Matter and Change

Question

Calculate the number of atoms in each element. a. 25.8 g of Hg b. 0.0340 g of Zn c. 150 g of Ar d. 0.124 g of Mg

Step-by-Step Solution

Answer

The number of atoms in each element is as follows: a. \(7.7409 \times 10^{22} \ atoms\) of Hg b. \(3.12986 \times 10^{20} \ atoms\) of Zn c. \(2.25935 \times 10^{24} \ atoms\) of Ar d. \(3.0707 \times 10^{21} \ atoms\) of Mg

Step 1: Find the molar masses of each element

Look up the molar masses of each element in the periodic table or a reference table. The molar masses are: - Hg (mercury): 200.59 g/mol - Zn (zinc): 65.38 g/mol - Ar (argon): 39.95 g/mol - Mg (magnesium): 24.31 g/mol

Step 2: Calculate the number of moles of each element

Use the given mass and molar mass to find the number of moles for each element: a. Hg: \(moles = \frac{25.8g}{200.59g/mol} = 0.1286 mol\) b. Zn: \(moles = \frac{0.0340g}{65.38g/mol} = 5.201 \times 10^{-4} mol\) c. Ar: \(moles = \frac{150g}{39.95g/mol} = 3.7547 mol\) d. Mg: \(moles = \frac{0.124g}{24.31g/mol} = 5.0995 \times 10^{-3} mol\)

Step 3: Convert moles to the number of atoms using Avogadro's number

Avogadro's number (N_A) is approximately \(6.022 \times 10^{23}\) atoms per mole. Multiply the number of moles by Avogadro's number to find the number of atoms: a. Hg: \(0.1286 \ mol \times 6.022 \times 10^{23} \ atoms/mol = 7.7409 \times 10^{22} \ atoms\) b. Zn: \(5.201 \times 10^{-4} \ mol \times 6.022 \times 10^{23} \ atoms/mol = 3.12986 \times 10^{20} \ atoms\) c. Ar: \(3.7547 \ mol \times 6.022 \times 10^{23} \ atoms/mol = 2.25935 \times 10^{24} \ atoms\) d. Mg: \(5.0995 \times 10^{-3} \ mol \times 6.022 \times 10^{23} atoms/mol = 3.0707 \times 10^{21} \ atoms\) Conclusion: The number of atoms in each element is as follows: a. \(7.7409 \times 10^{22} \ atoms\) of Hg b. \(3.12986 \times 10^{20} \ atoms\) of Zn c. \(2.25935 \times 10^{24} \ atoms\) of Ar d. \(3.0707 \times 10^{21} \ atoms\) of Mg

Key Concepts

Understanding Avogadro's Number

Avogadro's number is a fundamental constant in chemistry. It represents the number of atoms, ions, or molecules in one mole of a substance. The value is approximately \(6.022 \times 10^{23}\). This number is crucial when working with molecules and atoms because it allows scientists to count particles in a given substance by only considering moles.

For example, if you have one mole of mercury (Hg), it contains \(6.022 \times 10^{23}\) atoms of mercury. Using Avogadro's number simplifies the conversion between moles and atoms, which is essential for understanding and performing stoichiometric calculations.

It’s important to remember that Avogadro's number applies universally to all elements and compounds, allowing for consistent calculations regardless of the substance being analyzed.

Calculating the Number of Atoms

The number of atoms in a sample is calculated through a series of simple steps, typically involving the molar mass and Avogadro's number. First, you determine the number of moles in your substance. This is done by dividing the mass of your sample by the molar mass of the element or compound, which can be found in a periodic table.

After finding the number of moles, multiply it by Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol) to convert moles into atoms. This process transforms a mass measurement into an understandable number of individual atoms.

Consider, for example, the element mercury with a calculated 0.1286 moles. By multiplying this by Avogadro’s number, you get \(0.1286 \times 6.022 \times 10^{23} = 7.7409 \times 10^{22}\) atoms. Following these straightforward steps, you can determine the number of atoms in any given sample.

Introduction to Stoichiometry

Stoichiometry is the area of chemistry that deals with the quantitative relationships or ratios between reactants and products in chemical reactions. Such calculations are key in predicting how much of one substance you need or how much of a product you can make using reactants.

Stoichiometry relies heavily on balanced chemical reactions. Each element in a reaction must adhere to the law of conservation of mass. This means the number of atoms for each element should be the same on both sides of the equation.

In practice, stoichiometry uses moles as a counting unit to translate grams into moles and eventually into individual atoms, using Avogadro’s number. Understanding how stoichiometry works can allow you to predict how substances interact and how much of each substance is involved, providing a deeper understanding of how chemical reactions occur.