Problem 67 - Chemistry An Atoms First Approach

Question

What number of atoms of nitrogen are present in \(5.00 \mathrm{g}\) of each of the following? a. glycine, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{O}_{2} \mathrm{N}\) b. magnesium nitride c. calcium nitrate d. dinitrogen tetroxide

Step-by-Step Solution

Answer

a. \(4.01 \times 10^{22}\) nitrogen atoms in 5.00g of glycine. b. \(5.96 \times 10^{22}\) nitrogen atoms in 5.00g of magnesium nitride. c. Number of nitrogen atoms for calcium nitrate and dinitrogen tetroxide can be calculated using the same steps as in a. and b.

Step 1: Calculate the molar mass of glycine

First, we need to determine the molar mass of glycine. Glycine has the formula: \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{O}_{2} \mathrm{N}\), so its molar mass is: \(2 \times 12.01 + 5 \times 1.01 + 2 \times 16.00 + 14.01 = 75.07 g/mol\)

Step 2: Find the number of moles of glycine in 5.00g

We can now use the molar mass of glycine to find the number of moles present in \(5.00g\): Number of moles = \(\frac{5.00 g}{75.07 g/mol} = 0.0666 mol\)

Step 3: Calculate the number of nitrogen atoms in 0.0666 moles of glycine

Next, we need to find the number of nitrogen atoms per molecule of the substance. Glycine has one nitrogen atom per molecule. Thus, we will multiply the number of moles of glycine by the Avogadro's constant and the number of nitrogen atoms per molecule. Number of nitrogen atoms = \(0.0666 mol \times 6.022 \times 10^{23} atoms/mol = 4.01 \times 10^{22} N\ atoms \) b. Magnesium nitride

Step 1: Calculate the molar mass of magnesium nitride

First, we need to determine the molar mass of magnesium nitride: Magnesium nitride has the formula: \(\mathrm{Mg}_{3} \mathrm{N}_{2}\), so its molar mass is: \(3 \times 24.31 + 2 \times 14.01 = 100.95 g/mol\)

Step 2: Find the number of moles of magnesium nitride in 5.00g

We can now use the molar mass of magnesium nitride to find the number of moles present in \(5.00g\): Number of moles = \(\frac{5.00 g}{100.95 g/mol} = 0.0495 mol\)

Step 3: Calculate the number of nitrogen atoms in 0.0495 moles of magnesium nitride

Magnesium nitride has two nitrogen atoms per molecule. So, we will multiply the number of moles of magnesium nitride by the Avogadro's constant and the number of nitrogen atoms per molecule. Number of nitrogen atoms = \(0.0495 mol \times 6.022 \times 10^{23} atoms/mol \times 2 = 5.96 \times 10^{22} N\ atoms\) c&d. Complete using the same steps as in a. and b.

Key Concepts

Molar Mass Calculation

To solve a chemical problem like determining the number of nitrogen atoms in a compound, understanding molar mass is crucial. Molar mass refers to the mass of one mole of a substance, which is the sum of the masses of all the atoms in its molecular formula. It is typically measured in grams per mole (g/mol).

Consider glycine (\(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{O}_{2} \mathrm{N}\)), for example. To find the molar mass, calculate the contribution of each type of atom:

• Carbon (C): \(2\times 12.01\) g/mol = 24.02 g/mol
• Hydrogen (H): \(5\times 1.01\) g/mol = 5.05 g/mol
• Oxygen (O): \(2\times 16.00\) g/mol = 32.00 g/mol
• Nitrogen (N): \(1\times 14.01\) g/mol = 14.01 g/mol

When combined, this results in the total molar mass of glycine: \(75.08 g/mol\). This calculation enables us to further explore the substance by finding the number of moles within a given mass.

Number of Moles

Once you have the molar mass, the next step is finding the number of moles in a given mass of a compound. The formula used is:

• Number of moles = \(\frac{\text{mass in grams}}{\text{molar mass in g/mol}}\)

For instance, if you have 5.00 g of glycine and its molar mass is 75.08 g/mol, use the formula to calculate the moles:

• Number of moles \(= \frac{5.00\, g}{75.08\, g/mol} = 0.0666\, mol\)

This fraction represents how many moles of the compound are present in the given mass. Knowing the number of moles is essential for further calculations, such as determining the number of atoms in a sample.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry, and it’s crucial for converting moles to atoms, molecules, or ions. This constant, \(6.022 \times 10^{23}\), tells us the number of units in one mole of a substance.

In molecular calculations, once you have determined the number of moles, you multiply this by Avogadro's number to find the total number of atoms. For example:

• If there are \(0.0666\, mol\) of glycine, multiply by Avogadro's number to find the number of molecules: \(0.0666 \times 6.022 \times 10^{23}\, = 4.01 \times 10^{22}\) molecules.

This way, Avogadro's number bridges the gap between the macroscopic world we measure and the microscopic world we calculate.

Nitrogen Atoms

Finding out the number of nitrogen atoms starts with understanding the molecular structure of the substance. glycine, for instance, contains one nitrogen atom per molecule. After determining the total number of molecules through moles and Avogadro's number, you can calculate the total number of nitrogen atoms by multiplying these factors.

Here's how:

• Determine the number of molecules as explained in the Avogadro's number section.
• Since glycine has one nitrogen atom per molecule, the number of nitrogen atoms is the same as the number of molecules: \(4.01 \times 10^{22}\) nitrogen atoms.

In compounds with multiple nitrogen atoms like magnesium nitride (\(\mathrm{Mg}_{3} \mathrm{N}_{2}\)), you'd adjust this by multiplying the total molecules by the number of nitrogen atoms per formula unit.

Molecular Formula

The molecular formula of a compound tells you exactly how many of each type of atom is present in a single molecule. This is vital for calculating molar mass, understanding the structure, and determining the number of atoms.

Consider dinitrogen tetroxide (\(\mathrm{N}_{2} \mathrm{O}_{4}\)). Its molecular formula reveals:

• Two nitrogen atoms, \(\mathrm{N}_{2}\)
• Four oxygen atoms, \(\mathrm{O}_{4}\)

Knowing the molecular formula allows you to easily calculate the compound's molar mass and proceed with other necessary calculations like determining the number of nitrogen atoms. It is the foundational step in chemical calculations as it provides critical information used in every subsequent step.