Question

Calculate the molar mass of each of the following: (a) \(\mathrm{N}_{2} \mathrm{O}_{4}\) (b) \(\mathrm{C}_{4} \mathrm{H}_{9} \mathrm{OH}\) (c) \(\mathrm{MgSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}\) (d) \(\mathrm{Ca}\left(\mathrm{C}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\right)_{2}\)

Short answer

The molar masses are: (a) 92.02 g/mol, (b) 74.14 g/mol, (c) 246.52 g/mol, (d) 158.18 g/mol.

Step-by-step solution

Determine Atomic Masses

To begin, identify the atomic masses for each element: - Nitrogen (N): 14.01 g/mol - Oxygen (O): 16.00 g/mol - Carbon (C): 12.01 g/mol - Hydrogen (H): 1.01 g/mol - Magnesium (Mg): 24.31 g/mol - Sulfur (S): 32.07 g/mol - Calcium (Ca): 40.08 g/mol

Calculate Molar Mass of \(\text{N}_{2}\text{O}_{4}\)

Combine the atomic masses of each element multiplied by their respective quantities: - For \(\text{N}_{2}\): \(2 \times 14.01 = 28.02\) - For \(\text{O}_{4}\): \(4 \times 16.00 = 64.00\) - The molar mass of \(\text{N}_{2}\text{O}_{4}\): 28.02 + 64.00 = 92.02 g/mol

Calculate Molar Mass of \(\text{C}_{4}\text{H}_{9}\text{OH}\)

Combine the atomic masses of each element multiplied by their respective quantities: - For \(\text{C}_{4}\): \(4 \times 12.01 = 48.04\) - For \(\text{H}_{9}\): \(9 \times 1.01 = 9.09\) - For \(\text{OH}\): \(16.00 + 1.01 = 17.01\) - The molar mass of \(\text{C}_{4}\text{H}_{9}\text{OH}\): 48.04 + 9.09 + 17.01 = 74.14 g/mol

Calculate Molar Mass of \(\text{MgSO}_{4} \cdot 7 \text{H}_{2}\text{O}\)

First, calculate the mass of \(\text{MgSO}_{4}\): - Magnesium: 24.31 - Sulfur: 32.07 - Oxygen: \(4 \times 16.00 = 64.00\) - Total for \(\text{MgSO}_{4}\): 24.31 + 32.07 + 64.00 = 120.38 g/mol Finally, add the mass of 7 water molecules: - \(7 \times (2 \times 1.01 + 16.00) = 7 \times 18.02 = 126.14\) - Total molar mass of \(\text{MgSO}_{4} \cdot 7 \text{H}_{2}\text{O}\): 120.38 + 126.14 = 246.52 g/mol

Calculate Molar Mass of \(\text{Ca}(\text{C}_{2}\text{H}_{3}\text{O}_{2})_{2}\)

First, calculate the mass of \(\text{C}_{2}\text{H}_{3}\text{O}_{2}\): - Carbon: \(2 \times 12.01 = 24.02\) - Hydrogen: \(3 \times 1.01 = 3.03\) - Oxygen: \(2 \times 16.00 = 32.00\) - Total for one \(\text{C}_{2}\text{H}_{3}\text{O}_{2}\) group: 24.02 + 3.03 + 32.00 = 59.05 g/mol Then, multiply by 2 for the two acetate groups: - \(2 \times 59.05 = 118.10\) Finally, add the mass of calcium: - Calcium: 40.08 - Total molar mass of \(\text{Ca}(\text{C}_{2}\text{H}_{3}\text{O}_{2})_{2}\): 40.08 + 118.10 = 158.18 g/mol

Key concepts

Atomic Masses

To calculate the molar mass of a compound, we first need to know the atomic masses of the elements involved. Atomic masses represent the mass of a single atom of an element and are typically measured in grams per mole (g/mol). These values are crucial because they allow us to convert between the number of atoms and the mass of a substance. When you look up the atomic mass of an element on the periodic table, you often see a number with two decimal places, like 14.01 for nitrogen (N) or 16.00 for oxygen (O). These values come from averaging the masses of all the natural isotopes of that element.
Understanding and using atomic masses correctly is the foundation for many calculations in chemistry. Knowing the atomic masses helps you determine the molar mass of any chemical compound quickly and accurately.

Chemical Formulas

Chemical formulas are shorthand representations of compounds, showing the types and numbers of atoms involved. For example, \(\text{C}\_4\text{H}\_9\text{OH}\) represents butanol, indicating it has 4 carbon (C) atoms, 9 hydrogen (H) atoms, and 1 hydroxyl (OH) group.
When calculating the molar mass of a compound, chemical formulas guide us in identifying what elements are present and in what quantities. Let's take \(\text{N}\_2\text{O}\_4\) as another example: this formula tells us there are 2 nitrogen (N) atoms and 4 oxygen (O) atoms in one molecule of dinitrogen tetroxide.
By understanding chemical formulas, you can easily break down any compound into its atomic components, facilitating molar mass calculation.

Molecular Weight

Molecular weight, often synonymous with molar mass, is the sum of the atomic masses of all atoms in a molecule. To calculate it, you multiply the atomic mass of each element by the number of times that element appears in the molecule and then add these values together. For instance, to find the molar mass of \(\text{MgSO}\_4 \, \cdot \, 7 \text{H}\_2\text{O}\), you would:

• Calculate the mass of \(\text{MgSO}\_4\) by adding the masses of 1 magnesium (Mg) atom, 1 sulfur (S) atom, and 4 oxygen (O) atoms
  
• Calculate the total mass for 7 water molecules \(7 \times (2 \times 1.01 + 16.00) = 126.14\)
  
• Combine these two results to get the molar mass of the entire compound (120.38 + 126.14 = 246.52 g/mol)
  

Understanding molecular weight is critical for various applications, from stoichiometry in chemical reactions to determining the amounts of substances needed in industrial processes. This concept allows you to make accurate and precise calculations, ensuring successful outcomes in both academic and practical chemistry.