Problem 120 - General Chemistry: Principles and Modern Applications

Question

The compound \(\mathrm{C}_{7} \mathrm{H}_{7} \mathrm{NO}_{2}\) contains (a) 17 atoms per mole; (b) equal percents by mass of \(\mathrm{C}\) and \(\mathrm{H} ;\) (c) about twice the percent by mass of \(\mathrm{O}\) as of \(\mathrm{N} ;\) (d) about twice the percent by mass of \(\mathrm{N}\) as of \(\mathrm{H}\).

Step-by-Step Solution

Answer

Firstly, calculate the molar mass of C7H7NO2. Then, calculate the mass contribution of each element in the compound. Up next, compute the percent by mass of each element. Confirm if the given conditions in the exercise are met. If it's found that the percent by mass of Carbon is approximately equal to Hydrogen's, Oxygen's is about two times Nitrogen's, and Nitrogen's is about twice as much as Hydrogen's, then the exercise has been correctly solved.

Step 1: Compute the Molar Mass of the Compound

To compute the molar mass of the compound C7H7NO2, add up the atomic masses for each element according to its quantity given by the subscript in the molecular formula. The atomic masses of Carbon (C), Hydrogen (H), Nitrogen (N), and Oxygen (O) are roughly 12.01 g/mol, 1.008 g/mol, 14.007 g/mol, and 16.00 g/mol respectively. Therefore, the molar mass is computed as: Molar Mass = (7*12.01 g/mol) + (7*1.008 g/mol) + (1*14.007 g/mol) + (2*16.00 g/mol).

Step 2: Calculate the Mass of Each Element

Calculate the mass contribution of each element in the compound. For Carbon: Mass = (7*12.01 g/mol). For Hydrogen: Mass = (7*1.008 g/mol). For Nitrogen: Mass = (1*14.007 g/mol). For Oxygen: Mass = (2*16.00 g/mol).

Step 3: Calculate the Percent by Mass of Each Element

Now calculate the percent by mass of each element. The percent by mass is calculated by taking the mass of the element (calculated in Step 2) and dividing it by the molar mass (calculated in Step 1), then multiplying the result by 100. For example, the percent by mass for Carbon = [(mass of carbon) / (molar mass of C7H7NO2)] * 100

Step 4: Confirm the Given Conditions

Confirm that the percent by mass of Carbon is almost equal to that of Hydrogen, the percent by mass of Oxygen is about twice that of Nitrogen, and the percent by mass of Nitrogen is about twice that of Hydrogen. If these conditions are met, then the exercise has been solved correctly.

Key Concepts

Elemental Composition

Understanding the elemental composition of a compound allows us to grasp what elements are present and in what quantities. Each element in a chemical formula is represented by its chemical symbol followed by a subscript that indicates the number of atoms of that element in the molecule. For example, in the compound \(\mathrm{C}_{7} \mathrm{H}_{7} \mathrm{NO}_{2}\), the subscript numbers tell us there are 7 carbon atoms, 7 hydrogen atoms, 1 nitrogen atom, and 2 oxygen atoms in each molecule. This is crucial because it directly influences the molar mass calculation and subsequent percent by mass calculations.

• Carbon (C): 7 atoms
• Hydrogen (H): 7 atoms
• Nitrogen (N): 1 atom
• Oxygen (O): 2 atoms

Percent by Mass

The percent by mass of an element in a compound is a way of expressing the concentration of an element within that compound compared to the entire compound's mass. First, we calculate the molar mass of the entire compound by summing the product of the atomic masses of each element and their respective subscripts. For \(\mathrm{C}_{7} \mathrm{H}_{7} \mathrm{NO}_{2}\), the molar mass is calculated as follows:\[\text{Molar Mass} = (7 \times 12.01 \text{ g/mol}) + (7 \times 1.008 \text{ g/mol}) + (1 \times 14.007 \text{ g/mol}) + (2 \times 16.00 \text{ g/mol})\]Once the molar mass is found, the mass of each individual element is derived using its atomic mass and number of atoms. Finally, we calculate the percent by mass using the formula:\[\text{Percent by Mass} = \left(\frac{\text{Mass of the element}}{\text{Molar mass of compound}}\right) \times 100\]This helps in confirming points like whether the mass of one element is roughly equal to or twice another element in the compound.

Chemical Formula Analysis

Chemical formula analysis involves interpreting the relationships and assumptions made from the given formula and the calculated data such as percentages. For the compound \(\mathrm{C}_{7} \mathrm{H}_{7} \mathrm{NO}_{2}\), checking conditions like if oxygen's percentage is about twice that of nitrogen involves mathematical verification using the percent by mass calculations.
To confirm these conditions:

• Check if the percent by mass of Carbon approximates that of Hydrogen.
• Ensure that Oxygen has roughly twice the percent by mass of Nitrogen.
• Verify that the percent by mass of Nitrogen is about twice that of Hydrogen.

Through detailed examination of the calculated percentages from the molar masses, such insights verify the chemical's structural assumptions and properties.