Question

Calculate the percent by mass of the element listed first in the formulas for each of the following compounds. a. adipic acid, \(\mathrm{C}_{6} \mathrm{H}_{10} \mathrm{O}_{4}\) b. ammonium nitrate, \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) c. caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}\) d. chlorine dioxide, \(\mathrm{ClO}_{2}\) e. cyclohexanol, \(\mathrm{C}_{6} \mathrm{H}_{11} \mathrm{OH}\) f. dextrose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) g. eicosane, \(\mathrm{C}_{20} \mathrm{H}_{42}\) h. ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\)

Short answer

The percent by mass of the first element in each compound: a. Adipic acid, \(\mathrm{C}_6\mathrm{H}_{10}\mathrm{O}_4\): 49.32% Carbon b. Ammonium nitrate, \(\mathrm{NH}_{4} \mathrm{NO}_{3}\): 29.15% Nitrogen c. Caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\): 49.48% Carbon d. Chlorine dioxide, \(\mathrm{ClO}_{2}\): 52.56% Chlorine e. Cyclohexanol, \(\mathrm{C}_{6} \mathrm{H}_{11} \mathrm{OH}\): 72.68% Carbon f. Dextrose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\): 40.00% Carbon g. Eicosane, \(\mathrm{C}_{20} \mathrm{H}_{42}\): 85.03% Carbon h. Ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\): 53.30% Carbon.

Step-by-step solution

Determine the molar mass of each compound

Using the periodic table, find the molar mass of each element in the compound. Then, add these values up to find the molar mass of the entire compound.

Calculate the mass of the first element in each compound

Multiply the chemical symbol's subscript in the molecular formula for the first element by its atomic mass. This will give the mass of the first element in the compound.

Calculate the percent by mass of the first element in each compound

Divide the mass of the first element (found in Step 2) by the molar mass of the compound (found in Step 1) and multiply by 100 to find the percent by mass of the first element in each compound. This process should be completed for all the compounds: a. Adipic acid, \(\mathrm{C}_6\mathrm{H}_{10}\mathrm{O}_4\) Molar Mass: (6×12.01) + (10×1.01) + (4×16.00) = 72.06 + 10.10 + 64.00 = 146.16 g/mol Mass of C: 6×12.01 = 72.06 g Percent by mass of Carbon: \((\frac{72.06}{146.16})\times100\%=49.32\%\) b. Ammonium nitrate, \(\mathrm{NH}_{4} \mathrm{NO}_{3}\) Molar Mass: (1×14.01)+ (4×1.01)+(1×14.01)+(1×16.00)+(3×16.00)=14.01 + 4.04 + 14.01 + 16.00 + 48.00 = 96.06 g/mol Mass of N: 2×14.01 = 28.02 g Percent by mass of Nitrogen: \((\frac{28.02}{96.06})\times100\%=29.15\%\) c. Caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\) Molar Mass: (8×12.01) + (10×1.01) + (4×14.01) + (2×16.00) = 96.08 + 10.10 + 56.04 + 32.00 = 194.22 g/mol Mass of C: 8×12.01 = 96.08 g Percent by mass of Carbon: \((\frac{96.08}{194.22})\times100\%=49.48\%\) d. Chlorine dioxide, \(\mathrm{ClO}_{2}\) Molar Mass: (35.45)+(2×16.00) = 35.45 + 32.00 = 67.45 g/mol Mass of Cl: 35.45 g Percent by mass of Chlorine: \((\frac{35.45}{67.45})\times100\%=52.56\%\) e. Cyclohexanol, \(\mathrm{C}_{6} \mathrm{H}_{11} \mathrm{OH}\) Molar Mass: (6×12.01) + (11×1.01) + (1×16.00) = 72.06 + 11.11 + 16.00 = 99.17 g/mol Mass of C: 6×12.01 = 72.06 g Percent by mass of Carbon: \((\frac{72.06}{99.17})\times100\%=72.68\%\) f. Dextrose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) Molar Mass: (6×12.01) + (12×1.01) + (6×16.00) = 72.06 + 12.12 + 96.00 = 180.18 g/mol Mass of C: 6×12.01 = 72.06 g Percent by mass of Carbon: \((\frac{72.06}{180.18})\times100\%=40.00\%\) g. Eicosane, \(\mathrm{C}_{20} \mathrm{H}_{42}\) Molar Mass: (20×12.01) + (42×1.01) = 240.20 + 42.42 = 282.62 g/mol Mass of C: 20×12.01 = 240.20 g Percent by mass of Carbon: \((\frac{240.20}{282.62})\times100\%=85.03\%\) h. Ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) Molar Mass: (2×12.01) + (4×1.01) + (1×16.00) = 24.02 + 5.04 + 16.00 = 45.06 g/mol Mass of C: 2×12.01 = 24.02 g Percent by mass of Carbon: \((\frac{24.02}{45.06})\times100\%=53.30\%\)

Key concepts

Molar Mass

The concept of molar mass is crucial for understanding percent composition. Molar mass refers to the mass of one mole of a given substance. It is usually expressed in grams per mole (g/mol).
To calculate the molar mass of a compound:

• Identify all the elements in the chemical formula and note the number of each type of atom.
• Find the atomic mass of each element from the periodic table.
• Multiply the atomic mass by the number of atoms of that element in the compound.
• Sum all these values to get the molar mass of the compound.

This calculated molar mass becomes the basis for further computations, like determining the percent composition of an element in a compound. For example, in caffeine \( \mathrm{C}_8\mathrm{H}_{10} \mathrm{N}_4\mathrm{O}_2\), the molar mass is found by summing the contributions of carbon, hydrogen, nitrogen, and oxygen to get a total of 194.22 g/mol.

Atomic Mass

Atomic mass is the mass of a single atom of an element. It is typically given in atomic mass units (amu), where 1 amu is defined as one twelfth of the mass of a carbon-12 atom.
Every element's atomic mass can be found on the periodic table. This is crucial in calculating the molar mass of compounds.
When calculating how much mass an element contributes to a compound:

• Use the atomic mass of the element directly from the periodic table.
• Consider the number of atoms of that element present in the compound's chemical formula.
• Multiply the atomic mass by the number of atoms to get the total mass contributed by that element.

For example, in ethanol \( \mathrm{C}_2\mathrm{H}_5\mathrm{OH} \), the atomic mass of carbon is approximately 12.01 amu. With 2 carbon atoms in ethanol, the mass of carbon in the compound is calculated as 24.02 g.

Chemical Formula

A chemical formula represents the types and numbers of atoms in a molecule. It acts like a recipe indicating how many of each atom are present. For example, \( \mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6 \) for dextrose specifies:

• 6 atoms of carbon
• 12 atoms of hydrogen
• 6 atoms of oxygen

To use a chemical formula effectively in calculations:

• Identify the chemical symbol for each element involved.
• Interpret the subscript numbers to find out the ratio of each element within the compound.
• Use these numbers to calculate both the individual and total molar mass.

The chemical formula serves as the foundation for calculating the percent composition, where it's essential to determine how much of the compound's mass is due to a specific element.

Element Analysis

Element analysis refers to the examination of what elements make up a compound and in what proportions. This is key in chemistry for understanding the makeup and characteristics of substances.
In element analysis for percent composition, the focus is on the first-listed element:

• Identify the first element in the chemical formula.
• Calculate its total mass within the compound using its atomic mass and the number of atoms.
• Divide this mass by the molar mass of the compound and multiply by 100 to convert it into a percentage.

This process allows chemists to determine the proportion of any given element within a compound. For instance, in eicosane \( \mathrm{C}_{20} \mathrm{H}_{42} \), the percent mass of carbon is found to be 85.03%, illustrating the dominance of carbon in this hydrocarbon.