Problem 26 - Chemistry The Central Science In Si Units

Question

Calculate the percentage by mass of the indicated element in the following compounds: (a) hydrogen in methane, \(\mathrm{CH}_{4}\), the major hydrocarbon in natural gas; (b) oxygen in vitamin E, \(\mathrm{C}_{29} \mathrm{H}_{50} \mathrm{O}_{2} ;\) (c) sulphur in magnesium sulphate, \(\mathrm{MgSO}_{4}\), a substance used as a drying agent; (d) nitrogen in epinephrine, \(\mathrm{C}_{9} \mathrm{H}_{13} \mathrm{NO}_{3},\) also known as adrenalin, a hormone that is important for the fightor-flight response; \((\mathbf{e})\) oxygen in the insect pheromone sulcatol, \(\mathrm{C}_{8} \mathrm{H}_{16} \mathrm{O} ;(\mathbf{f})\) carbon in sucrose, \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11},\) the compound that is responsible for the sweet taste of table sugar.

Step-by-Step Solution

Answer

(a) 25.17%, (b) 7.43%, (c) 26.64%, (d) 7.65%, (e) 12.48%, (f) 42.11%

Step 1: Calculate the Molar Mass of Methane

Determine the molar mass of methane (\(\mathrm{CH}_{4}\)). Methane consists of 1 carbon atom and 4 hydrogen atoms. The molar mass of carbon is approximately 12.01 g/mol, and the molar mass of hydrogen is approximately 1.01 g/mol. Thus, the molar mass of methane is calculated as follows: \(\text{Molar mass of } \mathrm{CH}_{4} = (1 \times 12.01) + (4 \times 1.01) = 16.05 \text{ g/mol}.\)

Step 2: Calculate Percentage by Mass of Hydrogen in Methane

To determine the mass percentage of hydrogen in methane, use the formula: \[\text{Percentage by mass} = \left(\frac{\text{Total mass of hydrogen}}{\text{Molar mass of the compound}}\right) \times 100\%\]The total mass of hydrogen in \(\mathrm{CH}_{4}\) is \(4 \times 1.01 = 4.04\text{ g/mol}.\)Thus, \[\text{Percentage by mass of hydrogen} = \left(\frac{4.04}{16.05}\right) \times 100\% \approx 25.17\%\]

Step 3: Calculate the Molar Mass of Vitamin E

Vitamin E has the formula \(\mathrm{C}_{29} \mathrm{H}_{50} \mathrm{O}_{2}\). Compute its molar mass by adding the mass of each element:\[\text{Molar mass of } \mathrm{C}_{29} \mathrm{H}_{50} \mathrm{O}_{2} = (29 \times 12.01) + (50 \times 1.01) + (2 \times 16.00) = 430.76 \text{ g/mol}.\]

Step 4: Calculate Percentage by Mass of Oxygen in Vitamin E

Calculate the percentage by mass of oxygen using the formula:\[\text{Percentage by mass} = \left(\frac{\text{Total mass of oxygen}}{\text{Molar mass of the compound}}\right) \times 100\%\]The total mass of oxygen in \(\mathrm{C}_{29} \mathrm{H}_{50} \mathrm{O}_{2}\) is \(2 \times 16.00 = 32.00\text{ g/mol}.\)\[\text{Percentage by mass of oxygen} = \left(\frac{32.00}{430.76}\right) \times 100\% \approx 7.43\%\]

Step 5: Calculate the Molar Mass of Magnesium Sulphate

For magnesium sulphate (\(\mathrm{MgSO}_{4}\)), add the masses of magnesium, sulphur, and oxygen:\[\text{Molar mass of } \mathrm{MgSO}_{4} = 24.31 + 32.07 + (4 \times 16.00) = 120.37 \text{ g/mol}.\]

Step 6: Calculate Percentage by Mass of Sulphur in Magnesium Sulphate

Calculate the percentage by mass of sulphur:\[\text{Percentage by mass} = \left(\frac{\text{Total mass of sulphur}}{\text{Molar mass of the compound}}\right) \times 100\%\]\(\text{Total mass of sulphur} = 32.07 \text{ g/mol}.\)\[\text{Percentage by mass of sulphur} = \left(\frac{32.07}{120.37}\right) \times 100\% \approx 26.64\%\]

Step 7: Calculate the Molar Mass of Epinephrine

Epinephrine (\(\mathrm{C}_{9}\mathrm{H}_{13}\mathrm{NO}_{3}\)) consists of 9 carbons, 13 hydrogens, 1 nitrogen, and 3 oxygens. Calculate its molar mass:\[\text{Molar mass of } \mathrm{C}_{9} \mathrm{H}_{13} \mathrm{NO}_{3} = (9 \times 12.01) + (13 \times 1.01) + 14.01 + (3 \times 16.00) = 183.21 \text{ g/mol}.\]

Step 8: Calculate Percentage by Mass of Nitrogen in Epinephrine

Using the formula for mass percent, calculate the percentage by mass of nitrogen:\[\text{Percentage by mass} = \left(\frac{\text{Total mass of nitrogen}}{\text{Molar mass of the compound}}\right) \times 100\%\]\(\text{Total mass of nitrogen} = 14.01 \text{ g/mol}.\)\[\text{Percentage by mass of nitrogen} = \left(\frac{14.01}{183.21}\right) \times 100\% \approx 7.65\%\]

Step 9: Calculate the Molar Mass of Sulcatol

Determine the molar mass of sulcatol (\(\mathrm{C}_{8}\mathrm{H}_{16}\mathrm{O}\)):\[\text{Molar mass of } \mathrm{C}_{8} \mathrm{H}_{16} \mathrm{O} = (8 \times 12.01) + (16 \times 1.01) + 16.00 = 128.24 \text{ g/mol}.\]

Step 10: Calculate Percentage by Mass of Oxygen in Sulcatol

Now calculate the percentage by mass of oxygen in sulcatol:\[\text{Percentage by mass} = \left(\frac{\text{Total mass of oxygen}}{\text{Molar mass of the compound}}\right) \times 100\%\]\(\text{Total mass of oxygen} = 16.00 \text{ g/mol}.\)\[\text{Percentage by mass of oxygen} = \left(\frac{16.00}{128.24}\right) \times 100\% \approx 12.48\%\]

Step 11: Calculate the Molar Mass of Sucrose

For sucrose (\(\mathrm{C}_{12}\mathrm{H}_{22}\mathrm{O}_{11}\)), calculate its molar mass:\[\text{Molar mass of } \mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11} = (12 \times 12.01) + (22 \times 1.01) + (11 \times 16.00) = 342.30 \text{ g/mol}.\]

Step 12: Calculate Percentage by Mass of Carbon in Sucrose

Finally, calculate the percentage by mass of carbon in sucrose:\[\text{Percentage by mass} = \left(\frac{\text{Total mass of carbon}}{\text{Molar mass of the compound}}\right) \times 100\%\]\(\text{Total mass of carbon} = 12 \times 12.01 = 144.12 \text{ g/mol}.\)\[\text{Percentage by mass of carbon} = \left(\frac{144.12}{342.30}\right) \times 100\% \approx 42.11\%\]

Key Concepts

Molar Mass Calculation

The molar mass of a compound is the sum of the atomic masses of all the atoms present in its chemical formula. For example, when determining the molar mass of methane (CH1), you must consider the contributions of both carbon and hydrogen. One atom of carbon contributes approximately 12.01 g/mol, while the four atoms of hydrogen contribute about 4 7 1.01 g/mol. Thus, the molar mass of methane becomes 16.05 g/mol. This is a cornerstone step that enables us to further explore other properties of compounds, such as their percentage composition.

Chemical Formula

A chemical formula represents the types and numbers of atoms in a chemical compound. Understanding chemical formulas is vital, as they provide the basis for calculating molar masses and elemental composition. Take vitamin E's formula: C29 H50 O2. It specifies that the compound consists of 29 carbon atoms, 50 hydrogen atoms, and 2 oxygen atoms. Recognizing and interpreting these formulas allows us to calculate the total mass contributed by each element and, consequently, the compound's properties like mass percent.

Mass Percent

Mass percent, often called mass percentage, is the fraction of an element's mass compared to the total molar mass of the compound, expressed as a percentage. To find the mass percent, first calculate the total mass of the element in the compound, then divide it by the molar mass of the compound, and multiply by 100. For instance, in magnesium sulphate (MgSO4), if you need the mass percent of sulphur, use the expressed formula:

• Total mass of sulphur is 32.07 g/mol
• Molar mass of magnesium sulphate is 120.37 g/mol

Thus, the mass percent of sulphur becomes approximately 26.64%.

Elemental Analysis

Elemental analysis involves determining the quantitative composition of a compound by measuring the mass of each element. This technique is essential in fields like chemistry and biology, where understanding the detailed composition of substances is necessary. For example, by performing elemental analysis on substances like epinephrine (C9 H13 NO3), you gain insights into how different elements contribute to the compound's characteristics and reactivity. It also enables chemists to ensure compounds meet certain specifications, like purity or specific elemental ratios, which are crucial for scientific and industrial applications.