Problem 24 - Chemistry: The Central Science

Question

Calculate the percentage by mass of the indicated element in the following compounds: (a) carbon in acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}\), a gas used in welding; (b) hydrogen in ascorbic acid, \(\mathrm{HC}_{6} \mathrm{H}_{7} \mathrm{O}_{6}\), also known as vitamin \(\mathrm{C}\); (c) hydrogen in ammonium sulfate, \(\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}\), a substance used as a nitrogen fertilizer; (d) platinum in \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\), a chemotherapy agent called cisplatin; (e) oxygen in the female sex hormone estradiol, \(\mathrm{C}_{18} \mathrm{H}_{24} \mathrm{O}_{2} ;\) (f) carbon in capsaicin, \(\mathrm{C}_{18} \mathrm{H}_{27} \mathrm{NO}_{3}\), the com- pound that gives the hot taste to chili peppers.

Step-by-Step Solution

Answer

The percentage by mass of the indicated elements in the given compounds are as follows: (a) Carbon in acetylene \(\mathrm{C}_{2}\mathrm{H}_{2}\): \(92.24\%\) (b) Hydrogen in ascorbic acid \(\mathrm{HC}_{6}\mathrm{H}_{7}\mathrm{O}_{6}\): \(4.59\%\) (c) Hydrogen in ammonium sulfate \(\left(\mathrm{NH}_{4}\right)_{2}\mathrm{SO}_{4}\): \(4.58\%\) (d) Platinum in \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\): \(65.02\%\) (e) Oxygen in estradiol \(\mathrm{C}_{18}\mathrm{H}_{24}\mathrm{O}_{2}\): \(11.75\%\) (f) Carbon in capsaicin \(\mathrm{C}_{18}\mathrm{H}_{27}\mathrm{NO}_{3}\): \(70.79\%\)

Step 1: Calculate the molar mass of \(\mathrm{C}_{2}\mathrm{H}_{2}\)

Molar mass of \(\mathrm{C}_{2}\mathrm{H}_{2}\) = (2 x 12.01) + (2 x 1.01) = 26.04 g/mol

Step 2: Calculate the mass of carbon in the compound and find the percentage

Mass of carbon = 2 x 12.01 = 24.02 g Percentage of carbon in \(\mathrm{C}_{2}\mathrm{H}_{2}\) = \[\frac{24.02}{26.04} \times 100\% = 92.24\%\] (b) Calculate the percentage by mass of hydrogen in ascorbic acid, \(\mathrm{HC}_{6}\mathrm{H}_{7}\mathrm{O}_{6}\).

Step 1: Calculate the molar mass of \(\mathrm{HC}_{6}\mathrm{H}_{7}\mathrm{O}_{6}\)

Molar mass of \(\mathrm{HC}_{6}\mathrm{H}_{7}\mathrm{O}_{6}\) = (1.01) + (6 x 12.01) + (8 x 1.01) + (6 x 16) = 176.14 g/mol

Step 2: Calculate the mass of hydrogen in the compound and find the percentage

Mass of hydrogen = 8 x 1.01 = 8.08 g Percentage of hydrogen in \(\mathrm{HC}_{6}\mathrm{H}_{7}\mathrm{O}_{6}\) = \[\frac{8.08}{176.14} \times 100\% = 4.59\%\] (c) Calculate the percentage by mass of hydrogen in ammonium sulfate, \(\left(\mathrm{NH}_{4}\right)_{2}\mathrm{SO}_{4}\).

Step 1: Calculate the molar mass of \(\left(\mathrm{NH}_{4}\right)_{2}\mathrm{SO}_{4}\)

Molar mass of \(\left(\mathrm{NH}_{4}\right)_{2}\mathrm{SO}_{4}\) = (2 x (1.01 x 3)) + (32.06) + (4 x 16) = 132.14 g/mol

Step 2: Calculate the mass of hydrogen in the compound and find the percentage

Mass of hydrogen = 2 x 3 x 1.01 = 6.06 g Percentage of hydrogen in \(\left(\mathrm{NH}_{4}\right)_{2}\mathrm{SO}_{4}\) = \[\frac{6.06}{132.14} \times 100\% = 4.58\%\] (d) Calculate the percentage by mass of platinum in \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\).

Step 1: Calculate the molar mass of \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\)

Molar mass of \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\) = 195.08 + (2 x 35.45) + (2 x (1.01 x 3 + 14.01)) = 300.05 g/mol

Step 2: Calculate the mass of platinum in the compound and find the percentage

Mass of platinum = 195.08 g Percentage of platinum in \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\) = \[\frac{195.08}{300.05} \times 100\% = 65.02\%\] (e) Calculate the percentage by mass of oxygen in estradiol, \(\mathrm{C}_{18}\mathrm{H}_{24}\mathrm{O}_{2}\).

Step 1: Calculate the molar mass of \(\mathrm{C}_{18}\mathrm{H}_{24}\mathrm{O}_{2}\)

Molar mass of \(\mathrm{C}_{18}\mathrm{H}_{24}\mathrm{O}_{2}\) = (18 x 12.01) + (24 x 1.01) + (2 x 16) = 272.38 g/mol

Step 2: Calculate the mass of oxygen in the compound and find the percentage

Mass of oxygen = 2 x 16 = 32 g Percentage of oxygen in \(\mathrm{C}_{18}\mathrm{H}_{24}\mathrm{O}_{2}\) = \[\frac{32}{272.38} \times 100\% = 11.75\%\] (f) Calculate the percentage by mass of carbon in capsaicin, \(\mathrm{C}_{18}\mathrm{H}_{27}\mathrm{NO}_{3}\).

Step 1: Calculate the molar mass of \(\mathrm{C}_{18}\mathrm{H}_{27}\mathrm{NO}_{3}\)

Molar mass of \(\mathrm{C}_{18}\mathrm{H}_{27}\mathrm{NO}_{3}\) = (18 x 12.01) + (27 x 1.01) + (14.01) + (16) + (3 x 16) = 305.42 g/mol

Step 2: Calculate the mass of carbon in the compound and find the percentage

Mass of carbon = 18 x 12.01 = 216.18 g Percentage of carbon in \(\mathrm{C}_{18}\mathrm{H}_{27}\mathrm{NO}_{3}\) = \[\frac{216.18}{305.42} \times 100\% = 70.79\%\]

Key Concepts

Molar Mass Calculation

The first step in determining the percentage composition of any element in a compound is to calculate the molar mass of the entire compound. The molar mass is simply the sum of the masses of all the atoms present in a molecule of the compound. Each element in the periodic table has a unique atomic mass, typically expressed in grams per mole (g/mol).
For example, in the compound acetylene, \(\mathrm{C}_{2}\mathrm{H}_{2}\), the molar mass is calculated by adding the masses of carbon and hydrogen. There are two carbon atoms and two hydrogen atoms, so the molar mass calculation is:

• \(2 \times 12.01\) (molar mass of carbon) = 24.02 g/mol
• \(2 \times 1.01\) (molar mass of hydrogen) = 2.02 g/mol

Thus, the total molar mass of acetylene is:\(26.04\) g/mol.
This is a fundamental step, as it sets the stage for calculating the elemental mass percentages in the compound.

Elemental Mass Percentage

Elemental mass percentage allows us to understand what portion of a compound's mass comes from a specific element. After determining the molar mass of the compound, the next step is to find the mass of the target element within it. This helps to figure out how much of the molecule's total mass is due to that particular element.
To illustrate, consider vitamin C, or ascorbic acid: \(\mathrm{HC}_{6} \mathrm{H}_{7} \mathrm{O}_{6}\). To find the percentage of hydrogen, calculate the mass of all hydrogen atoms:

• Hydrogen mass: \(8 \times 1.01 = 8.08\) g

Then, to find the mass percentage of hydrogen, use the formula: \[ \text{Percentage} = \left(\frac{\text{mass of the element in the compound}}{\text{molar mass of the compound}}\right) \times 100\% \]Apply the numbers: \[ \frac{8.08}{176.14} \times 100\% = 4.59\% \]. This means that 4.59% of the total mass of ascorbic acid is made up of hydrogen.

Chemical Compounds

Chemical compounds are substances formed by the combination of different elements bonded together in fixed proportions. Understanding the structure of a compound is crucial for accurately calculating its molar mass and the corresponding percentage composition of its elements.
For instance, let's explore platinum in cisplatin: \(\mathrm{PtCl}_{2}\left(\mathrm{NH}_{3}\right)_{2}\). This compound consists of one platinum atom, two chlorine atoms, and two ammonia groups.
To compute its molar mass, you would sum the molar masses of all constituent atoms and groups:

• Mass of platinum: \(195.08\) g/mol
• Mass of chlorine: \(2 \times 35.45 = 70.9\) g/mol
• Mass of ammonia: \(2 \times (3 \times 1.01 + 14.01) = 34.06\) g/mol

This yields a total molar mass of \(300.05\) g/mol. Understanding how elements are organized within a compound helps elucidate the nature of its chemical reactions and interactions.

Mass Percentage Calculation

Mass percentage calculation is the process of finding the percentage of a specific element in a compound, which is an essential part of understanding the chemical composition of substances. It answers the question, "What fraction of the compound's mass does this particular element contribute?"
To calculate the mass percentage, divide the mass of the desired element by the molar mass of the compound, and then multiply by 100. This calculation is used because it is often important to know the relative amounts of each component, especially in chemical reactions and formulations.
Let's consider ammonium sulfate, \((\mathrm{NH}_{4})_{2}\mathrm{SO}_{4}\). To find the percentage of hydrogen, calculate as follows:

• Total mass of hydrogen: \(2 \times 4 \times 1.01 = 8.08\) g
• Total molar mass of compound: \(132.14\) g/mol

Thus, the percentage of hydrogen is: \[ \frac{6.06}{132.14} \times 100\% = 4.58\% \]. This provides insight into the compound's composition and potential reactivity.