Problem 65 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Cortisone is a molecular substance containing 21 atoms of carbon per molecule. The mass percentage of carbon in cortisone is \(69.98 \%\). What is the molecular mass of cortisone? (a) \(180.05\) (b) \(360.1\) (c) \(312.8\) (d) \(205.8\)

Step-by-Step Solution

Answer

The molecular mass of cortisone is approximately 360.1 grams/mole, which corresponds to option (b).

Step 1: Calculate the total mass of carbon in one molecule

The mass percentage of carbon in cortisone is given as 69.98%. To find the total mass of carbon in one molecule of cortisone, let the molecular mass of cortisone be represented by M. The mass of carbon per molecule of cortisone can be calculated by multiplying the mass percentage of carbon by the molecular mass (M), which gives us (69.98/100) * M.

Step 2: Calculate the mass contribution of the carbon atoms

Since there are 21 carbon atoms in cortisone and the atomic mass of carbon is approximately 12.01 g/mol, the total mass contribution of the carbon atoms can be found by multiplying the number of carbon atoms by the atomic mass of one carbon atom, which yields 21 * 12.01.

Step 3: Relate the mass contribution to the molecular mass

Now we have two expressions for the mass of carbon in cortisone: one from the mass percentage (Step 1) and one from the atomic mass (Step 2). Equate these to solve for the molecular mass (M) of cortisone. We set (69.98/100) * M equal to 21 * 12.01 and then solve for M.

Step 4: Calculate the molecular mass of cortisone

By solving the equation from Step 3 for M, we find M = (21 * 12.01) / (69.98/100). This gives us the molecular mass of cortisone.

Key Concepts

Mass Percentage

Understanding the mass percentage of an element within a compound is crucial in chemistry. It represents the fraction of the total mass of a substance that is occupied by a particular component and is expressed as a percentage. It is calculated by the formula:
\[ \text{Mass percentage} = \left( \frac{\text{Mass of component}}{\text{Total mass of the compound}} \right) \times 100 \% \]
In the context of the cortisone example, the mass percentage enables us to infer the mass of carbon relative to the whole molecule. For instance, if carbon constitutes 69.98% of cortisone's mass, for every 100 grams of cortisone, approximately 69.98 grams is carbon. This concept connects to stoichiometry, as determining proportions is fundamental in calculating reactants and products in chemical reactions.

Mole Concept

The mole concept is a bridge between the microscopic world of atoms and the macroscopic world we experience. A mole is a unit that measures the amount of substance and is based on the number of atoms in 12 grams of carbon-12, amounting to Avogadro's number (approximately \(6.022 \times 10^{23}\) entities). In stoichiometry and molecular mass calculations, it's essential to use the mole concept to convert between the mass of a substance and the number of moles. For example, knowing that one mole of carbon atoms has a mass of 12.01 grams lets us determine how many moles are in a given mass of carbon, and conversely, how much one mole of carbon contributes to the mass of a compound like cortisone.

Atomic Mass

The atomic mass of an element, measured in atomic mass units (amu) or grams per mole (g/mol), corresponds to the mass of an individual atom. On the periodic table, the atomic mass of carbon is approximately 12.01 g/mol. When calculating the molecular mass of a compound, understanding the atomic mass of each constituent element is necessary. We use the atomic mass to calculate how much a particular number of atoms of that element contributes to the total mass. With cortisone's 21 carbon atoms, we multiply their individual atomic mass by 21 to find their collective mass contribution in the molecule.

Stoichiometry

Stoichiometry, at its core, is about the quantitative relationships within a chemical reaction. It deals with the ratios of reactants and products and allows for the prediction of yields. When solving for the molecular mass of a substance like cortisone, stoichiometry principles are applied. By balancing mass contributions and percentages, stoichiometry enables the calculation of unknown masses through known quantities. This involves using the mole concept, atomic mass, and mass percentage to construct and solve equations relating the mass of individual elements to the total mass of a compound, as seen in the calculated molecular mass of cortisone.