Question

Topical hydrocortisone is often used to treat a variety of skin conditions, such as insect bites, eczema, and rashes. Each molecule of hydrocortisone contains 21 atoms of carbon (plus other atoms). The mass percentage of carbon in hydrocortisone is \(69.58 \%\). What is the molar mass of hydrocortisone?

Short answer

The molar mass of hydrocortisone is approximately \(362.5 g/mol\).

Step-by-step solution

Find the mass of carbon in one mole of hydrocortisone

Given that the mass percentage of carbon in hydrocortisone is 69.58%. Let the total mass of hydrocortisone in one mole be x grams. Then, the mass of carbon in one mole of hydrocortisone can be calculated as: Mass of carbon = (69.58 / 100) * x

Find the molar mass of carbon

We know that the molar mass of carbon is 12.01 g/mol. Since there are 21 carbon atoms in one molecule of hydrocortisone, the total mass of carbon in one mole of hydrocortisone is: Total mass of carbon = 21 * 12.01 g/mol = 252.21 g/mol

Set up an equation to solve for the molar mass of hydrocortisone

From step 1, we have: Mass of carbon = 0.6958 * x From step 2, we have: Mass of carbon = 252.21 g/mol Now we can combine these two equations: 0.6958 * x = 252.21 g/mol

Solve for the molar mass of hydrocortisone

To find the molar mass of hydrocortisone (x), we can simply divide both sides of the equation by 0.6958: x = 252.21 g/mol / 0.6958 x ≈ 362.5 g/mol So, the molar mass of hydrocortisone is approximately 362.5 g/mol.

Key concepts

Molar Mass Calculation

Understanding molar mass is critical when working in chemistry, especially when dealing with reactions and compounds. Molar mass is defined as the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is calculated by adding up the atomic mass of each element constituting the compound multiplied by the number of atoms of that element in one molecule of the compound.

In the exercise, calculating the molar mass of hydrocortisone involves knowing the mass percentage of carbon and the atomic mass of carbon. With these two pieces of information, you set up a proportion to solve for the unknown total mass of hydrocortisone. The steps taken involve both algebra and an understanding of stoichiometry - a concept that ties the molar masses of elements and compounds to quantitative aspects of chemical reactions.

Mass Percentage

The mass percentage is a way of expressing the concentration of an element in a compound or a component in a mixture. It is calculated as the mass of the component divided by the total mass of the mixture or compound, multiplied by 100%. For instance, a 69.58% mass percentage of carbon in hydrocortisone means that in every 100 grams of hydrocortisone, there are 69.58 grams of carbon.

To calculate the molar mass from the mass percentage, one must first determine the mass of the component (e.g., carbon) within the total mass of the compound for one mole. This step sets the groundwork for further calculations in stoichiometry and is instrumental when comparing the composition of different substances.

Mole Concept

The mole concept is a fundamental pillar in chemistry, providing a bridge between the micro world (atoms, molecules) and the macro world (grams, liters). A mole is defined as the amount of substance containing as many entities (atoms, molecules, ions, etc.) as there are atoms in exactly 12 grams of carbon-12. This number, known as Avogadro's number, is approximately \(6.022 \times 10^{23}\) entities per mole.

Understanding mole concept allows us to calculate the number of atoms, molecules, or ions in a given mass of a substance. It also enables chemists to count particles by weighing, which is the basis for stoichiometry. In the exercise we looked at, the mole concept helps understand how the mass of carbon relates to the molar mass of hydrocortisone through the known quantity of carbon atoms per molecule.

Atomic Mass

Atomic mass is the mass of a single atom, usually expressed in atomic mass units (amu) and it is roughly equivalent to the mass number of the element - the number of protons and neutrons. For example, the atomic mass of carbon is about 12.01 amu. When dealing with moles, atomic mass expresses how much one mole of an element weighs in grams per mole, mirroring the number of atomic mass units.

Knowing the atomic mass of carbon and the fact that hydrocortisone contains 21 atoms of carbon per molecule allows us to calculate the total mass contributed by carbon to the molar mass of hydrocortisone. This step is crucial because it relates a single element's characteristics to the overall molar mass of the compound it is a part of.