Question

The molar mass of caffeine is \(194.19 \mathrm{~g}\). Is the molecular formula of caffeine \(\mathrm{C}_{4} \mathrm{H}_{5} \mathrm{~N}_{2} \mathrm{O}\) or \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2} ?\)

Short answer

Caffeine is \(\text{C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2}\).

Step-by-step solution

Calculate Molar Mass of C4H5N2O

To determine the molecular formula, first, calculate the molar mass of \(\text{C}_{4}\text{H}_{5}\text{N}_{2}\text{O}\). Carbon (C) has an atomic mass of approximately \(12.01 \text{ g/mol}\), Hydrogen (H) \(1.01 \text{ g/mol}\), Nitrogen (N) \(14.01 \text{ g/mol}\), and Oxygen (O) \(16.00 \text{ g/mol}\).\[\text{Molar Mass of C}_{4}\text{H}_{5}\text{N}_{2}\text{O} = (4 \times 12.01) + (5 \times 1.01) + (2 \times 14.01) + (1 \times 16.00)\] Calculating gives: \[48.04 + 5.05 + 28.02 + 16.00 = 97.11 \text{ g/mol}\]

Calculate Molar Mass of C8H10N4O2

Next, calculate the molar mass of \(\text{C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2}\). Using the same atomic masses as before: \[\text{Molar Mass of C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2} = (8 \times 12.01) + (10 \times 1.01) + (4 \times 14.01) + (2 \times 16.00)\] Calculating gives: \[96.08 + 10.10 + 56.04 + 32.00 = 194.22 \text{ g/mol}\]

Compare Calculated Molar Masses to Given Molar Mass

The given molar mass of caffeine is \(194.19 \text{ g/mol}\). After calculations, the molar mass of \(\text{C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2}\) is \(194.22 \text{ g/mol}\), which closely matches the given molar mass, while \(\text{C}_{4}\text{H}_{5}\text{N}_{2}\text{O}\) is only \(97.11 \text{ g/mol}\).

Conclusion

The molecular formula of caffeine is \(\text{C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2}\) because its molar mass \(194.22 \text{ g/mol}\) is very close to the given molar mass \(194.19 \text{ g/mol}\).

Key concepts

Molar Mass Calculation

To solve problems involving molecular formulas, understanding molar mass calculation is essential. The molar mass of a compound is the sum of the atomic masses of all atoms in its molecular formula. It reflects the mass of one mole of that compound and is measured in grams per mole (g/mol).
When you calculate the molar mass of a compound, you multiply the number of each type of atom by its respective atomic mass. Sum up these values to get the total molar mass of the compound. For example, to find the molar mass of caffeine with a formula \(\text{C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2}\), you would:

• Multiply the atomic mass of carbon (12.01 g/mol) by 8.
• Multiply the atomic mass of hydrogen (1.01 g/mol) by 10.
• Multiply the atomic mass of nitrogen (14.01 g/mol) by 4.
• Multiply the atomic mass of oxygen (16.00 g/mol) by 2.

Then, add all these to get the total molar mass, which helps in identifying the molecular formula based on given data.

Atomic Mass

Atomic mass is a fundamental concept in chemistry, providing the mass of an atom of a chemical element. It is usually expressed in units of atomic mass units (amu), which are numerically equivalent to grams per mole (g/mol) when dealing with a mole-sized sample of the element.
Each element in the periodic table has an atomic mass, which takes into account the average mass of its isotopes, weighted by their natural abundance. For instance:

• Carbon has an atomic mass of approximately 12.01 g/mol.
• Hydrogen's atomic mass is about 1.01 g/mol.
• Nitrogen's atomic mass is 14.01 g/mol.
• Oxygen carries an atomic mass of 16.00 g/mol.

These atomic masses are essential when calculating the molar mass of compounds and for understanding the chemical composition of molecules such as caffeine.

Chemical Composition

The chemical composition of a compound provides insight into the types and amounts of atoms making it up. It's expressed as a molecular formula which indicates each type of atom present and how many of each are found in a single molecule. For example, caffeine’s molecular formula is \(\text{C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2}\).
This notation tells us that caffeine consists of:

• 8 carbon atoms.
• 10 hydrogen atoms.
• 4 nitrogen atoms.
• 2 oxygen atoms.

Understanding chemical composition is vital for determining properties like molar mass and to infer potential chemical reactivity. Knowing how to decipher chemical formulas enables chemists to calculate molar masses and predict the structures of compounds.

Caffeine Structure

Caffeine is a naturally occurring stimulant found in coffee, tea, and other sources. Its structural formula \(\text{C}_{8}\text{H}_{10}\text{N}_{4}\text{O}_{2}\) not only defines the number of each atom but also points to its arrangement in space, influencing its physical and chemical properties.
Caffeine belongs to a class of compounds known as xanthines and is closely related to other methylxanthines like theobromine and theophylline.
The molecular structure of caffeine can be broken down as follows:

• The presence of nitrogen atoms in caffeine contributes to its role as a central nervous system stimulant.
• Oxygen atoms play a key role in the molecule's chemical behavior, especially in solubility and interaction with other substances.
• Carbon and hydrogen atoms form the backbone of the molecule, holding the structure together.

This unique arrangement allows caffeine to interact with receptor sites in the brain, leading to its stimulating effects.