Problem 163 - Introductory Chemistry Atoms First

Question

The amino acid arginine is made up of \(41.37 \%\) by mass \(\mathrm{C}, 8.10 \%\) by mass \(\mathrm{H}, 32.16 \%\) by mass \(\mathrm{N}\), and \(18.37 \%\) by mass \(\mathrm{O}\). If the molar mass of arginine is approximately \(174 \mathrm{~g} / \mathrm{mol}\), what is its molecular formula?

Step-by-Step Solution

Answer

The molecular formula of arginine is C₃H₇N₂O.

Step 1: Determine the number of grams of each element from the mass percentages

To find out how much of each element arginine contains, we will use the mass percentage values given. Assuming a 100g sample, we can calculate the amount of each element in grams: - C: 41.37% × 100g = 41.37g - H: 8.10% × 100g = 8.10g - N: 32.16% × 100g = 32.16g - O: 18.37% × 100g = 18.37g

Step 2: Calculate the number of moles of each element

Now that we have the amount of each element in grams, we can convert them to moles using the molar masses of each element. - Molar mass of C: 12.01 g/mol - Molar mass of H: 1.01 g/mol - Molar mass of N: 14.01 g/mol - Molar mass of O: 16.00 g/mol Number of moles for each element: - Moles of C: \( \frac{41.37g}{12.01g/mol} = 3.44mol \) - Moles of H: \( \frac{8.10g}{1.01g/mol} = 8.02mol \) - Moles of N: \( \frac{32.16g}{14.01g/mol} = 2.30mol \) - Moles of O: \( \frac{18.37g}{16.00g/mol} = 1.15mol \)

Step 3: Determine the simplest whole number ratio between the numbers of moles of each element

To obtain the simplest whole number ratio, divide the number of moles of each element by the element with the lowest number of moles: - Ratio of C: \( \frac{3.44}{1.15} = 2.99 \approx 3 \) - Ratio of H: \( \frac{8.02}{1.15} = 6.97 \approx 7 \) - Ratio of N: \( \frac{2.30}{1.15} = 2.00 \approx 2 \) - Ratio of O: \( \frac{1.15}{1.15} = 1.00 \approx 1 \) The simplest whole number ratio is approximately 3:7:2:1.

Step 4: Write the molecular formula

Using the simplest whole number ratio obtained in step 3, we can write the molecular formula of arginine as: C₃H₇N₂O So, the molecular formula of arginine is C₃H₇N₂O.

Key Concepts

Amino Acids

Amino acids are the building blocks of proteins, which play crucial roles in numerous biological processes. They are organic compounds composed of carbon ( C ), hydrogen ( H ), nitrogen ( N ), oxygen ( O ), and sometimes sulfur ( S ).
Amino acids link together in specific sequences to form proteins, and each sequence determines the protein's structure and function. Arginine, the amino acid discussed in this exercise, is known for its involvement in protein synthesis, hormone regulation, and immune function.
Understanding the composition of amino acids is essential for studying their properties.

• Amino acids have a central carbon atom.
• They feature an amino group (- NH ₂), a carboxyl group (- COOH ), and a side chain or R-group, which varies among different amino acids.

Each amino acid's unique side chain helps define its role and properties in biological systems.

Mass Percentage

Mass percentage is a way to express the concentration of an element in a compound. It tells you what percentage of a compound's mass comes from a particular element.
When calculating the molecular formula, knowing the mass percentages of each element helps determine the empirical formula, which can be used to find the molecular formula.

• Mass percentage is calculated as: \( \frac{\text{mass of the element}}{\text{total mass of the compound}} \times 100 \)
• It's useful when analyzing compounds to find relationships between elements.

In the exercise above, the given mass percentages of carbon, hydrogen, nitrogen, and oxygen guide the calculation of the molecular formula for arginine, a vital process for understanding its composition and properties.

Molar Mass

Molar mass is a critical concept for chemists as it allows the conversion between grams and moles, bridging the gap between macroscopic and molecular scales.
Each element has its own molar mass, typically expressed in grams per mole (\text{g/mol}). This value is obtained from the atomic mass found on the periodic table.

• Carbon (C) has a molar mass of \(12.01 \text{ g/mol}\).
• Hydrogen (H) has a molar mass of \(1.01 \text{ g/mol}\).
• Nitrogen (N) has a molar mass of \(14.01 \text{ g/mol}\).
• Oxygen (O) has a molar mass of \(16.00 \text{ g/mol}\).

In the exercise, molar mass enables the conversion of grams to moles for each element in arginine, aiding in the subsequent calculation of the molecular formula.

Elements

Elements are pure substances consisting of only one type of atom, each indicated by a unique chemical symbol.
They are the building blocks of matter and combine in various ways to form compounds and molecules.

• Carbon ( C ), oxygen ( O ), nitrogen ( N ), and hydrogen ( H ) are essential elements in amino acids, each serving vital functions in biological molecules.
• These elements combine to determine the chemical and physical properties of the substances they compose.

Understanding the role of these elements in forming larger molecules, such as amino acids like arginine, is crucial for comprehending their biological significance and how they interact in living organisms.