Problem 67 - University Physics with Modern Physics

Question

How Many Atoms Are You? Estimate the number of atoms in the body of a \(50-\mathrm{kg}\) physics student. Note that the human body is mostly water, which has molar mass 18.0 \(\mathrm{g} / \mathrm{mol}\) and that each water molecule contains three atoms.

Step-by-Step Solution

Answer

A 50 kg person has approximately \(3.012 \times 10^{27}\) atoms in their body.

Step 1: Identify the composition of the body

The human body is primarily composed of water, which makes up about 60% of the body's weight. Therefore, we can estimate that a 50 kg person is made up of approximately 30 kg of water.

Step 2: Convert the mass of water to grams

To perform calculations in chemistry, it's common to convert mass into grams. Therefore, convert 30 kg of water to grams: \(30 \text{ kg} = 30,000 \text{ g}\).

Step 3: Calculate moles of water

Using the molar mass of water (18.0 g/mol), calculate the number of moles of water in the body. \(\text{Number of moles} = \frac{30,000 \text{ g}}{18.0 \text{ g/mol}} = 1666.67 \text{ moles}\).

Step 4: Determine number of molecules of water

Use Avogadro's number \(6.022 \times 10^{23}\) molecules/mol to find the number of molecules in the body. \(1666.67 \text{ moles} \times 6.022 \times 10^{23} \text{ molecules/mol} \approx 1.004 \times 10^{27}\text{ molecules}\).

Step 5: Calculate the total number of atoms

Since each water molecule contains three atoms (2 hydrogen and 1 oxygen), multiply the number of molecules by 3 to find the total number of atoms: \(1.004 \times 10^{27} \text{ molecules} \times 3 \approx 3.012 \times 10^{27}\text{ atoms}\).

Key Concepts

Molar Mass of Water

The concept of molar mass is essential in understanding chemical reactions and the composition of substances. In the context of water, the molar mass is the total mass of one mole of water molecules, which consists of two hydrogen atoms and one oxygen atom. The molar mass of hydrogen is approximately 1.0 g/mol, and for oxygen, it's 16.0 g/mol. Therefore, the molar mass of water (H₂O) is calculated as follows:

• Molar mass of H₂: 2 x 1.0 g/mol = 2.0 g/mol
• Molar mass of O: 1 x 16.0 g/mol = 16.0 g/mol
• Total molar mass of H₂O: 2.0 g/mol + 16.0 g/mol = 18.0 g/mol

This calculation is important because it helps determine the number of moles in a given mass of water, which is crucial for further calculations such as determining the number of molecules and ultimately, the number of atoms in a sample like the human body.

Avogadro's Number

Avogadro's number is a fundamental constant used in chemistry to describe the number of units (usually atoms or molecules) in a mole of any substance. Its value is approximately 6.022 x 10²³. This means that one mole of any element or compound contains exactly 6.022 x 10²³ atoms or molecules.

This constant is vital because it acts as a bridge between the atomic scale and the macroscopic scale that we can measure. For instance, when calculating the number of molecules in a given mass, as in the body's water content, Avogadro's number allows us to convert from moles to molecules. In our example of a 50 kg person, once we know the number of moles, we can use Avogadro's number to calculate the precise number of water molecules, leading us closer to finding the total number of atoms.

Human Body Composition

Human body composition plays a key role in understanding how many atoms comprise a person. Primarily composed of water, which makes up roughly 60% of body weight, the body's chemical composition also includes other elements like carbon, nitrogen, and calcium, among others.

For a simplified calculation, focusing on water is a helpful approach because the majority of our body's atoms are found there. By estimating that 60% of a 50 kg human body is water, we calculate that there are about 30 kg of water. This estimate helps simplify the process of determining the number of moles, molecules, and atoms, giving a substantial figure. However, it's crucial to remember that while water dominates, every other molecule in the body contributes to the total atom count, albeit in smaller quantities.

Chemical Calculations

Chemical calculations involve a series of mathematical steps to understand the composition of substances at a molecular level. These calculations often require converting mass to moles, determining the number of molecules, and finally, calculating the total number of atoms.

• First, convert the person's mass of body water from kilograms to grams, as it's easier to work in smaller units in chemistry.
• Next, use the molar mass of water to determine the number of moles of water. For example, 30,000 grams of water divided by 18.0 g/mol gives the moles of water available.
• Avogadro's number then helps convert moles to molecules, allowing us to see how many water molecules are present in the body.
• Finally, knowing each molecule has three atoms, we multiply the total number of molecules by 3 to find the number of atoms. This step-by-step approach breaks down complex processes into manageable calculations.

Understanding these steps gives insight not only into the specific problem of calculating atoms in the body but also into broader principles applicable in many areas of chemistry.