Question

Oxygen Atoms in People. Figure 5.5 shows that oxygen makes up about \(65 \%\) of the mass of a human being. A single oxygen atom has a mass of \(2.66 \times 10^{-26} \mathrm{kg}\). (a) Use this fact to estimate the number of oxygen atoms in your body. (Hint: If you know your weight in pounds, you can convert to kilograms by dividing by 2.2.) (b) Compare your answer to the number of stars in the observable universe (which is roughly \(10^{22}\)).

Short answer

The number of oxygen atoms in your body is about \(1.67 \times 10^{27}\), far greater than the \(10^{22}\) stars in the observable universe.

Step-by-step solution

Convert Body Weight to Kilograms

Determine your body weight in kilograms by dividing your weight in pounds by 2.2. For example, if you weigh 150 pounds, the calculation is \( \frac{150}{2.2} \approx 68.18 \text{ kg} \).

Calculate Mass of Oxygen in Body

Calculate the mass of oxygen in your body by multiplying your total body mass by the percentage of mass that is oxygen. If your body mass is 68.18 kg, then the mass of oxygen is \( 0.65 \times 68.18 = 44.32 \text{ kg} \).

Determine Number of Oxygen Atoms

To find the number of oxygen atoms, divide the mass of oxygen in your body by the mass of a single oxygen atom. Using the previous calculation: \( \frac{44.32}{2.66 \times 10^{-26}} \approx 1.67 \times 10^{27} \) oxygen atoms.

Compare to Stars in the Universe

The number of stars in the observable universe is approximately \(10^{22}\). The number of oxygen atoms in the body \((1.67 \times 10^{27})\) is far greater than the number of stars, indicating that your body contains many more oxygen atoms than there are stars in the observable universe.

Key concepts

Mass of Oxygen Atom

Oxygen is a fundamental component of the human body, making up a significant portion of our total mass. Each oxygen atom has a tiny mass of approximately \(2.66 \times 10^{-26} \text{ kg}\). This value indicates just how small a single oxygen atom is, which is crucial when considering the immense number of these atoms present in our bodies.

Knowing the mass of a single oxygen atom helps in calculating larger-scale biological compositions, by allowing us to scale from the atomic level to measure quantities in the kilograms seen in living organisms.

When dealing with atoms, these small mass numbers mean that even a small human body is made up of an astronomically high number of these microscopic particles. This tiny mass is the building block for calculating the total number of oxygen atoms in the human body.

Number of Atoms Calculation

Once you know the mass of a single oxygen atom, you can calculate how many such atoms are in your body. This is a multi-step process:

• First, convert your weight from pounds to kilograms by dividing by 2.2.
• Next, calculate the total mass of oxygen in your body using the fact that oxygen constitutes about \(65\%\) of the human body mass.
• Finally, divide the total mass of oxygen by the mass of a single oxygen atom to determine the total number of atoms.

For example, if a person's body weight converted to kilograms is 68.18 kg, then the mass of oxygen makes up 44.32 kg of that weight. This is derived from multiplying the weight by 0.65.

To find out how many oxygen atoms are in 44.32 kg of oxygen, divide by the mass of one oxygen atom: \( \frac{44.32}{2.66 \times 10^{-26}} \approx 1.67 \times 10^{27} \) oxygen atoms. This calculation shows just how extensive the quantity of oxygen atoms is, giving us a glimpse into the molecular makeup of living organisms.

Observable Universe

When comparing the number of oxygen atoms in the human body to the number of stars in the observable universe, the results are surprising yet enlightening. The observable universe contains roughly \(10^{22}\) stars, an incredibly vast number.

However, when we compute that a single person's body holds about \(1.67 \times 10^{27}\) oxygen atoms, this number dwarfs the count of stars. It emphasizes the difference in scales between atomic numbers and celestial counts.

This comparison highlights the incredible density and multitude of atoms within us, all packed into the space of a human body. It's a stark reminder of the vastness at both microscopic and cosmic levels, reinforcing the fact that our bodies are complex systems made up of impossibly small yet numerous particles.