Question

Atomic Numbers in Life. A typical bacterium has a volume of about 1 cubic micrometer. A typical atom has a diameter of about 0.1 nanometer. Approximately how many atoms are in a bacterium?

Short answer

A bacterium contains approximately \(1.91 \times 10^{12}\) atoms.

Step-by-step solution

Understand the dimensions

First, note the given measurements. The bacterium's volume is 1 cubic micrometer ( µm³), and the diameter of a typical atom is 0.1 nanometer (nm). We need to determine how many 0.1 nm-sized atoms will fit into the volume of a 1 µm³ bacterium.

Convert units

Convert the units so they are consistent. 1 micrometer is equal to 1000 nanometers. Thus, the volume of the bacterium in cubic nanometers is \((1 \, \mu m)^3 = (1000 \, nm)^3 = 10^9 \, nm^3\).

Calculate volume of an atom

Assuming spherical atoms, the volume of an atom with a diameter of 0.1 nm can be calculated using the formula for the volume of a sphere: \(V = \frac{4}{3} \pi r^3\), where the radius \(r = 0.05 \, nm\). Calculating gives \(V \approx \frac{4}{3} \pi (0.05)^3 \approx 5.24 \times 10^{-4} \, nm^3\).

Determine number of atoms

Divide the volume of the bacterium by the volume of one atom to estimate the number of atoms: \(\frac{10^9 \, nm^3}{5.24 \times 10^{-4} \, nm^3} \approx 1.91 \times 10^{12}\) atoms.

Key concepts

Atomic measurement

When we talk about atomic measurement, the size is often unfathomable because atoms are incredibly small. In this exercise, we are dealing with diameters measured in nanometers (nm), which are a billionth of a meter. Since a typical atom has a diameter of about 0.1 nm, comparing it to everyday objects helps illustrate its size. To give perspective: if an atom were the size of a basketball, a bacterium would be the size of a small building.
Understanding these measurements helps us grasp the minuscule yet foundational building blocks of matter. We use these measurements to comprehend how atoms assemble to form more complex structures, such as molecules, cells, and ultimately, all living organisms.

Unit conversion in science

Unit conversion is crucial in scientific calculations to ensure accuracy and consistency. In this problem, we convert micrometers to nanometers to keep our units consistent for calculation. For clarity:

• 1 micrometer (µm) = 1000 nanometers (nm)
• Volume in cubic micrometers = 1 µm³ = 10^9 nm³

Unit conversion ensures that calculations align with a common standard, avoiding errors that could arise from mismatched units.
Having a solid understanding of how to convert units between different metric scales is essential for scientists and engineers. It ensures that analyses are reliable, facilitating communication and understanding within the scientific community.

Volume calculation

To calculate the volume of spherical objects, like atoms, we use the formula for the volume of a sphere: \(V = \frac{4}{3} \pi r^3\). This formula shows us how the volume scales with its radius. In our case, the atomic radius is 0.05 nm, as it is half of the diameter.
Applying this formula, we found the volume of an atom to be approximately 5.24 imes 10^{-4} ext{ nm}^3. Such precise calculations are vital when dealing with microscopic entities, ensuring assumptions or errors do not accumulate into significant inaccuracies.
This volume calculation enables further insight into how densely packed or spacious these entities like bacteria might be, based on atom packing.

Bacteria composition

Bacteria are composed of countless atoms, arranged to perform complex biological processes. Understanding how many atoms fit into a bacterium involves not only math, but also insights into biological complexity.
With the given dimensions, a bacterium, having around 1.91 imes 10^{12} atoms, highlights the exquisite detail in even the most basic living entities. This atomic perspective helps us appreciate the minimum scale at which life operates.
Recognizing this scale is critical in fields such as microbiology and nanotechnology. It opens doors for innovations like targeted drug delivery or efficient biosensors, capitalizing on the intricacies of atomic and bacterial structures.