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Chapter 1: Problem 60

The human body contains about \(10^{14}\) cells, and the diameter of a typical cell is about \(10 \mu \mathrm{m}\). Like all ordinary matter, cells are made of atoms; a typical atomic diameter is \(0.1 \mathrm{nm}\). The number of atoms in the body is closest to \(\begin{array}{llll}\text { a. } 10^{14} & \text { b. } 10^{20} & \text { c. } 10^{30} . & \text { d. } 10^{40}\end{array}\).

Short Answer

Expert verified
The number of atoms in the body is closest to \(10^{30}\).

Step by step solution

01

Convert Units

The first step is to convert all sizes to the same units to make the comparison easier. The diameter of a cell is given as \(10 \mu \mathrm{m}\) (micrometers), and the diameter of an atom in nanometers (nm). There are 1000 nm in a \(\mu \mathrm{m}\) so the diameter of a cell can be converted to \(10^4\) nm, and the diameter of an atom stays as \(0.1 \mathrm{nm}\).
02

Calculate Volume Ratios

Since the cells and atoms are roughly spherical, the volume of a sphere formula can be used. The volume \(V\) of a sphere with radius \(r\) is \(\frac{4}{3}\pi r^3\). For simplicity, we can neglect the \(\frac{4}{3}\pi\) term as it cancels out in the ratio of volumes. Using the radius \(r\) (which is half the diameter) of a cell and an atom respectively, we can calculate the ratio of volumes to be \((\frac{10^4}{0.1/2})^3 = (2 \times 10^5)^3 = 8 \times 10^{15}\). This indicates that one cell can hold roughly \(8 \times 10^{15}\) atoms.
03

Estimate Total Number of Atoms

Finally, to estimate the total number of atoms in the body, we multiply the number of cells in the body by the number of atoms per cell. This gives \(10^{14} \times 8 \times 10^{15} = 8 \times 10^{29}\). The closest answer is therefore \(10^{30}\).

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The human body contains about \(10^{14}\) cells, and the diameter of a typical cell is about \(10 \mu \mathrm{m}\). Like all ordinary matter, cells are made of atoms; a typical atomic diameter is \(0.1 \mathrm{nm}\). The volume of a cell is about \(\begin{array}{ccc}\text { a. } 10^{-10} \mathrm{m}^{3} . & \text { b. } 10^{-15} \mathrm{m}^{3} . & \text { c. } 10^{-20} \mathrm{m}^{3} . \quad \text { d. } 10^{-30} \mathrm{m}^{3}\end{array}\)

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