Problem 3 - University Physics with Modern Physics

Question

Estimate how many electrons there are in your body. Make any assumptions you feel are necessary, but clearly state what they are. (Hint: Most of the atoms in your body have equal numbers of electrons, protons, and neutrons.) What is the combined charge of all these electrons?

Step-by-Step Solution

Answer

There are approximately \(1.4 \times 10^{28}\) electrons in the body, with a total charge of \(-2.24 \times 10^{9}\) coulombs.

Step 1: Estimating the number of atoms in the body

Assume the average mass of a human body is 70 kg. A large portion of the human body mass consists of water, and water (H2O) molecules contain two hydrogen atoms and one oxygen atom. Hydrogen and oxygen together are about 99% of the atoms in the body. Assuming roughly 10% of body composition is carbon, nitrogen, etc., estimate the total number of atoms by approximating the body composition as 70% water. The molecular weight of water is 18 g/mol, so the number of moles in the body is \( \frac{70,000 \text{ g}}{18 \text{ g/mol}} \approx 3889 \text{ moles} \). Using Avogadro's number (\( 6.022 \times 10^{23} \text{ atoms/mol} \)), calculate the total number of water molecules/atoms: \( 3889 \times 6.022 \times 10^{23} \approx 2.34 \times 10^{27} \) atoms.

Step 2: Estimating the number of electrons per atom

Assume all atoms in the body have approximately the same number of electrons as protons and assume they are electrically neutral. The most common elements (hydrogen, oxygen, carbon, nitrogen) have atomic numbers corresponding to the number of electrons: Hydrogen (1 electron), Oxygen (8 electrons), Carbon (6 electrons), and Nitrogen (7 electrons). Averaging these, the estimated electrons per atom can be around 5-7 electrons/atom. Take an average of approximately 6 electrons/atom.

Step 3: Calculating total number of electrons

Multiply the number of atoms by the average number of electrons per atom to estimate total electrons: \( 2.34 \times 10^{27} \text{ atoms} \times 6 \text{ electrons/atom} \approx 1.4 \times 10^{28} \text{ electrons} \).

Step 4: Calculating the total charge of these electrons

Each electron has a charge of \(-1.6 \times 10^{-19} \text{ Coulombs }\). Multiply the total number of electrons by the charge per electron to find the total charge: \( 1.4 \times 10^{28} \times (-1.6 \times 10^{-19}) = -2.24 \times 10^{9} \text{ Coulombs} \). This is the total charge, considering all electrons carry a negative charge.

Key Concepts

Electron Count

The concept of electron count is fundamental in understanding atomic structure. Electrons are negatively charged particles surrounding the nucleus of an atom. They are important in chemical reactions and determine the atom's properties. To estimate how many electrons are in your body, it's helpful to know about the common elements and atoms it contains.

Consider that elements like hydrogen, oxygen, carbon, and nitrogen make up most of the body. Each of these elements has a certain number of electrons:

• Hydrogen has 1 electron.
• Oxygen has 8 electrons.
• Carbon has 6 electrons.
• Nitrogen has 7 electrons.

A rough estimate can average around 6 electrons per atom in your body. By multiplying this average with the total number of atoms in a typical body, you can make a reasonable approximation of the number of electrons.

Molecular Composition

Molecular composition involves understanding which molecules primarily make up a substance—in this case, the human body. Water (H₂O), with its two hydrogen atoms and one oxygen atom, comprises about 70% of the human body's composition.

To estimate the molecules in the body, think about water being a significant portion. The molecular weight of water is 18 grams per mole. When you know the body's approximate mass (say, 70 kg), you can calculate the moles of water it contains:

• Convert the body mass to grams: 70,000 grams.
• Divide by the molecular weight: \(\frac{70,000}{18} \approx 3889 ext{ moles of water}\)

This gives us a starting point for determining how many atoms—and therefore, how many electrons—are in the body by multiplying by Avogadro's number.

Avogadro's Number

Avogadro's Number is a cornerstone of chemistry, providing the link between macroscopic and microscopic scales. Defined as \(6.022 \times 10^{23} \text{ atoms/mol} \), it tells us how many atoms or molecules are in one mole of a substance.

When estimating the number of atoms in the human body, Avogadro's Number becomes crucial. Given a certain number of moles (like the estimated 3889 moles of water in a 70 kg body), multiplying this by Avogadro's Number gives the total number of molecules:

• \(3889 \times 6.022 \times 10^{23} \approx 2.34 \times 10^{27} \text{ atoms}\)

This immense number illustrates how tiny individual atoms are, and how many it takes to make up something as substantial as a human body.

Charge Calculation

Charge calculation involves determining the total charge carried by electrons. Since each electron carries a charge of \(-1.6 \times 10^{-19} \text{ Coulombs} \), knowing the total number of electrons means you can calculate the total charge they produce.

For example, if you estimate around \(1.4 \times 10^{28} \text{ electrons}\) in your body (found by multiplying the number of atoms by the average number of electrons per atom), then the total charge is:

• Multiply the number of electrons by the electron's charge:\(1.4 \times 10^{28} \times (-1.6 \times 10^{-19}) = -2.24 \times 10^{9} \text{ Coulombs}\)

This results in a large negative charge, which is counterbalanced by the positive charges of protons in the body, maintaining electrical neutrality.