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Chapter 22: Problem 4

Glass of Water Calculate the number of coulombs of positive charge in \(250 \mathrm{~cm}^{3}\) of (neutral) water (about a glassful).

Short Answer

Expert verified
The charge is approximately 1.34 脳 10鈦� C.

Step by step solution

01

Understand the problem

We need to calculate the number of coulombs of positive charge in 250 cm鲁 of water. Since the water is neutral, we can use the properties of water molecules and their composition to derive the answer.
02

Determine the number of water molecules

First, calculate the number of moles of water in 250 cm鲁 (or 250 grams, since the density of water is 1 g/cm鲁). The molar mass of water (H鈧侽) is about 18 g/mol.\[ \text{Number of moles} = \frac{250 \text{ g}}{18 \text{ g/mol}} \]
03

Calculate the moles

Find the number of moles from the above expression:\[ \text{Number of moles} = \frac{250}{18} \text{ mol} 鈮� 13.89 \text{ mol} \]
04

Find the number of water molecules

Use Avogadro's number (\text{N_A} = 6.022 脳 10虏鲁 molecules/mol) to find the total number of water molecules in 13.89 mol:\[ \text{Number of molecules} = 13.89 \text{ mol} 脳 6.022 脳 10^{23} \text{ molecules/mol} \]
05

Calculate the number of molecules

Calculate the total number of water molecules:\[ \text{Number of molecules} 鈮� 8.36 脳 10^{24} \text{ molecules} \]
06

Determine the number of protons per water molecule

Each water molecule (H鈧侽) has 2 hydrogen atoms (with one proton each) and 1 oxygen atom (with 8 protons), totaling:\[ 2 \text{ (H)} + 8 \text{ (O)} = 10 \text{ protons per molecule} \]
07

Calculate the total number of protons

Multiply the number of water molecules by the number of protons per molecule:\[ \text{Total protons} = 8.36 脳 10^{24} \text{ molecules} 脳 10 \text{ protons/molecule} = 8.36 脳 10^{25} \text{ protons} \]
08

Find the charge per proton

Knowing the charge of one proton is approximately 1.602 脳 10^{-19} C, calculate the total charge:\[ \text{Total charge} = 8.36 脳 10^{25} \text{ protons} 脳 1.602 脳 10^{-19} \text{ C/proton} \]
09

Calculate the total charge

Finally, calculate the total charge in coulombs:\[ \text{Total charge} 鈮� 1.34 脳 10^{7} \text{ C} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Molar Mass
The molar mass is the mass of one mole of a given substance. For water (H鈧侽), it consists of two hydrogen atoms and one oxygen atom. The atomic masses are approximately 1 g/mol for hydrogen and 16 g/mol for oxygen.
  • For H鈧�, the mass is 2 g/mol.
  • For one O, the mass is 16 g/mol.
Consequently, the molar mass of H鈧侽 is approximately 18 g/mol.
This means one mole of water, which is about 6.022 脳 10虏鲁 molecules (thanks to Avogadro's number), weighs 18 grams. Understanding the molar mass is essential for converting grams of a substance into moles, a crucial step in our calculations.
Avogadro's Number
Avogadro's number is a constant that signifies the number of constituent particles (usually atoms or molecules) in one mole of a substance. Its value is approximately 6.022 脳 10虏鲁 particles per mole.
This means:
  • 1 mole of water = 6.022 脳 10虏鲁 water molecules.
  • 1 mole of hydrogen atoms = 6.022 脳 10虏鲁 hydrogen atoms.
  • Any substance in 1 mole = 6.022 脳 10虏鲁 entities of that substance.
In our problem, knowing Avogadro's number allows us to determine how many water molecules are present when we have a specific number of moles. We used it to find that 13.89 moles of water correspond to 8.36 脳 10虏4 molecules of water.
Protons
Protons are positively charged particles found in the nucleus of an atom. Every element has a unique number of protons, which defines its atomic number. Here are the numbers relevant to our exercise:
  • Hydrogen (H) has 1 proton.
  • Oxygen (O) has 8 protons.
Therefore, each water molecule (H鈧侽) contains:
  • 2 protons from the two hydrogen atoms.
  • 8 protons from the one oxygen atom.
So, each water molecule has a total of 10 protons. By multiplying the total number of water molecules by the number of protons per molecule, we derive the total number of protons in our sample of water.
Coulombs
Coulombs (C) measure electric charge. In our problem, we need to find the total positive charge in a glass of water. Each proton carries a positive charge of approximately 1.602 脳 10鈦宦�9 coulombs.
To find the total charge:
  • First, calculate the number of protons in the water.
  • Second, multiply the number of protons by the charge of one proton.
For our 250 cm鲁 (or 250 grams) of water, the total positive charge is found by multiplying 8.36 脳 10虏5 protons by 1.602 脳 10鈦宦�9 C/proton, which equals approximately 1.34 脳 10鈦� coulombs.
Neutral Water
Neutral water means that it has no overall charge; the positive and negative charges are balanced. In a water molecule (H鈧侽):
  • The hydrogen atoms each contribute a single proton with a positive charge.
  • The oxygen atom, which has 8 protons, also contributes enough electrons to balance these protons' positive charges.
Because water is neutral, the number of positive charges (protons) equals the number of negative charges (electrons). Understanding this balance is important. It's why we only calculate the charge from protons in this exercise, knowing that the electrons' negative charge will cancel out these when considering neutral water completely.

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Most popular questions from this chapter

Force on Each A point charge of \(+3.00 \times 10^{-6} \mathrm{C}\) is \(12.0 \mathrm{~cm}\) from a second point charge of \(-1.50 \times 10^{-6} \mathrm{C}\). Calculate the magnitude of the force on each charge.

Two Equally Charged Two equally charged particles, held \(3.2 \times 10^{-3} \mathrm{~m}\) apart, are released from rest. The initial acceleration of the first particle is observed to be \(7.0 \mathrm{~m} / \mathrm{s}^{2}\) and that of the second to be \(9.0 \mathrm{~m} / \mathrm{s}^{2}\). If the mass of the first particle is \(6.3 \times 10^{-7} \mathrm{~kg}\), what are (a) the mass of the second particle and (b) the amount of charge on each particle?

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