Two small copper spheres each have radius 1.00 mm . (a) How many atoms does each sphere contain?

Two small copper spheres each have radius 1.00 mm . (a) How...

Two small copper spheres each have radius 1.00 $\mathrm{mm} .$ (a) How many atoms does each sphere contain? (b) Assume that each copper atom contains 29 protons and 29 electrons. We know that electrons and protons have charges of exactly the same magnitude, but let's explore the effect of small differences (see also Problem 21.81 ). If the charge of a proton is $+e$ and the magnitude of the charge of an electron is 0.100$\%$ smaller, what is the net charge of the each sphere and what force would one sphere exert on the other if they were separated by 1.00 $\mathrm{m} ?$

Key Concepts

Avogadro's Number

Avogadro's Number is a fundamental constant that represents the number of atoms or molecules in one mole of a substance. It is used to convert between the mass of a material and the number of its constituent atoms, allowing us to determine how many atoms are present in an object knowing its mass and molar mass.

Density and Molar Mass

Density and molar mass are crucial properties used to relate the physical mass of a material to its chemical content. By using the density of a substance and its molar mass, one can calculate the number of moles present in a given volume, which is necessary for determining the total number of atoms in that volume.

Spherical Volume Calculation

The volume of a sphere is calculated using the formula (4/3)?r³, where r is the radius. This concept is used in calculating the total volume—and hence the mass—of a spherical object, which is then used in conjunction with density to derive further properties such as the number of atoms contained.

Net Electric Charge

Net electric charge refers to the imbalance between the total positive charge (from protons) and the total negative charge (from electrons) in a system. Even a minute difference in the magnitudes of charge per particle can lead to a measurable total net charge when multiplied over an astronomical number of atoms, impacting the system's overall electrical behavior.

Coulomb’s Law

Coulomb’s Law describes the electrostatic force between two point charges. It states that the magnitude of the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This fundamental principle is used to calculate the electric force experienced by objects carrying net charge.

Transcript

00:01 So we know the mass can be equal to density times of volume, which is equal to 4 pi r q over 3 times row.

00:07 And the mass can be equal to mole times the motor mass, which is nm.

00:11 So where nm is equal to 4 pi rq over 3 times row.

00:14 So we have mole is equal to 4 pi rq over 3m times row.

00:20 So now we can determine the total amount of addens, which is capital in here, which is equal to 4 pi r cube over 3m times row and n times the abrogadros number.

00:29 And these are the information given by the question.

00:32 So 1 millimeter is 1 times 102 -93 meter.

00:35 And this is for radius.

00:37 And the motor mass of copper is 63 .546 gram per mole, which is 63 .546 times 10 -2 -93 kilogram per more.

00:45 And the density of a carbon is 8 .95 times 10 to power 3 kilogram per meter cube.

00:49 Evergarsal numbers given as 6 .02 times 10 to power of 23.

00:54 So eventually we can determine the addens, the amount of addens, which is, you eventually we have 3 .55 times 10 to the power of 20.

01:03 Okay, so each sphere contains 3 .55 times 10 to power 20 atoms.

01:08 And for question b, well, we know in each item there are 29 protons and 29 electrons.

01:15 So the total amount of electrons and protons in the copper sphere is 29 times the total amount of atoms, which is 1 .03 times 10 to the power of 22.

01:25 And we're saying that the charge on the electron is 0 .1 % smaller than the charge on proton.

01:33 And we know the charge on electron is negative.

01:35 So therefore we have 1 minus 0 .1 % times negative e...

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