An American penny is 97.5 % zinc and 2.5 % copper and has a mass of 2.5 g. (a) Use the approxim

An American penny is 97.5 % zinc and 2.5 % copper and has a...

An American penny is $97.5 \%$ zinc and $2.5 \%$ copper and has a mass of $2.5 \mathrm{~g}$. (a) Use the approximation that a penny is pure zinc, which has an atomic mass of $65.38 \mathrm{~g} / \mathrm{mol},$ to estimate the number of electrons in a penny. (Each zinc atom has 30 electrons.) (b) Estimate the net charge on all of the electrons in one penny. (c) The net positive charge on all of the protons in a penny has the same magnitude as the charge on the electrons. Estimate the force on either of two objects with this net magnitude of charge if the objects are separated by $2 \mathrm{~cm}$. (d) Estimate the number of leaves on an oak tree that is 60 feet tall. (e) Imagine a forest filled with such trees, arranged in a square lattice, each $10 \mathrm{~m}$ distant from its neighbors. Estimate how large such a forest would need to be to include as many leaves as there are electrons in one penny. (f) How does that area compare to the surface area of the earth?

Key Concepts

Scaling and Dimensional Analysis

Scaling and dimensional analysis are techniques used to estimate real-world quantities by using known relationships and conversion factors between different units. These methods allow one to relate microscopic particle counts to macroscopic physical dimensions, ensuring consistency and plausibility in approximations.

Fermi Estimation

Fermi estimation involves making approximate calculations to arrive at an order-of-magnitude estimate when precise data is unavailable or unnecessary. This approach relies on assumptions and scaling arguments to simplify complex problems into more tractable, approximate values.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges. It states that the force is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them. This law is central in calculating the magnitude of forces that charged objects exert on each other.

Fundamental Charge

The concept of fundamental charge refers to the charge of a single electron or proton, which is a constant value. This constant is essential for computing the net charge of a collection of subatomic particles and is a foundational aspect in the study of electromagnetism.

Atomic Structure and Electron Count

Understanding atomic structure is crucial to calculating the number of subatomic particles in a material. Knowing that atoms consist of electrons, protons, and neutrons—and that each element has a characteristic number of electrons (equal to its atomic number) in its neutral state—allows one to estimate the total electron count from the number of atoms present in a sample.

Molar Mass and Avogadro's Number

This concept connects the mass of a substance with the number of atoms or molecules it contains. Using the atomic or molecular weight alongside Avogadro's number allows one to convert a given mass into the number of particles, which is a fundamental tool in chemistry and physics for relating microscopic properties to macroscopic quantities.

Transcript

00:01 Hello, everyone.

00:01 In this problem, we're asked to do a bunch of estimations regarding an american penny.

00:06 So we're told that the penny is made up of mostly zinc and copper.

00:13 And we're told that we can assume for the purposes of this problem is entirely made of sync.

00:18 So we're also told that it has a mass of 2 .5 grams.

00:22 We're told that the molar mass of zinc is 65 .38 grams per moles.

00:28 And we also know everganderos number tells us how many atoms there are in a mole.

00:33 So it's 6 .022 times 1033.

00:36 And we're also told that each atom, each zinc atom, contains 30 electrons.

00:43 So the first part of the equation is asking us to estimate how many electrons there are in a penny.

00:50 So the way we do this is first we get how many grams or how many moles there are.

00:59 In a given in a given penny so the way we do that is that we know that there's 2 .5 grams right that's the mass of the penny and then we know that one mole of zinc so 6 .022 times and 323 atoms of zinc contain 65 or they have a mass of 65 .38 grams so to get how many moles there are in a zinc, how many moles of zinc atoms, we're going to divide or take the ratio of m over m .a.

01:35 So that gives us how many moles there are in a zinc, or sorry, in a paddy.

01:44 And then we're going to multiply that by ever -gradron number, because that tells us how many atoms there are.

01:49 Right.

01:49 So m over m -a tells us how many moles there are.

01:54 M -over -a times n -a tells us how many atoms.

01:56 And a tells us how many atoms there are and then we multiply that by the electron density which is 30 electrons per atom to get that all in all there should be around 6 .9 times 10 to 23 electrons in any given penny.

02:12 Then in part b we're asked to find what is the net charge that such a penny has or what's the net sorry what's the what's the total of charge in the electrons in a penny and so that's just multiplying the number of electrons times the charge of an electron.

02:27 Which is 1 .6 minus 1 .60 times 10 to the 19 coulums.

02:33 And so when applying the two numbers together, it gives you that it should be around minus 1 .1 times 10 to the 5 coulams.

02:41 Then part c is asking us to just calculate what happens if we assume that all the point charges or all the charges, all the positive and all the negative charges in the penny are put onto point charges that are separated by 0 .2 centimeters.

03:02 So here we are assuming that r here is 0 .02 meters, which is 2 centimeters.

03:14 And we just have to use goulomb's law for that.

03:16 So we're going to put in k is 9 .0 times 1039.

03:22 Q1 and q2 are both qe.

03:25 Actually, q1 is minus 1 .1 times 335, and q2 should be then 1 .1 times that 3 to 5 so that they're opposite charges.

03:33 But we're only really interested in the magnitude of this quantity.

03:38 So this opposite sign is where this minus sign here comes from, but we're only interested in the absolute value, so the sign doesn't matter.

03:45 And so once you put in all the values, we find that the force that these point charges would experience would be of a magnitude of 2 .8 times 10 to 23 meters, which is a humongous force...

Please give Ace some feedback