Two small aluminum spheres, each having mass 0.0250 0.0250 0.0250 kg, are separated by 80.0 80.0 80.0 cm. How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 26.982 g/mol, and its atomic number is 13 13 .)

Hey, everyone in this problem, we have two tiny iron balls to be used in an electrostatic experiment that have a mass of 0.055 kg. Each The balls are placed 65 cm, per, Were asked to determine the number of electrons on each neutral ball. We're told that iron has an atomic mass of 55.85 g per mole. An atomic number of 26. Let's just start out with some of this information that we're given. What does it tell us? Okay. Well, we're told the atomic number is 26. What does that tell us? Okay. Well, the atomic number tells us the number of protons we have per adam. Okay. So this tells us that we have 26 protons per Adam. Now, the question is asking us about electrons, not about protons. How do these relate while we're told that this is a neutral ball, this is a neutral ball, which means that the number of electrons is equal the number of protons. Okay, which is equal to 26. Okay. So we have the number of electrons. Her item Is equal to 26. Alright. So now what are we trying to find? We're trying to find the number of electrons purple. Okay. The number of electrons purple. Well, this is gonna be equal to the number of electrons per Adam. We just found times. Okay. Well, we want electrons in the numerator and ball in the denominator. So in order to eliminate this unit of atom or this atom from the denominator, we need to multiply by the number of atoms we're going to divide by purple. Okay. So we have the number of electrons for Adam times the number of atoms for ball is going to give us the number of electrons purple. We've already found this first quantity. So now we need to find the second quantity. Now, the number of Adam's purple, how can we find that? Well, we can find the number of more through the following. Okay, recall that N is equal to M over M. So the number of moles is equal to the mass divided by the molar mass here, the atomic mass And we're told the mass of one ball is 0. kg. Okay. Now, our molar mass or atomic mass is given in grams per mole. So let's convert our kilograms into grams. We're gonna multiply by 1000 g per kilogram. We divide by our atomic mass which is 55. grams per mole. And this is going to leave us with 0.9, eight, 4, 7, 8 more. Alright. So we know how many more we have. What about Adams? Okay. So the number of atoms purple is going to be equal to and the moller the number of mole times avocados number 6. times 10 to the 23. Okay, Adams Permal and this is more purple. And so we're looking at our units here we have mold per bowl times atoms per mole, okay. The unit of mold cancels and we're left with atoms per ball, which is what we were trying to find. Alright, so this is going to be 5.9284 times 10 to the 23 atoms. Alright. So now we figured out how many atoms are in each ball and we figured out how many electrons were in niigata. So we multiply the two together, we have the number of electrons purple and I'm going to rewrite the equation I wrote before. Okay. This is going to be the number, go back to remind you the number of electrons per atom times the number of atoms purple. Now recall that the number of electrons per atom that we found was 26, correct? And the number of atoms per ball was 5.92, 8, 4 times 10 to the 23 atoms. And if we multiply the two together, this is going to give us 1.54, times 10 to the 25 electrons. Mhm. Alright. So if we go back up to our answer traces. We see that we found answer choice. D okay. We have 1.54 times 10 to the 25 electrons in each neutral ball. Thanks everyone for watching. I hope this video helped see you in the next one.

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Problem 8a

Textbook Question

Two small aluminum spheres, each having mass $0.0250$ kg, are separated by $80.0$ cm. How many electrons does each sphere contain? (The atomic mass of aluminum is $26.982$ g/mol, and its atomic number is $13$ .)

Verified step by step guidance

First, determine the number of moles of aluminum in each sphere. Use the formula: $ \text{moles} = \frac{\text{mass of sphere}}{\text{molar mass of aluminum}} $. Convert the mass of the sphere from kg to g by multiplying by 1000.

Calculate the number of moles: $ \text{moles} = \frac{25.0 \text{ g}}{26.982 \text{ g/mol}} $.

Next, use Avogadro's number to find the total number of aluminum atoms in each sphere. Avogadro's number is $ 6.022 \times 10^{23} \text{ atoms/mol} $. Multiply the number of moles by Avogadro's number.

Since each aluminum atom contains 13 electrons (as given by the atomic number), multiply the total number of atoms by 13 to find the total number of electrons in each sphere.

Finally, express the result as the total number of electrons in each sphere, ensuring all units are consistent and calculations are clear.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Atomic Structure

Atoms are the basic units of matter, consisting of a nucleus surrounded by electrons. The atomic number of an element, like aluminum, indicates the number of protons in its nucleus, which is equal to the number of electrons in a neutral atom. Understanding atomic structure is crucial for determining the number of electrons in a given mass of an element.

Mole Concept

The mole is a fundamental unit in chemistry that represents a specific number of particles, typically atoms or molecules. One mole contains Avogadro's number of particles, approximately 6.022 x 10^23. This concept allows us to convert between the mass of a substance and the number of atoms or molecules it contains, which is essential for calculating the number of electrons in the spheres.

Mass-to-Mole Conversion

To find the number of moles in a given mass of a substance, we use the formula: moles = mass (g) / molar mass (g/mol). For aluminum, with a molar mass of 26.982 g/mol, this conversion helps determine how many moles of aluminum are present in the spheres, which is necessary to calculate the total number of electrons.

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