Polonium crystallizes with a primitive cubic structure. It has a density of 9.4 g/cm 3 , a radius of 167 pm, and a molar mass of 209 g/mol. Calculate the number of atoms in one mole of Polonium.

Polonium crystallizes with a primitive cubic structure. Remember, another name for a simple cubic unit cell is a primitive cubic unit cell. It has a density of 9.4 grams per centimeters cubed, a radius of 167 picometers, and a molar mass of 209 grams per mole. Here we need to calculate the number of atoms in 1 mole of polonium. Alright. So when it comes to any of these cubic unit cells, we can say that their densities are equal to the number of atoms per unit cell, which we'll abbreviate as n, times the molar mass of the element divided by the volume, which will just be our edge length cubed, times our number of atoms, which is NA. At this moment, when we say n a like this, n a here represents Avogadro's number. And realize with Avogadro's number, we say that 1 mole of any substance is equal to 6.022 times 10 to the 23. Here we're dealing with 1 mole of our element, which would mean that we're dealing with 6.022 times 10 to the 23 atoms of that element. So without doing any calculations, we should see that the answer is c. But let's just humor ourselves and work this question out. We're trying to isolate this basic, number of atoms, So let's just get it by itself. We're gonna multiply both sides by volume times number of atoms. So when we do that we're gonna get here volume times number of atoms times density equals the number of atoms per unit cell times the molar mass. Divide both sides here by volume and density, so we can isolate the number of atoms. So then when we do that we're going to get here that our number of atoms equals the number of atoms per unit cell times the molar mass of the element divided by volume times density. So let's start plugging in the values that we know. Since this is a simple cubic unit cell, we know that it is 1 atom per 1 unit cell. And the molar mass we're told is 209 grams per mole. The volume the volume here, we're gonna want to get that volume in centimeters cubed because the density is in centimeters cubed. So we have 167 picometers and what we're gonna do here is we're gonna convert picometers into centimeters. Remember, 1 pico is 10 to the negative 12 and 1 centi is 10 to the negative 2. So this is 1.67 times 10 to the negative 8 centimeters. And then here, we would say that remember, for a civil cubic unit cell, our edge length is equal to 2 times the radius. So that's 2 times the answer that we just got. So this equals here 3.34 times 10 to the negative 8 centimeters, and then remember volume equals a cubed. So we just take the cube of this number and that will give us our volume. So when we do that we're gonna get here, I'm gonna write it down here, 3.72 60 times 10 to the negative 23 centimeters cubed. Our density we're told is 9.4 grams per centimeters cubed. And what cancels out? We have centimeters cubes cancel out, and then what else do we have out? Grams cancel out. And what we'll have as our answer is the number of atoms per mole of a unit cell. When we punch all that in, we're going to get roughly 5 point 967 times 10 to the 23, which we can round up to 6.0 times 10 to the 23, which shows that c is our answer.

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13. Liquids, Solids & Intermolecular Forces

Simple Cubic Unit Cell

General Chemistry

13. Liquids, Solids & Intermolecular Forces

Simple Cubic Unit Cell

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Multiple Choice

Polonium crystallizes with a primitive cubic structure. It has a density of 9.4 g/cm³, a radius of 167 pm, and a molar mass of 209 g/mol. Calculate the number of atoms in one mole of Polonium.

6.0×10^-23

3.2×10¹¹

6.0×10²³

1.7×10¹⁵

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Verified step by step guidance

Understand that the problem is asking for the number of atoms in one mole of Polonium, which is essentially Avogadro's number. This is a fundamental constant in chemistry, approximately 6.022 x 10^23 atoms/mol.

Recognize that the given density, radius, and molar mass are not directly needed to find Avogadro's number, as it is a known constant. However, these values are typically used to calculate the number of atoms in a unit cell or to verify the structure type.

Recall that a primitive cubic structure has one atom per unit cell. This information is useful if you were calculating the number of atoms in a given volume or verifying the density.

If you were to use the density, radius, and molar mass to find the number of atoms in a unit cell, you would use the formula: \( \text{Density} = \frac{\text{Molar Mass}}{\text{Volume of Unit Cell} \times \text{Avogadro's Number}} \). However, this is not necessary for this problem.

Conclude that the number of atoms in one mole of any element, including Polonium, is Avogadro's number, which is approximately 6.022 x 10^23 atoms/mol. This is a constant value and does not change with the element.