The Avogadro Project aims to define the kilogram with...

The Avogadro Project aims to define the kilogram with respect to a precise value of the Avogadro constant, $L$. For this task, spheres of ultrapure silicon are being made. Let the mass of a sphere be $M,$ and its volume be $V .$ (a) Write down an expression for the density of silicon. (b) What is the mass of one atom of silicon if the molar mass is $M_{\mathrm{r}} ?$ (c) The structure of elemental silicon is described in terms of a repeating unit called the unit cell. If the volume of the unit cell is $V_{c}$ and it contains $n$ atoms, how many atoms are present in the entire sphere? (d) What is the mass of the sphere of silicon in terms of $M_{r}, L, V, V_{c}$ and $n ?$ (e) Rearrange the equation from part (d) to show how a value of the Avogadro constant can be determined.

The Avogadro Project aims to define the kilogram with respect to a precise value of the Avogadro constant, $L$. For this task, spheres of ultrapure silicon are being made. Let the mass of a sphere be $M,$ and its volume be $V .$ (a) Write down an expression for the density of silicon. (b) What is the mass of one atom of silicon if the molar mass is $M_{\mathrm{r}} ?$
(c) The structure of elemental silicon is described in terms of a repeating unit called the unit cell. If the volume of the unit cell is $V_{c}$ and it contains $n$ atoms, how many atoms are present in the entire sphere?
(d) What is the mass of the sphere of silicon in terms of $M_{r}, L, V, V_{c}$ and $n ?$ (e) Rearrange the equation from part (d) to show how a value of the Avogadro constant can be determined.

Key Concepts

Scaling from Unit Cell to Bulk Material

Scaling from the unit cell to the entire sample involves determining the number of unit cells (or atoms based on the unit cell contents) in the macroscopic volume of the material. This concept bridges the crystal lattice structure with the observed bulk properties.

Unit Cell

The unit cell is the smallest repeating structural unit in a crystalline material, such as a silicon crystal. Its volume and the number of atoms it contains provide essential information about the overall crystal structure and are used to scale up to the macroscopic properties of the material.

Algebraic Rearrangement for Determining Constants

This involves manipulating equations that relate measurable quantities (such as mass, volume, and geometrical parameters of the crystal) to isolate and determine a fundamental constant. Such rearrangements are essential in experimental physics for extracting precise values from indirect measurements.

Molar Mass

Molar mass is the mass of one mole of a substance, linking the measurable mass of a sample to the number of atoms or molecules it contains. It allows conversion between the macroscopic scale and the mass of an individual atom when divided by Avogadro’s constant.

Density

Density is a fundamental property defined as the mass of a substance divided by its volume. This concept is crucial for relating the macroscopic measurements of an object (like a sphere) to its inherent material properties.

Avogadro's Constant

Avogadro's constant is a key fundamental constant that represents the number of atoms or molecules in one mole of a substance. It is pivotal for converting between bulk molar quantities and the properties of individual atoms.

Transcript

00:03 All right.

00:04 For this question, we're asked just a series of questions about a sample of silicon trying to identify exactly what avagadro's constant is.

00:15 And one of the hardest things about the specific question is they've got some a little bit of funky variables they're using.

00:23 So for avagadro's constant, they're using the capital l.

00:27 And so really, so they're talking about avagadro's project trying to define the kilogram based off of the.

00:33 Value.

00:34 And so we're really just answering a series of questions that's really just testing our ability to use units and unit conversions and stuff like that.

00:41 So the first part, we got to write down an expression for the density of silicon.

00:47 Well, we know that density in general uses the symbol row and density is always defined as a mass over a volume.

00:56 And in this specific case, in the problem statement, it defines mass of the sphere of silicon as capital m.

01:03 And the volume of that sphere is just v.

01:06 All right.

01:07 So that's all a is.

01:09 That's not too bad.

01:10 All right.

01:10 How about b? let's see.

01:12 Part b, they want us to figure out the mass of one atom of silicon if the molar mass is m sub r, capital m sub r.

01:23 And so i approach these problems kind of all in the same way.

01:25 I use dimensional analysis.

01:26 So i say, okay, if i have one atom of silicon, all right, and i got to convert that into grams.

01:33 Into mass.

01:34 So i'm going to have two steps.

01:36 I'm going to have the first step where i have to go to moles and then a second step where i have to mass.

01:41 Right.

01:42 Anyway, so you have atoms in the bottom and moles in the top.

01:47 And i know that there are one avagadro's number worth of moles in every atom.

01:55 Right.

01:55 So we'll say l for our avagadro's constant for every one mole.

02:00 All right.

02:01 And then on the next step, we say, okay, i've got moles of silicon now, right? because my atoms of silicon cancel.

02:07 We've got moles of silicon.

02:09 And then up here i'll put my grams of silicon.

02:12 And i know that one mole of silicon weighs one molar mass of silicon, that capital m sub r they give us in the problem statement.

02:21 And so if we do all this, we'll see that, you know, and they don't want us to actually find that we're just writing out expressions.

02:28 We can very easily, we know what avagadra's constant is, and we know what the molar mass of silicon is.

02:33 Both those are easy to find, but really they just want us to look through and come up with an expression for it.

02:39 So our expression would be m subr over l.

02:45 Right, so that's the expression for how much one atom weighs.

02:51 Part c asks us to find the talk about unit cell, and they ask us to find how many atoms are in one unit cell...

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