Consider the densities and atomic radii of the noble gases at 25^∘ C : a. Estimate the dens

Consider the densities and atomic radii of the noble gases...

Consider the densities and atomic radii of the noble gases at $25^{\circ} \mathrm{C} :$ $\begin{array}{l}{\text { a. Estimate the densities of argon and xenon by interpolation }} \\ {\text { from the data. }} \\ {\text { b. Estimate the density of the element with atomic number } 118} \\ {\text { by extrapolation from the data. }}\end{array}$ $\begin{array}{l}{\text { c. Use the molar mass of neon to estimate the mass of a neon }} \\ {\text { atom. Then use the atomic radius of neon to calculate the average}} \\ {\text {density of a neon atom. How does this density compare }} \\ {\text { to the density of neon gas? What does this comparison suggest }} \\ {\text {about the nature of neon gas? }}\end{array}$$\begin{array}{l}{\text { d. Use the densities and molar masses of krypton and neon to }} \\ {\text { calculate the number of atoms of each element found in a }} \\ {\text { volume of } 1.0 \text { L. Use these values to estimate the number of }} \\ {\text { atoms present in } 1.0 \text { L of Ar. Now use the molar mass of argon }} \\ {\text { to estimate the density of Ar. How does this estimate com- }} \\ {\text { pare to that in part a? }}\end{array}$

Consider the densities and atomic radii of the noble gases at $25^{\circ} \mathrm{C} :$
$\begin{array}{l}{\text { a. Estimate the densities of argon and xenon by interpolation }} \\ {\text { from the data. }} \\ {\text { b. Estimate the density of the element with atomic number } 118} \\ {\text { by extrapolation from the data. }}\end{array}$ $\begin{array}{l}{\text { c. Use the molar mass of neon to estimate the mass of a neon }} \\ {\text { atom. Then use the atomic radius of neon to calculate the average}} \\ {\text {density of a neon atom. How does this density compare }} \\ {\text { to the density of neon gas? What does this comparison suggest }} \\ {\text {about the nature of neon gas? }}\end{array}$$\begin{array}{l}{\text { d. Use the densities and molar masses of krypton and neon to }} \\ {\text { calculate the number of atoms of each element found in a }} \\ {\text { volume of } 1.0 \text { L. Use these values to estimate the number of }} \\ {\text { atoms present in } 1.0 \text { L of Ar. Now use the molar mass of argon }} \\ {\text { to estimate the density of Ar. How does this estimate com- }} \\ {\text { pare to that in part a? }}\end{array}$

Key Concepts

Interpolation

Interpolation is a mathematical method used to estimate unknown values within the range of a discrete set of known data points. In the context of physical measurements like gas densities, interpolation allows one to determine approximate values for a gas (e.g., argon or xenon) by using the relationship shown in the provided data for gases with nearby atomic properties.

Extrapolation

Extrapolation is the technique of estimating the value of a variable outside the range of known data points. For instance, by extending the trend observed in the densities of noble gases, one can predict the density of an element with a higher atomic number than those in the data set. This method involves using the established pattern to make an informed guess beyond the available measurements.

Density

Density is a physical property defined as mass per unit volume. It can be calculated for a bulk material, like a gas under certain conditions, or for individual atoms if one considers the atomic mass and the volume estimated from the atomic radius. Comparing these different densities helps in understanding the physical state and structure of matter.

Molar Mass

Molar mass is the mass of one mole of a substance, and it provides a bridge between the macroscopic measurements and the microscopic scale of individual atoms or molecules. By using the molar mass of a gas, one can calculate the mass of a single atom, which is essential for determining properties like the intrinsic density of an atom.

Atomic Radius

Atomic radius represents the size of an atom, often approximated by the distance from the nucleus to the outer boundary of the electron cloud. It is crucial in estimating the volume occupied by an atom and subsequently in calculating the density of matter at the atomic scale.

Number Density and Avogadro's Number

Number density refers to the number of atoms or molecules per unit volume, and it is calculated using the molar mass and density of the gas along with Avogadro's number. This concept is essential for relating the macroscopic properties of a gas, such as density, to the microscopic count of its constituent particles.

Macroscopic vs Atomic Density

Macroscopic density is the overall density of a substance as measured in bulk, while atomic density refers to the density within a single atom, calculated using the atomic mass and estimated volume from its radius. Comparing these densities illustrates the significant empty space that exists in gases, highlighting that atoms themselves are much denser than the gas as a whole due to vast distances between them.

Transcript

00:01 Given the densities in atomic radia of the novel gases at 25 degrees celsius, for a, we're asked to estimate the densities of argon and xenon by interpolation from the data.

00:13 So for a, we need to come up with a graph.

00:17 So the graph that we're going to come up with is going to be, we'll create a data table.

00:25 We're going to plot atomic number versus the density for the elements given.

00:34 So we have helium, atomic number two, and its density is 0 .18.

00:40 And this would be grams per liter.

00:45 We have the element neon, and its density given is 0 .90.

00:52 We have the element krypton, atomic number 36.

00:56 Density is 3 .75.

00:58 And we have the element radon, atomic number 86.

01:03 Density is 9 .73.

01:06 So if we do a plot here of these values, we do get a linear relationship.

01:16 And if we go ahead and i will take a picture of this plot for us.

01:36 So there's our plots.

01:37 Not in excel and we have a equation of the line given to us as well.

01:46 So estimating this, we need to estimate the density of argon.

01:56 So the density for argon will be equal to 0 .1146 times the atomic number.

02:11 Atomic number is 18 minus .1975 and we find that this would be equal to 1 .86 grams per liters so there's the estimated density of argon and let's calculate the estimated density of xenon using the formula .1146.

02:32 Xenon has an atomic number of 54 minus .1975 and we can estimate the density of xenon to be 6 .0 grams per liter.

02:47 For b we're as to estimate the density of an undiscovered element and we'll call this undiscovered element x so the density of this undiscovered element x we're told that it has an atomic number of 118 minus 0 .1975 and according to our formula here this density of this undiscovered element x would be equal to 13 .3 grams per liter.

03:19 For c, the molar mass of, use the molar mass of neon to estimate the mass of a neon atom, then use the atomic radius to calculate the average density, and how does this density compare to the density of neon gas? and what does this say about the nature of neon gas? so for c, the molar mass for neon, for neon, is equal to 20 .18 grams per mole.

03:53 And we can start with the mass of a neon atom.

04:07 So 20 .18 grams per mole times one mole to 6 .02 times 10 to the 23rd.

04:19 Atoms in one mole and this would be equal to 3 .35 times 10 to the negative 23 grams per atom so that's the mass of one neon atom.

04:34 We're also told you we can look up the atomic radius for neon is equal to 70 picometers and converting this into meters this is equal to 7 .0 times 10 to of the negative 11 meters.

04:53 So the volume of a neon atom, as a sphere, it's 4 3rd pi are cubed.

05:03 4 3rd pi 7 .0 times 10 to the negative 11 meters cubed.

05:08 And this would be equal to 1 .4 times 10 of the negative 30 meters cubed.

05:19 And now let's calculate the density for neon.

05:22 Because we have the mass and the volume...

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