Garrett J. answered • 03/07/21
General Sciences Tutor for High School and College
To start this question, you first need to find the radius of a zirconium atom, as the value given to you in the question is the diameter and it is asking for the radius. To do this, you would just divide the diameter by 2.
3.10A ÷ 2 = 1.55A
Now that you have this you can start to solve the questions.
The first thing that is asked is to find the radius of the zirconium atom in meters. To do this, you must remember that 1 angstrom (A) is equal to 1 × 10-10 meters (m).
To solve, you would just divide the radius by 1010. 1.55A ÷1010 = 1.55 × 10-10m
Next you need to find the radius of the zirconium atom in nanometers (nm). You can solve this section in two ways, by converting angstroms to nanometers (1A = 0.1nm) or by converting your answer in meters to nanometers (1m = 109nm). For simplicity, I am just going to convert angstroms to nanometers so it would look like:
1.55 ÷ 10 = 0.155nm
The next part of the question asks you to find how many atoms would be lined up to span 1.10mm. Remember that when lining up spherical atoms, the distance between them is equal to the diameter. So to solve this you will need the diameter of the atom (3.10A) and the conversion rate of angstroms to millimeters (1A = 1 × 10-7mm). The first step to find this would be to convert your diameter from angstroms to millimeters.
3.10A ÷ 107 = 3.1 × 10-7mm
Next, just divide the distance of the line by this value to get the number of atoms in the line.
1.10mm ÷ 3.1 × 10-7mm = 3.55 × 106 zirconium atoms
The last section asks for you to find the volume of a zirconium atom in cubic centimeters (cm3). To do this you must remember that the formula for the volume of a sphere is V = (4/3)πr3 and that 1A = 1 × 10-8. To solve you would first convert the radius of the zirconium atom into centimeters.
1.55A ÷ 108 = 1.55 ×10-8cm
Now just plug in the values into the spherical volume formula:
V = (4/3)πr3
V = (4/3)π × (1.55×10-8)3 = 1.56 × 10-23cm
