One edge of a copper cube is carefully measured and found to be 2.162 cm. An atom of copper is a sphere 0.2560 nanometers in diameter. Assuming that the volume of the cube contains nothing but copper atoms, how many copper atoms are contained in the cube? (the volume of a cube is the length of one edge raised to the third power. the volume of a sphere is 1.333 x n x the radius of the sphere raised to the third power)
Factor label method
Well, buckle up for a funny ride through scientific calculations! Let's factor and label our way to the answer.
First things first, we need to calculate the volume of the copper cube. We know that the edge of the cube is 2.162 cm, so its volume is gonna be edge cubed.
Volume of the cube = (2.162 cm)^3
Now it's time to convert the units, because we need to be consistent here. We'll use a conversion factor of 1 cm = 10^-7 meters, because that's how scientists like it.
Volume of the cube = (2.162 cm)^3 x (10^-7 m/cm)^3
If we crank up the calculator, we get the volume of the cube to be a long number as a result. Don't worry, I won't bore you with all those digits.
Now let's move on to the copper atom. We know that its diameter is 0.2560 nanometers, but we need the radius for math purposes, which is half the diameter. So let's divide it by 2. Isn't it fun playing with halves?
Radius of the copper atom = 0.2560 nm / 2
Now, the volume of the sphere (the copper atom) is given by the formula, 1.333 x π x (radius of the sphere)^3. We'll plug in the numbers and have some serious pi fun.
Volume of the sphere = 1.333 x π x (0.2560 nm / 2)^3
After another calculator party, we have the volume of one copper atom, which, believe me, is a tiny number.
Now, to find how many copper atoms are in the cube, we just divide the volume of the cube by the volume of one copper atom. Et voila!
Number of copper atoms = Volume of the cube / Volume of one copper atom
And that, my friend, is the answer to your question! I hope you had as much fun as I did during this factor label method madness.
To find the number of copper atoms contained in the cube, we need to calculate the volume of the cube and then divide it by the volume of a single copper atom.
Step 1: Calculate the volume of the cube.
Given that the length of one edge of the cube is 2.162 cm, we can calculate the volume using the formula:
Volume of cube = (length of edge)^3
Substituting the given value:
Volume of cube = (2.162 cm)^3
Step 2: Convert the volume to nm^3.
Since the diameter of a copper atom is given in nanometers, we need to convert the volume of the cube into nanometers as well.
1 nm = 0.0000001 cm
So, 1 nm^3 = (0.0000001 cm)^3
To convert the volume of the cube to nanometers, we divide the volume in cubic centimeters by the conversion factor.
Volume of cube in nm^3 = (Volume of cube in cm^3) / (0.0000001 cm)^3
Step 3: Calculate the volume of a single copper atom.
Given that the diameter of the copper atom is 0.2560 nanometers, we can calculate the radius by dividing the diameter by 2.
Radius of copper atom = 0.2560 nm / 2
Using the formula for the volume of a sphere:
Volume of sphere = (4/3) * π * (radius)^3
Substituting the given value:
Volume of sphere = (4/3) * 3.1415 * (0.2560 nm / 2)^3
Step 4: Calculate the number of copper atoms in the cube.
To find the number of copper atoms, we divide the volume of the cube by the volume of a single copper atom:
Number of copper atoms = (Volume of cube in nm^3) / (Volume of sphere)
Simply follow the steps above to perform the calculations.
To find the number of copper atoms in the cube, you need to determine the volume of the cube and then calculate how many copper atoms can fit into that volume.
Step 1: Calculate the volume of the cube
The volume of a cube is given by the formula: volume = edge^3
Given that the length of one edge of the copper cube is 2.162 cm, we can substitute this value into the formula:
volume = (2.162 cm)^3
Step 2: Convert the edge length to nanometers
Since the diameter of a copper atom is given in nanometers, we need to convert the edge length of the cube from centimeters to nanometers.
1 cm = 10^-7 meters = 10^-7 x 10^9 nanometers = 10^2 nanometers
So, 2.162 cm = 2.162 x 10^2 nanometers
Step 3: Calculate the volume of the cube in nanometers
Now, substitute the value into the equation:
volume = (2.162 x 10^2 nanometers)^3 = 10.063 x 10^6 nanometers^3
Step 4: Calculate the volume of one copper atom
The volume of a sphere is given by the formula: volume = (4/3)π(radius)^3
Given that the diameter of the copper atom is 0.2560 nanometers, we can calculate its radius by dividing the diameter by 2:
radius = diameter / 2 = 0.2560 nanometers / 2 = 0.1280 nanometers
Now, substitute the radius value into the formula:
volume of one copper atom = (4/3)π(0.1280 nanometers)^3
Step 5: Calculate the number of copper atoms in the cube
To find the number of copper atoms in the cube, divide the volume of the cube by the volume of one copper atom:
number of copper atoms = (volume of the cube) / (volume of one copper atom)
number of copper atoms = (10.063 x 10^6 nanometers^3) / [(4/3)π(0.1280 nanometers)^3]
Now, plug in the values and calculate the result using a calculator.