# Titanium metal has a body-centered cubic unit cell. The density of Titanium is 4.50 g/cm^3. Calculate the edge length of the unit cell and a value for the atomic radius of titanium.

Ok, I know how to find the density using the atomic radius, but I'm having trouble working backwards using the density to find the atomic radius). I'm not really sure how to find the volume. And I know that the pythagorean theorem will help me find the radius but I'm missing some steps in between.

## To find the edge length of the unit cell and the atomic radius of titanium, we can use the given density and the assumption that titanium has a body-centered cubic (BCC) unit cell.

Step 1: Determine the volume of the unit cell.
In a body-centered cubic (BCC) unit cell, there are actually two atoms. The volume of the unit cell can be determined by dividing the molar mass of titanium by its density.

Since the molar mass of titanium is 47.87 g/mol, we can use this value:

Volume of unit cell = Molar mass / Density
= 47.87 g/mol / 4.50 g/cm^3

Step 2: Convert the volume to cm^3.
In order to ensure that the units are consistent, we need to convert the molar mass from g/mol to g/cm^3.

To convert g/mol to g/cm^3, we divide the result of Step 1 by Avogadro's number (6.022 x 10^23) to get the volume of the unit cell in cm^3.

Volume of unit cell (in cm^3) = Volume of unit cell (in g/mol) / Avogadro's number

Step 3: Calculate the edge length of the unit cell.
Since the titanium unit cell is a cube, we know that the edge length (a) is given by:

a = (Volume of unit cell)^(1/3)

Step 4: Determine the atomic radius.
In a body-centered cubic (BCC) unit cell, the atomic radius (r) is related to the edge length (a) by the equation:

a = 4 * r / (sqrt(3))

Now we can substitute the value of 'a' from Step 3 into this equation to solve for the atomic radius.

By following these steps, you will be able to calculate the edge length of the unit cell and the atomic radius of titanium.

## To calculate the edge length of the unit cell and the atomic radius of titanium, let's break it down step by step.

First, we need to find the volume of the unit cell. Since the unit cell of titanium is body-centered cubic, it consists of eight atoms located at the corners and one atom at the center. The atoms touch each other along the diagonal of the cube.

The volume of a body-centered cubic unit cell can be calculated using the equation:
Volume = (Edge Length)^3

Next, to find the atomic radius, we can use the fact that the atoms touch each other along the diagonal of the cube. The diagonal can be found using the Pythagorean theorem.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the case of a body-centered cubic unit cell, the diagonal is the hypotenuse, and the edge length of the cube is one side. We can use this information to find the diagonal and ultimately the atomic radius.

Let's put this into practice:

1. Calculate the edge length of the unit cell (l):
To find the edge length, we need to rearrange the volume formula mentioned earlier. Given that the density of titanium is 4.50 g/cm^3 and the molar mass of titanium is approximately 47.867 g/mol, we can use the following equation to find the edge length:

Edge Length (l) = (Molar Mass / (Avogadro's Number * Density))^(1/3)

Avogadro's Number is a constant with a value of approximately 6.022 × 10^23 mol^(-1).

2. Calculate the diagonal of the unit cell:
Since the diagonal of the unit cell is related to the edge length (l), we can use the Pythagorean theorem to calculate it. In a body-centered cubic structure, the diagonal length (d) can be calculated as follows:

Diagonal (d) = l * sqrt(3)

3. Calculate the atomic radius (r):
Since the atoms touch along the diagonal of the unit cell, the atomic radius (r) is half of the diagonal length. Therefore, we can calculate the atomic radius:

Atomic Radius (r) = d / 2

By following these steps, you should be able to calculate the edge length of the unit cell and the value for the atomic radius of titanium.

## I would be willing to bet you are stuck on just one step. Here it is,

1 mol Ti atoms = 47.88 g (check that on your periodic table)and that mass contains 6.022 x 10^23 atoms. Use that to calculate the mass of 1 atom of Ti, from that you have m =volume x density which allows you to calculate the volume. Take the cube root of V to obtain a, the length of the edge of the cube. Can you take it from here. The radius, if you need help there, is (1/4)*(sqrt 3)*a. I hope this helps.