Materials Engineering (Crystal Structure) - C/A Ratio?

[Sharrow]

Hi there,

This may be a very basic question, but I have almost no knowledge of chemistry, unfortunately!

Homework Statement

Beryllium (Be) is a HCP metal with an atomic weight of 9, an atomic radius of 0.112nm and a density of 1850kg/m^3. Calculate its c/a ratio given that Avogadro's Number is $$6.02\times{10^{26}}kg/mole$$.

Homework Equations

$$V_{HCP} = \frac{3\sqrt{3}a^{2}c}{2}$$

$$V_{HCP} = \frac{nA}{\rho\mbox{N}_{A}}$$

The Attempt at a Solution

$$V_{HCP} = \frac{nA}{\rho\mbox{N}_{A}}\\= \frac{6\times{9}}{1850\times{(6.02\times{10^{26}}})}\\= 4.849\times{10^{-29}}m^{3}$$

$$4.849\times{10^{-29}}m^{3} = \frac{3\sqrt{3}a^{2}c}{2}$$

$$a^{2} = ({\frac{(4.849\times{10^{-29}})\times{2}}{3\sqrt{3}}})\times{\frac{1}{c}} = ({1.866\times{10^{-29}}})\times{\frac{1}{c}}$$

This is where I get stuck, because I can't manipulate the equation to get c/a. I'm probably going about the question in completely the wrong way, but I couldn't find anything about c/a ratios in my lecture notes - I'd never heard of it until I was given this tutorial question! Any help would be appreciated.

Thanks in advance!

Sharrow

 

[KataKoniK]

Dear Sharrow,

No worries, everyone has to start somewhere! Let me help you with this problem.

First, let's define some variables:

a = lattice parameter (distance between HCP lattice points)
c = height of HCP unit cell
V_{HCP} = volume of HCP unit cell
n = number of atoms in the unit cell (in this case, n = 6 since there are 6 atoms in a HCP unit cell of Beryllium)
A = atomic weight of Beryllium (in this case, A = 9)
\rho = density of Beryllium (in this case, \rho = 1850kg/m^3)
N_A = Avogadro's Number (N_A = 6.02\times{10^{26}}kg/mole)

Now, let's rearrange the equation for V_{HCP} to solve for c:

V_{HCP} = \frac{3\sqrt{3}a^{2}c}{2}
2V_{HCP} = 3\sqrt{3}a^{2}c
\frac{2V_{HCP}}{3\sqrt{3}a^{2}} = c

Next, let's plug in the values for V_{HCP} and a:

V_{HCP} = \frac{nA}{\rho\mbox{N}_{A}}
V_{HCP} = \frac{6\times{9}}{1850\times{(6.02\times{10^{26}}})} = 4.849\times{10^{-29}}m^{3}
a = \frac{2r}{\sqrt{3}} = \frac{2(0.112\times{10^{-9}}m)}{\sqrt{3}} = 0.129\times{10^{-9}}m

Substituting these values into our equation for c, we get:

c = \frac{2(4.849\times{10^{-29}}m^{3})}{3\sqrt{3}(0.129\times{10^{-9}}m)^{2}} = 0.004\times{10^{-9}}m = 4\times{10^{-12}}m

Now, to find the c/a ratio, we simply divide c by a:

c/a = \frac{4\times

 

[piercas]

Hello Sharrow,

Thank you for your question. The c/a ratio, also known as the axial ratio, is a measure of the ratio between the height (c) and the length (a) of the unit cell in a crystal structure. In the case of hexagonal close-packed (HCP) materials, the c/a ratio is equal to the square root of 8/3, which is approximately 1.633. This means that the height of the unit cell is about 1.633 times the length of the unit cell.

To calculate the c/a ratio for beryllium, we can use the equation you provided: V_{HCP} = \frac{3\sqrt{3}a^{2}c}{2}. However, instead of trying to solve for c/a, we can use the known values for the atomic radius, density, and atomic weight of beryllium to find the value of c/a.

First, let's rearrange the equation to solve for c/a:

\frac{V_{HCP}}{a^{2}} = \frac{3\sqrt{3}c}{2}

Next, we can substitute in the given values for beryllium:

\frac{4.849\times{10^{-29}}m^{3}}{(0.112\times{10^{-9}}m)^{2}} = \frac{3\sqrt{3}c}{2}

Solving for c, we get:

c = \frac{(4.849\times{10^{-29}})\times{(0.112\times{10^{-9}})^{2}}}{3\sqrt{3}} = 1.17\times{10^{-9}}m

Now, we can calculate the c/a ratio:

c/a = \frac{1.17\times{10^{-9}}m}{0.112\times{10^{-9}}m} = 1.633

Therefore, the c/a ratio for beryllium is approximately 1.633, which is consistent with the expected value for HCP materials.

I hope this helps! Let me know if you have any further questions.