Magnesium crstallizes in the hcp arrangment. The dimensions of the unit cell are height 520om, length on an edge, 320. Calculate the density of Mg (s) and compare it with the measured value of 1.738 g/cm^3
this is what i did :
m= 2* 24.30g/6.022*10^23 = 8.07*10^-23
v= l^2*h
v= (320)^2*520= .0053248
d=m/v
d= 8.07*10^-23/. 0053248
d= 1.51* 10^-20 cm
BUT the answer is suppose to be 1.75
I think I made a mistake on the volume :S
You don't list units for all of the measurements but I assume the 320 and 520 are picometers. You must convert the volume you have, which I assume is in cubic picometers, OR convert to cm before you begin. I highly recommend that you convert 320 to cm and convert 520 to cm.
320 pm x (1 m/10^12 pm) x (100 cm/1 m) = 3.2 x 10^-8 cm BEFORE doing the calculations. See if that helps.
sorry yes they were in pm I corrected my units:
m= 2* 24.30g/6.022*10^23 = 8.07*10^-23
v= l^2*h
v=(3.2*10^-8cm)^2* 5.2*10^-8cm v=5.32*10^-23cm
d=m/v
d= 8.07*10^-23/5.32*10^-23
d= 1.52cm
> it's still 1.52 where as the correct answer is suppose to be 1.75g/cm^3. Is my method of calculation correct?
Is your volume formula consistent with the hexagonal close packed crystal structure? That might be the cause of your disagreement.
See http://www.chm.davidson.edu/ChemistryApplets/Crystals/UnitCells/hcp.html
that just pretty much explains how to calculate height. and the length = 2r
It says a lot more than that. The base area of a unit cell is not the square of 2r. It is a hexagon
To find the volume of a HCP structure given the height and the lenght, you should use this formula:
l^2*h*((3)^1/3)/2)
To calculate the density of magnesium (Mg), we need to first determine the volume of the unit cell and then divide it by the mass of the unit cell.
Given:
Height of unit cell = 520 Ångstrom (Å) = 520 x 10^-10 meters
Length of an edge of the unit cell = 320 Å = 320 x 10^-10 meters
Measured density of Mg = 1.738 g/cm^3
To find the volume of the unit cell, we need to consider the shape of the unit cell. In this case, it is mentioned that magnesium crystallizes in the hexagonal close-packed (hcp) arrangement. In hcp, the unit cell is shaped like a prism, with a hexagonal base and height. The volume (V) of this prism can be calculated by multiplying the area of the base (A) with the height (h).
For an hcp arrangement, the base of the unit cell is a hexagon. The formula to calculate the area of a hexagon is A = (3√3 x a^2) / 2, where 'a' is the length of one side of the hexagon (which is the same as the length of an edge of the unit cell in our case).
Using the length of an edge of the unit cell (320 Å), we can calculate the area of the hexagonal base:
A = (3√3 x a^2) / 2
= (3√3 x (320Å)^2) / 2
= (3√3 x 102400 Å^2) / 2
= (3 x 1.732 x 102400 Å^2) / 2
= 527378.476 Å^2
Now, we can calculate the volume (V) of the unit cell:
V = A x h
= 527378.476 Å^2 x 520 Å
= 274251057.92 Å^3
Since the measured density is given in grams per cubic centimeter (g/cm^3), we need to convert the volume from Å^3 to cm^3. There are 1 x 10^24 Å^3 in 1 cm^3.
Converting the volume:
V = 274251057.92 Å^3 x (1 cm^3 / 1 x 10^24 Å^3)
= 2.7425105792 x 10^-16 cm^3
Now, we can calculate the density (d) by dividing the mass (m) of the unit cell by the volume (V) of the unit cell.
Given that the mass of the unit cell (m) is 2 x 24.30 g (since magnesium has an atomic mass of 24.30 g/mol and the unit cell contains 2 magnesium atoms):
m = 2 x 24.30 g
= 48.60 g
d = m / V
= 48.60 g / (2.7425105792 x 10^-16 cm^3)
= 1.7718974 x 10^17 g/cm^3
Comparing this calculated density with the measured density (1.738 g/cm^3), we can see that there is a discrepancy. It is possible that there was an error in the calculations or measurements.