An organic compound crystallizes in an orthorhombic system with 2 molecules per unit cell. If the density of the crystal is #"1.419 g/cm"^3#, what is the molar mass of the compound?

The unit cell dimensios are #12.05*10^(-8)"cm"#, #15.05*10^(-8)"cm"#, and #2.69*10^(-8)"cm"#.

1 Answer
May 14, 2017

Answer:

#"208 g mol"^(-1)#

Explanation:

The first thing you need to do here is to figure out the volume of a single unit cell by using

#color(blue)(ul(color(black)(V = L xx l xx h)))#

In your case, you have

  • #L = 15.05 * 10^(-8)# #"cm"#
  • #l = 2.69 * 10^(-8)# #"cm"#
  • #h = 12.05 * 10^(-8)# #"cm"#

The volume of a single unit cell will thus be

#V = 15.05 * 10^(-8)color(white)(.)"cm" * 2.69 * 10^(-8)color(white)(.)"cm" * 12.05 * 10^(-8)color(white)(.)"cm"#

#V = 4.878 * 10^(-22)# #"cm"^3#

Now, the compound's molar mass tells you the mass of exactly #1# mole of the compound. As you know, a mole is defined as

#color(blue)(ul(color(black)("1 mole" = 6.022 * 10^(23)color(white)(.)"molecules"))) -># Avogadro's constant

Since you know that each unit cell holds #2# molecules, you will need to calculate the volume occupied by

#6.022 * 10^(23) color(red)(cancel(color(black)("molecules"))) * "1 unit cell"/(2color(red)(cancel(color(black)("molecules")))) = 3.011 * 10^(23)# #"unit cells"#

To do that, use the volume of a single unit cell

#3.011 * 10^(23)color(red)(cancel(color(black)("unit cells"))) * (4.878 * 10^(-22)color(white)(.)"cm"^3)/(1color(red)(cancel(color(black)("unit cell")))) = 146.9# #"cm"^3#

So, you know that #1# mole of molecules occupies #"146.9 cm"^3# and that each #"1 cm"^3# of this compound has a mass of #"1.419 g"#.

This means that the mass of #1# mole will be

#146.9 color(red)(cancel(color(black)("cm"^3))) * overbrace("1.419 g"/(1color(red)(cancel(color(black)("cm"^3)))))^(color(blue)("the density of the compound")) = "208.45 g"#

Therefore, you can say that this organic compound has a molar mass of

#color(darkgreen)(ul(color(black)("molar mass = 208 g mol"^(-1))))#

The answer is rounded to three sig figs, the number of sig figs you have for one of the dimensions of the unit cell.