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Two images are shown. The first image shows a cube with black dots at each corner and a red dot in the center of each face of the cube. This cube is stacked with seven others that are not colored to form a larger cube. The second image is composed of eight small green spheres that form the corners of a cube with six other small green spheres located in the faces of the cube. Eight larger green spheres are spaced inside the cube and all of the spheres are connect to one another by lines. The name under this image reads “C a F, subscript 2, face-centered unit cell.”
Calcium fluoride, CaF 2 , forms an FCC unit cell with calcium ions (green) at the lattice points and fluoride ions (red) occupying all of the tetrahedral sites between them.

Calculation of ionic radii

If we know the edge length of a unit cell of an ionic compound and the position of the ions in the cell, we can calculate ionic radii for the ions in the compound if we make assumptions about individual ionic shapes and contacts.

Calculation of ionic radii

The edge length of the unit cell of LiCl (NaCl-like structure, FCC) is 0.514 nm or 5.14 Å. Assuming that the lithium ion is small enough so that the chloride ions are in contact, as in [link] , calculate the ionic radius for the chloride ion.

Note: The length unit angstrom, Å, is often used to represent atomic-scale dimensions and is equivalent to 10 −10 m.

Solution

On the face of a LiCl unit cell, chloride ions contact each other across the diagonal of the face:

Three images are shown. The first shows a cube of alternating green and purple spheres. A smaller cube within that cube is outlined and a larger version of it appears next. This figure is a grey cube that appears to be made up of spheres. There are small spaces between each sphere. There is a right triangle outlined in this cube and a larger version of it appears next. This right triangle has two sides labeled “a,” and the hypotenuse, which spans two half-circles and one full one is labeled, “r, 2 r, and r.”

Drawing a right triangle on the face of the unit cell, we see that the length of the diagonal is equal to four chloride radii (one radius from each corner chloride and one diameter—which equals two radii—from the chloride ion in the center of the face), so d = 4 r . From the Pythagorean theorem, we have:

a 2 + a 2 = d 2

which yields:

( 0.514 nm ) 2 + ( 0.514 nm ) 2 = ( 4 r ) 2 = 16 r 2

Solving this gives:

r = ( 0.514 nm ) 2 + ( 0.514 nm ) 2 16 = 0.182 nm ( 1.82 Å ) for a Cl radius .

Check your learning

The edge length of the unit cell of KCl (NaCl-like structure, FCC) is 6.28 Å. Assuming anion-cation contact along the cell edge, calculate the radius of the potassium ion. The radius of the chloride ion is 1.82 Å.

Answer:

The radius of the potassium ion is 1.33 Å.

Got questions? Get instant answers now!

It is important to realize that values for ionic radii calculated from the edge lengths of unit cells depend on numerous assumptions, such as a perfect spherical shape for ions, which are approximations at best. Hence, such calculated values are themselves approximate and comparisons cannot be pushed too far. Nevertheless, this method has proved useful for calculating ionic radii from experimental measurements such as X-ray crystallographic determinations.

X-ray crystallography

The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography    . Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring atoms in crystals (on the order of a few Å).

When a beam of monochromatic X-rays strikes a crystal, its rays are scattered in all directions by the atoms within the crystal. When scattered waves traveling in the same direction encounter one another, they undergo interference , a process by which the waves combine to yield either an increase or a decrease in amplitude (intensity) depending upon the extent to which the combining waves’ maxima are separated (see [link] ).

Questions & Answers

what is catenation
Oladuji Reply
The property of carbon to form long chain with other atom!
Lareb
hydrocarbons can be classified as..1.Aliphatic compounds 2.cyclic compounds.under aliphatic compounds there are two types saturated hydrocarbons(alkanes) and unsaturated hydrocarbons(alkenes and alkynes).
Niroshan Reply
thanks but i have also heard of aromatic hydrocarbons
emmanuel
so am kinda confused
emmanuel
hello i have big problems in understanding organic chemistry
emmanuel Reply
what are the main types of hydrocarbon
emmanuel
how many elements are in the periodic table
emmanuel Reply
118
daniel
please what are the main types of hydrocarbons
emmanuel
why Rutherford uses the gold foil instead of other metals?
Lareb Reply
Rutherford chose gold was because its extremely malleable. One can stretch gold foil until it is only a few atoms thick in places, which is not possible with aluminum. If the foil were too thick, there would be no transmission of particles at all; the whole point was to demonstrate that most alpha
daniel
wjat does Rutherford mean?
Asali
Ernest Rutherford was the scientist that preformed the experiment.
daniel
although other metals are also present which are more melleable!?so
Lareb
gold
daniel
what is a balanced equation 4 trioxonitrate (V)acid and sodium hydroxide?
Marcel Reply
proved ur Worth: If A is a of trioxonitrate(V)acid,HNO3' of unknown concentration .B is a standard solution of sodium hydroxide containing 4.00g per dm cube of solution.25cm cube portions solution B required an average of 24.00cm cube of solution A for neutralization,using 2 drops of methyl orange.
Marcel
calculate the concentration of solution B in moles per dm cube
Marcel
calculate the concentration of solution A and B in moles per DM cube
Marcel
finally calculate the concentration in g/dm cube of HNO3 in solution A (H=1,N=14,O=16,Na=23)
Marcel
calculate the standard enthalpy of formation for propane(C3H8) from the following data; 1), C3H8+5O2->3CO2+4H2O; -222.0kJ/mol 2), C+O2->CO2;-395.5kJ/mol 3),H2+O->H2O; 285.8kJ/mol
Josephine
let eventually of formation of propane = X X + (-222)=3×(-395.5)+4×(-286) rearrange to find X
Paul
wat is electrolysis?
Mgbachi Reply
it is the chemical decomposition of a substance when electric current is passed through it either in molten form or aqueous solution
Nuru
list the side effect of chemical industries
Chelsea Reply
how do you ionise an atom
Rabeka Reply
many ways ,but one of them is when the atom becomes heated to a certain temperature the surface electron becomes too energetic and leaves the atom because the attraction between the nucleus and the electron becomes overpowered by the energetic eletron
sunday
also hitting of two atoms can cause transfer of surface electrons
sunday
and when this transfers occur the atom becomes ionised
sunday
who is doing Cape chemistry tomorrow?
caramel Reply
What is hybridization
edmondnti Reply
the mix between different breeds of species in one
Jared
it is the blending of orbitals.
stanley
the mixing of orbital
caramel
are covalent bonds influenced by factors such as temperature and pressure?
patrick Reply
what is catalyst used for mirror test
Sanjay Reply
when an atom looses electron, what does it become?
Abdullahi Reply
it's oxidized and called an ion
Anora
thanks
Abdullahi
Now, I get it
Abdullahi
cation
Anora
can you give an example please, if you don't mind
Abdullahi
a positive ion,become positively charged/a cation.
Janis
sodium plus one is simple cation is exmpl
ajmal
Taking Sodium as example..... it carries a positive charge which means it is positively charged.....when it gains an electron, it is reduced cuz an electron is negatively charged.....also when an atom looses an electron, it becomes positively charged and when it gains, it becomes negatively charged.
Nuru
typically, ionization is the process where an atom looses or gains electron(s) to form ion(s) either a positively or negatively
Nuru
what is copper
Bryan Reply
just an element
Power
Cu
daniel

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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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