CHEM 105 DISCUSSION SECTION, 7-NOV-02

Involved examinations and interpretations of X-ray diffraction photographs of (1) sodium chloride and (2) sucrose. In addition, mention was made of end-of-semester student assignments.

(1) EXAMINATION AND INTERPRETATION OF X-RAY DIFFRACTION PHOTOGRAPHS OF CRYSTALLINE SODIUM CHLORIDE.

(a) Determination of unit cell edge length.

measurement of distance between left-right diffraction "spots" = 70.7 mm (mean of two student readings)

measurement of distance between fudicial marks on copies of film = 27.2 mm

use these measurements to calculate d* and d...

d* = 0.1791(Angstrom)^-1

1/d* = d = unit cell edge length = 5.583 Angstrom

A published value for this length is 5.62779 Angstrom.

What is the % disagreement between these two values?

% disagreement = 0.80 %, i.e., less than one per cent.

(Not bad for such a quick ruler/pencil measurement.)

(b) Number of unit cells present in one mole of crystalline sodium chloride? (given Density = 2.18 g/cm³ )

Volume of one mole NaCl:

= (23+35.5 g/mole)/(2.18 g/cm³) = 26.83 cm³/mole

Volume of one unit cell = (5.628 Angstrom)³ = 178.26 Angstrom³

Number unit cells in one mole:

= (26.83 cm³/mole) / [(178.26 Angstrom³)*(10^-24 cm³ / 1 Angstrom³)]

= 1.505 x 10⁺²³ unit cells / mole

(Were you expecting Avogadro's number? Why isn't this Avogadro's number?)

The answer is correct, this is the number of unit cells per mole for sodium chloride.

However, each unit cell contains a net of 4 sodium cations, and 4 chloride anions. {Eight sodium cations at corners contribute 1/8 each to the cell for a total of ONE. Six sodium cations at face-centers contribute 1/2 each to the cell for a total of THREE. Net number of sodium cations per unit cell is FOUR. Twelve chloride anions at center of cell edges contribute 1/4 each to the cell for a total of THREE. An additional chloride anion is present in the center of the cell; it contributes all of itself to the cell, i.e., ONE. Net number of chloride anions per unit cell is also FOUR. Equal numbers of cations and anions present in the unit cell is the real reason why the empirical formula of sodium chloride is 1 : 1.}

So let's change the question to: What is the number of sodium ions per mole?

= (4 Na cations / unit cell) x (1.505 x 10⁺²³ unit cells / mole) = 6.02 x 10⁺²³ Na cations / mole

This is exactly how Avogadro's number was first evaluated in 1912, using X-ray data to obtain volume of the unit cell, density to find the volume of one mole, and a knowledge of the contents of the unit cell to determine the net number of ions (or atoms, or molecules) present in the unit cell. Avogadro's Number indicates the number of ions, atoms or molecules present in one mole of a substance.

(c) Sizes of Ions

Knowledge of the structure of sodium chloride, i.e., knowing WHERE ions are located and the DISTANCE of the unit cell edge length, allows calculation of the sizes of ions. Sodium chloride crystallizes in a face-centered cubic unit cell. Chloride anions are located at all corners of the unit cell, and also all face-centers; they are in contact along the face diagonal. FOUR chloride anion radii span the face diagonal of the unit cell. Accordingly,

four chloride anion radii = [sqrt(2)*5.628 Angstroms]

radius of chloride anion = 1.99 Angstroms

Two chloride anion radii and two sodium cation radii span an edge of the unit cell. Accordingly,

two sodium cation radii + two chloride anion radii = 5.628 Angstroms

radius of sodium cation = 0.89 Angstroms.

(2) EXAMINATION AND INTERPRETATION OF X-RAY DIFFRACTION PHOTOGRAPHS OF CRYSTALLINE SUCROSE.

(a) Diffraction spots are closer together (i.e., smaller separations) on sucrose films as compared to sodium chloride films. Conclusion: unit cell edge lengths for sucrose are larger than for sodium chloride.

(b) Ruler measurements of spacings between diffraction spots along three axial lengths show none are the same. Conclusion: the unit cell of sucrose has three independent edge lengths.

(c) One film shows axes inclined at a angle measured (by protractor) to be between 102 and 103 degrees. Conclusion: the sucrose unit cell belongs to the MONOCLINIC crystal system.

(d) A table of published values for the sucrose crystal reveals the following unit cell parameters:

a = 10.8633, b = 8.7050, c = 7.7585 Angstroms, and beta = 102.945 degrees.

Unit cell volume can be calculated from the expression: abc[sin(beta)] = 715 Angstrom³ .

A MAGE model of sucrose was displayed. There are TWO molecules of sucrose per unit cell.

(e) Assignment (for next week Thursday): Calculate the density of crystalline sucrose.