Lab Report on Qualitative and quantitative analysis of alkali halide mixture by powder X-ray diffraction

The lab report below was submitted as part of the coursework for CM3292 Advanced Experiements in Analytical and Physical Chemistry. Please do not plagiarise from it as plagiarism might land you into trouble with your university. Do note that my report is well-circulated online and many of my juniors have received soft copies of it. Hence, please exercise prudence while referring to it and, if necessary, cite this webpage.
 

1. Aim

This experiment uses X-ray powder diffraction technique to identify crystal structures, determine the structural parameters for alkali halides and quantitatively analyse the composition of alkali halide mixtures.

2. Results and calculations

Using the sample A1 of pure KCl for calculations, the following results were obtained:

2θ / °	sin2θ	h2 + k2 + l2	Integer	Simple cubic plane
28.559	0.060837	1.000000	1	100
40.770	0.121331	1.994358	2	110
50.403	0.181308	2.980212	3	111
58.869	0.241502	3.969629	4	200
66.604	0.301458	4.955148	5	210
73.854	0.360957	5.933148	6	211

Given h2 + k2 + l2 = (4a2/λ2)sin2θ, and that λ = 1.5405Å, and that gradient = λ2 / 4a2= 0.0600,

Solving, a = 3.1445Å

For a simple cubic model, there is only 1 molecule of KCl per unit cell.

Density using obtained parameter = 1 ×74.556.022 × 1023 ×(3.1445 × 10-8)3 = 3.981g/cm3

Comparing this obtained density value to the macroscopic density of 1.984 g/cm3, the obtained value does not coincide. Therefore, KCl does not have a simple cubic structure.

Using the FCC model for KCl instead, the following results were obtained:

2θ  / °	sin2θ	h2 + k2 + l2	FCC integer	FCC plane
28.559	0.060837	1.000000	4	200
40.770	0.121331	1.994358	8	220
50.403	0.181308	2.980212	12	222
58.869	0.241502	3.969629	16	400
66.604	0.301458	4.955148	20	420
73.854	0.360957	5.933148	24	422

Following similar calculations as done above for the simple cubic model,

 

λ2 / 4a2= 0.0150, a = 6.289Å

For FCC model, there are 4 molecules per unit cell,

Density = 4 ×74.5516.022 × 1023 ×(6.289 × 10-8)3 = 1.9908 g / cm3

This calculated value using the FCC model is closer to the given value of 1.984 g/cm3. Hence, the Bravais lattice for KCl is face-centred cubic.

Given that the ionic radius of K+ ion is 1.34Å,

Ionic radius of Cl- ion = 3.1445Å - 1.34Å =1.805Å

The experimental radius of Cl- is close to its reported value of 1.81 Å [2].

The following results were obtained for sample A5 of pure NaCl:

2θ  / °	sin2θ	h2 + k2 + l2	FCC integer	FCC plane
27.535	0.056636	1.000000	3	111
32.059	0.076249	1.346307	4	200
45.583	0.150062	2.649607	8	220
56.626	0.224949	3.971860	12	222
66.355	0.299466	5.287578	16	400
75.404	0.373999	6.603593	20	420

Gradient of the line = λ2 / 4a2 = 0.0186, a = 5.648Å

Density = 4 ×58.4436.022 × 1023 ×(5.648 × 10-8)3 = 2.044g / cm3

This calculated value compares favorably to the literature value of 2.165g / cm3, suggesting that NaCl has a face-centered cubic lattice.

From the earlier calculated radius of Cl- = 1.805Å,

Ionic radius of Na+ ion = (5.648/2) – 1.805 = 1.019Å

The experimental radius of Na+ is close to its reported value of 1.020 Å [2], suggesting that the experiment is highly accurate. The Cl- ions are larger than the Na+ ions. The Cl- ions are represented by the larger spheres in the above diagram.

Qualitative analysis

	Mixtures of NaCl and KCl
	Mass / gram	Weight / %	(200) peak intensity	(200) peak intensity %
Sample	NaCl	KCl	NaCl	NaCl	KCl	NaCl	KCl
A1	0.0000	2.0002	0.00	0	1577	0.00	100.00
A2	0.5051	1.5001	25.19	4538	5827	43.78	56.22
A3	1.0004	1.0097	49.86	2785	1596	63.57	36.43
A4	1.5024	0.5008	75.00	3437	854	80.10	19.90
A5	2.0008	0.0000	100.00	12096	0	100.00	0.00
Unknown A		7973	6974	53.34	46.66

Peak intensity of NaCl in unknown sample = 57.37%

Using the graph, the weight percentage of NaCl in unknown sample = (57.37 – 10.17) / 0.946

     = 49.89% NaCl by weight

Weight percentage of KCl in unknown sample = 100 – 49.89 = 50.11% KCl by weight

3. Discussion

In this experiment, Powder X-Ray Diffraction (XRD) is used to determine the crystallographic structures of different alkali halides. The crystalline powder of these samples will diffract incident X-ray beams if their crystallites are coincidentally oriented such that it obeys the Bragg equation.

Principles of XRD

*

Diagram showing X-ray diffraction by a plane [4]

From the right figure, the path difference of the X-ray beams is given by EF + FG. They are equidistant, and can be calculated by EF = FG = dhklsinθ. For constructive interference to occur, the waves must be at integer value difference from each other, and Bragg’s law can be written as nλ = 2 dhklsinθ. For a cubic unit cell system, dhkl2 =a2h2+ k2+ l2, and Bragg’s law can thus be rewritten as h2 + k2 + l2 = (4a2/λ2)sin2θ. In the machine used, the disc is gradually rotated over a range of angles to obtain the different diffraction patterns (bottom right diagram).

*

Diagram showing the operation by the machine in determining the angles [4]

In cubic systems, the first X-ray diffraction signal in the spectrum is due to the diffraction by the plane with the lowest Miller indices. For a cubic unit cell, the following are possible:

Simple cubic (100) plane, h2 + k2 + l2 = 1

BCC (110) plane, h2 + k2 + l2 = 2

FCC (111) plane, h2 + k2 + l2 = 3

As such, the value of sin2 q/ sin2 qmin has to be multiplied with the appropriate integer value to determine the Bravais lattice type.

The position of the atoms and the scattering factor affects the peak intensity. For KCl, the scattering factor is the same for the cations and anions. The following structure factors apply to FCC lattices:

Fhkl = 4(f+ + f-) for all even values of Miller indices

Fhkl = 4(f+ - f-) for all odd values of Miller indices

Fhkl = 0 for mix of odd and even Miller indices

where f+ and f- represent the scattering factor of the cations and anions respectively. It thus follows that the reflection intensity will only be recorded for even values of Miller indices since the odd ones will cancel out. This accounts for the systematic absence recorded where some reflections expected were not observed.

X-Ray Diffractograms of NaCl and KCl

The angles θ (given by the position of the lines) due to the same set of Miller planes are slightly larger in NaCl. This is because the NaCl unit cell is slightly smaller than that of KCl. The more striking difference is the absence of certain lines like (111), (311), (511) etc. in KCl although these lines are present in powder pattern of NaCl. From systematic absence rules for FCC crystals, we know that the lines with mixed indices should be absent for both NaCl and KCl. This is indeed so. Further, for NaCl it can be observed that the successive orders of (111) planes (these are 111, 222, 333, 444 etc.) are alternately weak and strong. For example, the reflection from (111) is weak and that from (222) is strong and so on. In the binary fcc compounds like NaCl or KCl, the (111) planes are alternately occupied by Na+ (or K+) and Cl– ions. If the scattered waves from two or more such planes containing only sodium are in phase and intensify the reflection, the planes containing sodium will be out of phase with planes containing chlorine and will interfere destructively thus diminishing the intensity. This is the reason why (111) reflection in NaCl is weak and (222) reflection is strong. For KCl, the alternate (111) planes have potassium and chlorine. These two ions have the same number of electrons and identical scattering power. So far as X-ray is concerned, K+ and Cl – are identical. Since the radiation scattered by these planes are out of phase, the reflected waves will get completely annihilated and the reflections from (111), (333) etc. will not be seen[3].

At the end of the XRD, it was observed that the disc A1 had some purple colouration and disc A5 had a yellowish brown colouration. This is due to phosphorescene. When the X-ray beams were incident on the compounds for a long period of time, they were excited to the triplet state from the singlet state despite this being a spin forbidden transition. Similarly, it would take a long period of time for the excited triplet state to lose energy to return to the ground singlet state. In the midst of doing so, visible light is given off, thus allowing us to observe colouration on the plates. Comparing with literature values, NaCl absorbs at about 450nm and KCl absorbs at about 550nm. Thus a complementary colour of yellow and purple were observed for NaCl and KCl respectively. The other discs did not have as distinct a colouration as A1 and A5, possibly due to the mixture of 2 different salts.

The experimental radius of Na+ (1.019 Å) is close to its reported value of 1.02 Å [2]. The experimental radius of Cl- (1.805 Å) is close to its reported value of 1.81 Å [2], suggesting that the experiment is highly accurate.

Limitations of XRD

Initial calculations of h2 + k2 + l2 for KCl seemed to show that it has a simple cubic unit cell. This can be explained by the same number of electrons on the cation and anion, and the same ability to disperse the X-ray beams. Consequently, the X-ray beams were not able to differentiate between the two different elements and KCl appears to have a simple cubic lattice when it is actually a FCC lattice.

While X-ray diffraction is sufficient in determining the atomic positions and the Bravais lattice type, it has its limitations too. The bond length as determined by X-ray diffraction is distorted by the electron cloud, and will be shorter than actual. A more accurate alternative would be to employ neutron diffraction.

4. Conclusion

The results are summarised as follows:

Salt	Molecular weight (g/mol)	Macroscopic Density (g/cm3)	Bravais Lattice Type	Unit Cell Dimension (Å)	Crystallographic density (g/cm3)
NaCl	58.443	2.165	Face-centered cubic	5.648	2.172
KCl	74.551	1.984	Face-centered cubic	6.289	1.991

The ionic radii of Na+ ion and Cl- ion are 1.019Å and 1.805Å respectively. The weight percentage of the unknown sample was determined to be 49.89% weight NaCl and 50.11% weight KCl.

5. Exercise

1. In both KCl and NaCl, the Cl- ions can be viewed as lying on the (111) plane. In between these (111) planes, layers of the cations (K+ or Na+) lie. The reflection of the X-ray beams from Cl- would be completely out of phase with the reflection by Na+. As such, destructive interference would take place.

For KCl, both K+ and Cl- ions have 18 electrons. They are isoelectronic. As such, they disperse X-ray beams to a similar extent. Destructive interference would then occur, resulting in the cancellation of the reflections of the X-ray beams. Thus, the (111) reflection appears to be absent.

For NaCl and KBr, the cations have fewer electrons than the anions. The abilities of the ions to scatter X-rays are different. As such, their reflections of X-rays will only partially cancel out each other and give a weak peak. [4] Specifically, the (111) reflection in NaCl corresponds to one wavelength of path difference between the neighbouring (111) planes. [5]

2. Given the values for tungsten metal, the following results were obtained:

   2θ  / °	sin2θ	h2 + k2 + l2	Simple Cubic integer	BCC integer
   40.262	0.118451	1	1	2
   58.251	0.236900	1.999983115	2	4
   73.184	0.355350	2.999974673	3	6
   86.996	0.473797	3.999940904	4	8
   100.632	0.592250	4.999957788	5	10
   114.923	0.710700	5.999949346	6	12

   
   
   Using the simple cubic model and the equation h2 + k2 + l2 = (4a2/λ2)sin2θ,
   Gradient = λ2 / 4a2 = 0.1184, a = 2.238Å
   Density = 1 ×183.846.022 × 1023 ×(2.238 × 10-8)3 = 27.234g / cm3
   This calculated value differs greatly from the literature density value. Hence, the Bravais lattice of tungsten is not simple cubic.
   Using the BCC model instead,
   Gradient = λ2 / 4a2 = 0.0592, a = 3.166Å
   Density = 2 ×183.846.022 × 1023 ×(3.166 × 10-8)3 = 19.240g / cm3
   This calculated value is close to the literature value for density of tungsten. Therefore, it can be determined that tungsten has a body centred cubic Bravais lattice and a unit cell length of 3.166Å.
   6. References
   1. Shriver & Atkins, Inorganic Chemistry, 4th edition, Oxford.
   2. Jim Clark, Atomic and Ionic Radius, 2012.
   3. Lesley E. Smart, Elaine A. Moore, Solid State Chemistry, 3rd edition, CRC.
   4. University of Gaziantep, X-Ray Diffraction in Crystal, 2011.
   5. Skoog, D. A., West, D. M., Holler, F. J., & Crouch, S. R. . Fundamaentals of Analytical Chemistry, 2004.