Challenge Problem: According to Kenny et al.,3 Avogadro’s number NA may be calculated from...

Challenge Problem: According to Kenny et al.,3 Avogadro’s number NA may be calculated from thefollowing equation using measurements on a sphere fabricated from an ultrapure single crystal of silicon:

where

NA = Avogadro’s number

n = the number of atoms per unit cell in the crystal lattice of silicon = 8

MSi = the molar mass of silicon

V = the volume of the silicon sphere

m = the mass of the sphere

a = the crystal lattice parameter =

(a) Derive the equation for Avogadro’s number.

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(b) From the recent data assembled by Andreas et al.4 on Sphere AVO28-S5 in the table below, calculate the density of silicon and its uncertainty. You may wish to delay the uncertainty calculations until you have studied Chapter 6.

Variable	Value	Relative uncertainty
Sphere volume, cm3	431.059059	23 × 10–9
Sphere mass, g	1000.087560	3 × 10–9
Molar mass, g/mol	27.97697026	6 × 10–9
Lattice spacing d(220), pm	543.099624	11 × 10–9

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(c) Calculate Avogadro’s number and its uncertainty.

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(d) We have presented data for only one of the two silicon spheres used in these studies. Look up the data for Sphere AVO28-S8 cited in note 3 and calculate a second value for NA. After you have studied Chapter 7, compare your two values for NA, and decide whether the difference in the two numbers is statistically significant. If difference between the two values is not statistically significant, calculate a mean value for Avogadro’s number determined from the two spheres and the uncertainty of the mean.

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(e) Which of the variables in the table have the most significant influence on the value that you calculated and why?

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(f) What experimental methods were used to make the measurements shown in the table?

(g) Comment on experimental variables that might contribute to the uncertainty in each measurement.

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(h) Suggest ways that the determination of Avogadro’s number might be improved.

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(i) Use a search engine to locate the NIST website on fundamental physical constants. Look up the accepted value of Avogadro’s number and its uncertainty (2010 or later) and compare them with your computed values. Discuss any differences and suggest possible causes for the discrepancies.

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(j) What technological innovations of the past several decades have led to the easy availability of ultrapure silicon? What steps have been taken in recent years to minimize errors associated with impurities in the silicon used to fabricate the near-perfect spheres?5

3M. J. Kenny et al., IEEE Trans. Instrum. Meas., 2001, 50, 587, DOI: 10.1109/19.918198.

4B. Andreas et al., Phys. Rev. Lett., 2011, 106, 030801, DOI: 10.1103/PhysRevLett.106.030801.

5P. Becker et al, Meas. Sci. Technol., 2009, 20, 092002, DOI:10.1088/0957-0233/20/9/092002.