Question

According to Kenny et al., ${ }^{3}$ Avogadro's number $N_{A}$ may be calculated from the following equation using measurements on a sphere fabricated from an ultrapure single crystal of silicon: $$ N_{\mathrm{A}}=\frac{m M_{\mathrm{Si}} V}{m a^{3}} $$ where $$ \begin{aligned} N_{\mathrm{A}}=& \text { Avogadro's number } \\ n=& \text { the number of atoms per unit cell in the } \\ & \text { crystal lattice of silicon }=8 \\ M_{\mathrm{S}}=& \text { the molar mass of silicon } \\ V=& \text { the volume of the silicon sphere } \\ m=& \text { the mass of the sphere } \\ a=& \text { the crystal lattice parameter }=\\ & d(220) \sqrt{2^{2}+2^{2}+0^{2}} \end{aligned} $$ (a) Derive the equation for Avogadro's number. (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 . $$ \begin{array}{lcc} \text { Variable } & \text { Value } & \text { Relative uncertainty } \\ \hline \text { Sphere volume, } \mathrm{cm}^{3} & 431.059059 & 23 \times 10^{-9} \\ \text { Sphere mass, g } & 1000.087560 & 3 \times 10^{-9} \\ \text { Molar mass, g/mol } & 27.97697026 & 6 \times 10^{-9} \\ \text { Lattice spacing } d(220), \mathrm{pm} & 543.099624 & 11 \times 10^{-9} \\ \hline \end{array} $$ (c) Calculate Avogadro's number and its uncertainty. (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 $N_{\mathrm{A}}$. After you have studied Chapter 7, compare your two values for $N_{\mathrm{A}}$, 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. (e) Which of the variables in the table have the most significant influence on the value that you calculated and why? (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. (h) Suggest ways that the determination of Avogadro's number might be improved. (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. (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?

Name: Problem 41, Chapter 4: Fundamentals of Analytical Chemistry
Uploaded: 2020-09-28T15:05:12-08:00
Duration: 5 min 36 s
Channel: Lottie Adams

   According to Kenny et al., ${ }^{3}$ Avogadro's number $N_{A}$ may be calculated from the following equation using measurements on a sphere fabricated from an ultrapure single crystal of silicon:
$$
N_{\mathrm{A}}=\frac{m M_{\mathrm{Si}} V}{m a^{3}}
$$
where
$$
\begin{aligned}
N_{\mathrm{A}}=& \text { Avogadro's number } \\
n=& \text { the number of atoms per unit cell in the } \\
& \text { crystal lattice of silicon }=8 \\
M_{\mathrm{S}}=& \text { the molar mass of silicon } \\
V=& \text { the volume of the silicon sphere } \\
m=& \text { the mass of the sphere } \\
a=& \text { the crystal lattice parameter }=\\
& d(220) \sqrt{2^{2}+2^{2}+0^{2}}
\end{aligned}
$$
(a) Derive the equation for Avogadro's number.
(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 .
$$
\begin{array}{lcc}
\text { Variable } & \text { Value } & \text { Relative uncertainty } \\
\hline \text { Sphere volume, } \mathrm{cm}^{3} & 431.059059 & 23 \times 10^{-9} \\
\text { Sphere mass, g } & 1000.087560 & 3 \times 10^{-9} \\
\text { Molar mass, g/mol } & 27.97697026 & 6 \times 10^{-9} \\
\text { Lattice spacing } d(220), \mathrm{pm} & 543.099624 & 11 \times 10^{-9} \\
\hline
\end{array}
$$
(c) Calculate Avogadro's number and its uncertainty.
(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 $N_{\mathrm{A}}$. After you have studied Chapter 7, compare your two values for $N_{\mathrm{A}}$, 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.
(e) Which of the variables in the table have the most significant influence on the value that you calculated and why?
(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.
(h) Suggest ways that the determination of Avogadro's number might be improved.
(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.
(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?

Fundamentals of Analytical Chemistry

Douglas A. Skoog,… 9th Edition

Chapter 4, Problem 41

Instant Answer

Solved by Expert Lottie Adams

09/28/2020

Step 1

** To derive the equation for Avogadro's number $ N_A $, we start with the given equation: \[ N_A = \frac{m M_{\mathrm{Si}} V}{m a^3} \] We know that the number of atoms $ n $ in a unit cell of silicon is 8. The volume of the unit cell $ V_{\text{cell}} $ Show more…

Show all steps

Thanks for your feedback!

According to Kenny et al., ${ }^{3}$ Avogadro's number $N_{A}$ may be calculated from the following equation using measurements on a sphere fabricated from an ultrapure single crystal of silicon: $$ N_{\mathrm{A}}=\frac{m M_{\mathrm{Si}} V}{m a^{3}} $$ where $$ \begin{aligned} N_{\mathrm{A}}=& \text { Avogadro's number } \\ n=& \text { the number of atoms per unit cell in the } \\ & \text { crystal lattice of silicon }=8 \\ M_{\mathrm{S}}=& \text { the molar mass of silicon } \\ V=& \text { the volume of the silicon sphere } \\ m=& \text { the mass of the sphere } \\ a=& \text { the crystal lattice parameter }=\\ & d(220) \sqrt{2^{2}+2^{2}+0^{2}} \end{aligned} $$ (a) Derive the equation for Avogadro's number. (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 . $$ \begin{array}{lcc} \text { Variable } & \text { Value } & \text { Relative uncertainty } \\ \hline \text { Sphere volume, } \mathrm{cm}^{3} & 431.059059 & 23 \times 10^{-9} \\ \text { Sphere mass, g } & 1000.087560 & 3 \times 10^{-9} \\ \text { Molar mass, g/mol } & 27.97697026 & 6 \times 10^{-9} \\ \text { Lattice spacing } d(220), \mathrm{pm} & 543.099624 & 11 \times 10^{-9} \\ \hline \end{array} $$ (c) Calculate Avogadro's number and its uncertainty. (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 $N_{\mathrm{A}}$. After you have studied Chapter 7, compare your two values for $N_{\mathrm{A}}$, 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. (e) Which of the variables in the table have the most significant influence on the value that you calculated and why? (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. (h) Suggest ways that the determination of Avogadro's number might be improved. (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. (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?

Play audio

Feedback

Lottie Adams and 95 other educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Avogadro's Number

Avogadro's number is a fundamental constant that represents the number of constituent particles, usually atoms or molecules, in one mole of a substance. It serves as a bridge between the microscale of atoms and molecules and the macroscale of bulk matter. The derivation of Avogadro's number in experiments such as those using silicon spheres involves linking the measured mass and volume of the sphere to the known lattice structure of the solid, thereby reinforcing its role in quantifying matter at an atomic level.

Crystal Lattice and Unit Cell

The crystal lattice is the periodic arrangement of atoms in a crystalline solid, and the unit cell is the smallest repeating structure that fully describes the lattice. In contexts where physical constants are derived through crystallography, understanding the number of atoms per unit cell and the lattice parameter is crucial because these determine the material’s density and serve as the foundation for calculations linking microscopic properties to macroscopic measurements.

Density Determination

Density is defined as the mass per unit volume of a substance and is a critical property in material characterization. Determining the density of a crystalline sphere involves precise measurements of its mass and volume. In experiments aiming to calculate fundamental constants, the accuracy of the density measurement directly influences the reliability of the derived values, thereby emphasizing the importance of high-precision metrological techniques.

Uncertainty Analysis

Uncertainty analysis is the process of quantifying the errors associated with measurements and propagating these uncertainties through calculations. In experiments with high precision requirements, such as the determination of Avogadro's number, recognizing and accurately estimating the contribution of measurement errors from parameters like mass, volume, and lattice spacing is essential. This allows for an assessment of the reliability and statistical significance of the final evaluated quantities.

Experimental Metrology Techniques

Experimental metrology involves the precise measurement of physical quantities using advanced apparatus and standardized methods. In the context of determining Avogadro's number with silicon spheres, techniques such as interferometry for lattice spacing measurement, high-accuracy mass comparisons, and precision volumetry are employed. These methodologies ensure that the data acquired is both accurate and reproducible, which is vital when connecting experimental results to fundamental physical constants.

Material Purity and Technological Advances

The use of ultrapure silicon in the fabrication of nearly perfect crystalline spheres is critical for minimizing systematic errors in experiments. Advances in material science and technology, including processes for refining and purifying silicon, help reduce impurities that could alter the crystal lattice structure or introduce defects. This improved material quality enhances the precision of measurements and the accuracy of derived constants, reflecting the impact of technological innovations in modern experimental physics.

Please give Ace some feedback