2.48: How Many Bibles Can Fit on the Head of a Pin
- Page ID
- 158383
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)On December 29, 1959 Richard Feynman gave an address (Thereʹs Plenty of Room at the Bottom) in which he calculated how many Encyclopedia Britannicas could fit on the head of a pin. Legend identifies this event as the beginning of the field of theoretical nanotechnology.
To illustrate to general chemistry students how small nanoscopic entities, such as atoms, are I calculate with the following simple model how many bibles can fit on the head of a pin.
The model assumes that the pinhead surface is made up of Fe atoms packed in a simple squaric array. It further assumes that letters would be made by placing adatoms in the pockets created by the surface Fe atoms using a scanning tunneling microscope, and that 100 pinhead surface Fe atoms would be required to form a letter.
The first step is to calculate the number of Fe atoms on the pinhead.
\[ \begin{matrix} \text{Radium of pinhead:} & R_{PH} = \frac{1}{32} in & R_{PH} = 7.94 \times 10^8 pm \\ \text{Area of pinhead:} & \text{Area}_{PH} = \pi R_{PH}^2 & \text{Area}_{PH} = 1.98 \times 10^{18} \text{pm}^3 \\ \text{Radius of Fe:} & \text{R}_{Fe} = 126 \text{pm} & \text{R}_{Fe} = 1.26 \times 10^{-10} \text{pm} \\ \text{Area of Fe atom:} & \text{Area}_{Fe} = \pi \text{R}_{Fe}^2 & \text{Area}_{Fe} = 4.99 \times 10^4 \text{pm}^2 \end{matrix} \nonumber \]
Effective area of Fe atom:
\[ \begin{matrix} \text{EffectiveArea}_{Fe} = 4 \text{R}_{Fe}^2 \\ \text{EffectiveArea}_{Fe} = 6.35 \times 10^4 \text{pm}^2 \end{matrix} \nonumber \]
Fe atoms per pinhead:
\[ \begin{matrix} \text{FeAtomsPerPinHead} = \frac{ \text{Area}_{PH}}{ \text{EffectiveArea}_{Fe}} & \text{FeAtomsPerPinHead} = 3.12 \times 10^{13} \end{matrix} \nonumber \]
A typical family Bible consists of 1,000 pages with an average of 5,000 characters and spaces per page. If it takes 100 Fe atoms to define a character, how many Bibles can fit on the head of a pin?
\[ \begin{matrix} \text{PagesPerBible} = 1000 & \text{CharactersPerPage} = 5000 & \text{FeAtomsPerCharacter} = 100 \end{matrix} \nonumber \]
Fe atoms required per bible:
\[ \begin{matrix} \text{FeAtomsPerCharacter CharactersPerPage PagesPerBible} = 5 \times 10^8 \\ \text{BiblesPerPinHead} = \frac{ \text{FeAtomsPerPinHead}}{ \text{FeAtomsPerCharacter CharactersPerPage PagesPerBible}} \\ \text{BiblesPerPinHead} = 6.2 \times 10^4 \end{matrix} \nonumber \]
Define picometer: \( \text{pm} = 10^{-12} \text{m}\)