A simple estimation of the size of the molecules using a pencil lead
Ricardo Medel-Esquivel, Isidro Gómez-Vargas, J. Alberto Vázquez, Ricardo García-Salcedo
AA simple estimation of the size of the molecules using a pencil lead
Ricardo Medel Esquivel,
1, a
Isidro G´omez Vargas, J. Alberto V´azquez, and Ricardo Garc´ıa Salcedo Instituto Polit´ecnico Nacional, CICATA-Legaria, Ciudad de M´exico, CP 11500, M´exico. Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, Cuernavaca, Morelos, 62210, M´exico (Dated: January 13, 2020)
PACS numbers:
One of the main topics of elementary physics is theidea that every material is composed of ”little particlesthat move around in perpetual motion, attracting eachother when they are a little distance apart, but repellingupon being squeezed into one other”[1]. These particlescould be atoms or molecules. Atoms are the smallest partinto which any material can be divided. Whereas whenseveral atoms are joined together, molecules are formed.Some interesting experiments to estimate the size ofsuch atoms or molecules have been done that do not in-volve sophisticated equipment. One of these early ex-periments was conducted by Lord Rayleigh (1842-1919),which consisted of a small drop of oil spread to form acircular patch on the surface of the water. With a fewsimple calculations it is possible to determine the size ofthe oil molecule composition and therefore to provide anestimate of the diameter of the carbon atom[2, 3].The main aim of this article is that students, at thebasic level of education, gain a quantitative understand-ing of the size of molecules by performing a simple ex-periment easily designed within the classroom. All theyneed is a pencil lead, millimeter paper and a measuringinstrument. Of course, we assume that all molecules areapproximately the same size[4].The pencil lead is composed of graphite molecules (thefourth most abundant chemical element in the Universe[5]), which we can imagine as identical spherical particles.These molecules form the macroscopic structure of thepencil, which has the shape of an elongated cylinder andhence its volume is given by: V C = πR H, (1)where R is the radius of the pencil lead and H its heightas we can see in Figure 1.If we draw a line on a sheet of millimeter paper, keep-ing the pencil straight, part of the pencil material willhave moved to the paper matrix, forming a very thinparallelepiped or tiny height box, whose volume is givenby: V B = 2 RLh, (2)where L is the length of the line, 2 R is the width of theline and h is the height of the box. a Electronic address: [email protected]
While the volume of graphite spent after drawing theline is: V (cid:48) C = πR ( H − H (cid:48) ) . (3)Under this assumption the material is deposited en-tirely on the surface of the paper, without loss, then wecan match the volume spent on the pencil lead would beequal to the volume deposited on the paper. See Figure1. Fig.
1: It is assumed that the volume of the graphite cylinderis distributed in n identical boxes, whose height h gives usan estimate of the maximum size a molecule could have. Indetail, it is considered that h is not the height of a singlemolecule but of many. And to make the effect more visible, it is possible todraw n equal lines, of known length. Then, the followingequality is satisfied V (cid:48) C = V B , therefore from (2) and (3): h = π R ( H − H (cid:48) )2 nL . (4)This height h can be considered as an upper boundfor the size of the graphite molecules, considered that isnot the height of a single molecule but of many. Taking n large enough to significantly reduce the length of thepencil lead is possible to calculate this estimate numeri-cally.According to the manufacturer’s specifications the HBpencil lead have H = 60 mm and 2 R = 0 . a r X i v : . [ phy s i c s . e d - ph ] J a n Fig.
2: It is shown the ”lead” (A) before drawing the linesand the ”lead” (B) after draw some lines in the millimeterpaper. The measure of what has been consumed of graphiteis that it is used to calculate the volume. the millimeter paper, starting from above. We performed the test with n = 50 y L = 100 mm (Fig. 2), and ob-tained that H (cid:48) = 59 . h = π . mm (0 . mm )2(50)(100 mm ) = 0 . mm = 3 . × − m. (5)This result is reasonable as a higher level for the sizeof the molecules, the individual size of the so-calledgraphene sheets has been systematically measured andvaries from 2 to 20 nm (2 × − m to 2 × − m ) [7]. Theorder of magnitude of the result does not change signif-icantly when the number of lines drawn or their lengthis increased. Similar results can be found by performinganalog experiments to the one presented here, althoughthose are more sophisticated and require more equipmentto carry them out [6].) [7]. Theorder of magnitude of the result does not change signif-icantly when the number of lines drawn or their lengthis increased. Similar results can be found by performinganalog experiments to the one presented here, althoughthose are more sophisticated and require more equipmentto carry them out [6].