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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 57 7 Dimensions Let A be the atomic mass of the atom; it is (roughly) the number of pro- a a tons and neutrons in the nucleus. Although A is called a mass, it is dimensionless. Each atom occupies a cube of side length a ∼ 3 Å, and has mass Am proton . The density of the substance is mass Am proton ρ = volume ∼ (3 Å ) 3 . You do not need to remember or look up m proton if you multiply this fraction by unity in the form of N A / N A , where N A is Avogadro’s number: Am proton N A . ρ ∼ (3 Å ) 3 × N A The numerator is A g, because that is how N A is defined. The denominator is 3 10 − 23 cm 3 × 6 10 23 = 18 . · · So instead of remembering m proton , you need to remember N A . However, N A is more famil- iar than m proton because N A arises in chemistry and physics. Using N A also emphasizes the connection between microscopic and macroscopic values. Carrying out the calculations: ρ ∼ A g cm − 3 . 18 The table compares the estimate against reality. Most every- Element ρ estimated ρ actual day elements have atomic masses between 15 and 150 , so the Li . 39 . 54 density estimate explains why most densities lie between 1 and H 2 O 1 . 1 . 10 g cm − 3 . It also shows why, for materials physics, cgs units are more convenient than SI units are. A typical cgs density of Si 1 . 56 2 . 4 a solid is 3 g cm − 3 , and 3 is a modest number and easy to re- Fe 3 . 11 7 . 9 member and work with. However, a typical SI density of a solid Hg 11 . 2 13 . 5 3000 kg m − 3 . Numbers such as 3000 are unwieldy. Each time Au 10 . 9 19 . 3 you use it, you have to think, ‘How many powers of ten were U 13 . 3 18 . 7 there again?’ So the table tabulates densities using the cgs units of g cm − 3 . I even threw a joker into the pack – water is not an element! – but the density estimate is amazingly accurate. 7.4.3 Physical interpretation The previous method, dimensional analysis, is mostly mathematical. As a second computa- tion of a , we show you a method that is mostly physics. Besides checking the Bohr radius, it provides a physical interpretation of it. The Bohr radius is the radius of the orbit with the lowest energy (the ground state). The energy is a sum of kinetic and potential energy. This division suggests, again, a divide-and-conquer approach: first the kinetic energy, then the potential energy. 6.055 / Art of approximation 58 What is the origin of the kinetic energy? The electron does not orbit in any classical sense....
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apr04b - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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