Droplets and atoms. Benjamin Schumacher Department of Physics Kenyon College. Bright Horizons 35 (July, 2018)

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1 Droplets and atoms Benjamin Schumacher Department of Physics Kenyon College Bright Horizons 35 (July, 2018)

2 Part I: Einstein's other great idea

The old revolution Birth of modern physics (1900-1930) Atomic

relativity Cosmology 19th Century synthesis (1800-1830) Wave

3 The old revolution Birth of modern physics ( ) Atomic nuclei & radioactivity Quantum physics Special and general relativity Cosmology 19th Century synthesis ( ) Wave optics Atomic theory Thermodynamics Electric and magnetic fields

4 Atomic explanations Chemical mass ratios 8 g oxygen Elements combine to form compounds only in specific ratios of mass 1 g + 9 g hydrogen water Atoms of different masses combine to form molecules O 16 u + H H 1 u each H O 18 u H H 2 O

5 Atomic explanations Gas volumes Equal volumes of different gases have different masses (at same T, P). 16 g oxygen 1 g hydrogen ~ 10 liters Avogadro's law Equal volumes of gases contain the same number of molecules (at same T, P). Some gases (H 2, O 2, N 2 ) are diatomic molecules.

6 Atomic explanations Crystal structure Solids form crystals with regular shapes. Only certain crystal shapes occur. Crystals are regular 3-D arrangements of molecules.

7 Atomic explanations Heat and pressure Matter has thermal energy (heat); gases exert pressure. heat pressure Atoms and molecules have random (i.e., disordered) thermal motions. Typical air molecule speed: v ~ 500 m/s (compare v s = 345 m/s) Pressure is due to huge numbers of tiny molecular impacts.

8 But how tiny is tiny? Atoms are very small. But exactly how small? N A = Avogadro's number (huge) (number of H-atoms in 1 g of H) 1 u = atomic mass unit (tiny) (mass of 1 H-atom) a = atomic radius (tiny) (atomic spacing in crystal lattice) Karlsruhe Congress (September, 1860) Today we can directly image atoms using techniques like scanning tunneling microscopy. (graphite: a = nm)

Ernst Mach (late 19th C) Perhaps atoms are only an "abstract model" with no real existence.

9 Alternate worlds How would things be different if atoms were 10 times smaller or larger (and thus more or less numerous)? Ernst Mach (late 19th C) Perhaps atoms are only an "abstract model" with no real existence. N A =, u = 0, a = 0 Josef Lodschmidt (1865) More subtle physical properties (e.g., viscosity) depend on the size of atoms and can be used to estimate it.

10 A mysterious motion Robert Brown (1827) Tiny particles suspended in water or air display an erratic jittering motion. Brownian motion does not depend on the composition of particles (dust, pollen, etc.), but is more for smaller grains.

11 A mysterious motion Robert Brown (1827) Tiny particles suspended in water or air display an erratic jittering motion. Brownian motion does not depend on the composition of particles (dust, pollen, etc.), but is more for smaller grains.

12 A mysterious motion Robert Brown (1827) Tiny particles suspended in water or air display an erratic jittering motion. Brownian motion does not depend on the composition of particles (dust, pollen, etc.), but is more for smaller grains.

Annus mirabilis Albert Einstein (1905): Six amazing papers Light quanta and photoelectric effect (Nobel, 1921) Special theory of relativity Mass and energy in

13 Annus mirabilis Albert Einstein (1905): Six amazing papers Light quanta and photoelectric effect (Nobel, 1921) Special theory of relativity Mass and energy in relativity (E = mc 2 ) Three papers on statistical physics: Ph.D. thesis Viscosity of liquids with suspended particles Detailed explanation of Brownian motion

14 A game of chance Flip a coin. Heads and tails are equally probable. If heads, win W dollars. If tails, win 0 dollars. On average, each coin toss wins W/2 dollars.

15 The square-root rule Setting: Repeat a random process. On average, event X will occur N times. How much does the actual number differ? Typical variation: ± " Usually within " of N Seldom farther than ~3 " Example: Geiger counter "clicks" from a radioactive sample. On average, we expect 100 clicks/min. We observe: 99, 106, 107, 92, 85, 101, = 10 click, click-click, click,...

16 A game of chance Flip a coin. Heads and tails are equally probable. If heads, win W dollars. If tails, win 0 dollars. On average, each coin toss wins W/2 dollars. 10 games with W = $1: " = 5 5 = 2.24 Average winnings $5 Typical variation $2-3

17 A game of chance Flip a coin. Heads and tails are equally probable. If heads, win W dollars. If tails, win 0 dollars. On average, each coin toss wins W/2 dollars games with W = $0.01: # = = 22.4 Average winnings $5 Typical variation $

18 Buffeted by molecules A floating dust grain is hit by molecules from all sides. These tiny impulses cancel out on average. Atoms larger, N A smaller: Actual collisions may not cancel out as nearly. More Brownian motion. Einstein: By studying Brownian motion, we can measure N A! N A (huge) (Perrin, 1908) Atoms smaller, N A larger: Actual collisions tend to cancel out pretty well. Less Brownian motion.

19 Part II: A rainy day in Cambridge

20 What is a droplet? A droplet is a cluster of particles held together by short-range forces. Example: Water drop Molecular forces are short-range. Example: Atomic nucleus Nuclear forces are short-range.

21 What is a droplet? A droplet is a cluster of particles held together by short-range forces. Non-example: Star cluster Gravitational forces are long-range.

22 Droplet density Interactions between particles How close the particles are to each other Density of the droplet Big and small droplets of a given type have about the same density. Water drop: r = 10 3 kg/m 3 Nucleus: r = kg/m 3

23 Forces and "bonds" e Consider two nearby (interacting) particles We must add energy to pull the particles apart. Nearby particles are joined by a bond. We must input an energy e in order to break the bond and pull them apart.

24 Binding energy droplet + energy cloud E b = binding energy = energy needed to break all of the bonds in the droplet

25 How many bonds? N small Each particle is bound to every other particle.! " $ %! " $ $ N large Most particles are inside the droplet, bound to the same number of neighbors.! " $! " $ = '()*+,)+

26 Binding energy = boiling energy The energy required to boil a large sample of liquid is proportional to the amount of liquid. We need about 2500 J to boil 1 cm 3 of water. clock power meter boiling water scale

27 Nuclear binding energy We cannot boil a nucleus. But we can measure its mass very exactly! ¹! " = $% & ' The mass of a nucleus is slightly less than the total mass of the protons and neutrons separately. binding energy speed of light Mass defect This looks okay to me.

28 The curve of binding energy Binding energy per nucleon! " #! " # #! " # = &'()*+(* # Number of nucleons What we expect

29 The curve of binding energy Binding energy per nucleon! " # H Fe # Number of nucleons U What we expect Actual curve

30 The curve of binding energy Binding energy per nucleon! " # H Fe # Number of nucleons U About half of the nucleons are positively charged protons. Protons repel each other via long-range electric force. Thus, nuclei with many protons are less strongly bound (lower E b ).

31 The curve of binding energy Binding energy per nucleon! " # H Fe # Number of nucleons U We can release energy by creating more strongly bound nuclei (larger E b ) Fusion: Lighter nuclei combine to form larger ones. Fission: Heavier nuclei split into smaller fragments.

32 Energy at the edge Liquid droplet (lots of particles) Interior particles are surrounded by neighbors Particles near outer surface have fewer "bonds" to neighbors overall binding energy E b is less

33 Surface tension Any actual droplet will have a little extra energy ("missing" binding energy) because of particles at the surface.! " = $ % surface tension Falling liquid droplets are spherical. Why? This shape minimizes surface area (and thus E s ) Weizsäcker and Bethe (1935) include surface tension in the nuclear binding energy formula.

34 Measuring surface tension Water drops on a horizontal non-wetting surface: Drop tries to minimize its total energy, including both surface tension and gravity. Smaller drops all have about the same shape. Larger drops ("puddles") all have about the same depth. We can measure surface tension by measuring depth of puddles.

35 Puddles on paper water wax paper glass plate graph paper Pour a known volume of water on the wax paper. Use the graph paper to find the area of the resulting puddle. Determine the depth of the puddle.

36 From my kitchen 10 cm 3 of water Graph paper has 1 cm 2 grid Puddle occupies about 30 cm 2 Puddle depth is about 3.3 mm My estimated value of surface tension: s = J/m 2 (Fairly close to textbook value.)

37 energy (J) For a droplet of any radius R, we can find E b R 3 and E s R radius (m)

38 10 9 energy (J) E b = binding energy E s = surface energy Something funny at R ~ meters. Surface tension cancels binding energy! radius (m)

39 10 9 energy (J) E b = binding energy Size of molecules: ~ m E (distance s = surface scale for molecular energy forces) Something funny at R ~ meters. Surface tension cancels binding energy! radius (m)

40 Moral(s) of our story Moral #1 It might seem that atoms are so small that it does not actually matter how small they are. But this is not true humanly visible phenomena do depend on exactly how big atoms are! Bridging the gap: 1. Random fluctuations matter, not just averages! 2. Apparently different things (boiling energy, surface tension) are aspects of the same atomic-scale physics!

41 Moral(s) of our story Moral #2 We can get a lot of mileage out of a very simple idea that a droplet is a cluster of particles held together by short-range forces.

What he said Richard Feynman "If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would

42 What he said Richard Feynman "If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis... that all things are made of atoms little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence... there is an enormous amount of information about the world, if just a little imagination and thinking are applied." (Feynman Lectures on Physics, Lecture 1)

43 Moral(s) of our story Moral #2 We can get a lot of mileage out of a very simple idea that a droplet is a cluster of particles held together by short-range forces. Moral #3 By extrapolating known physical laws, we can guess where new physics must lie.

44 Where the new physics is energy E b = binding energy ~10-10 m E s = surface energy droplet size new physics?

45 Finis

46 Surface tension from puddle depth Puddle area: # Puddle depth: h Volume of water:! = # h h gravity: g 10 m/s 2 Total energy: % & + % ( = ½ * + h +,! h Energy is minimized: h - = 2, * +

The structure of Atom III

The structure of Atom III Atomic Structure If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement