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2. The Concept of the Atom Our present knowledge about the size and internal structure of atoms is the result of a long development of ideas and concepts that were initially based both on philosophical

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2. The Concept of the Atom Our present knowledge about the size and internal structure of atoms is the result of a long development of ideas and concepts that were initially based both on philosophical speculations and on experimental hints, but were often not free of errors. Only during the 19th century did the increasing number of detailed and carefully planned experiments, as well as theoretical models that successfully explained macroscopic phenomena by the microscopic atomic structure of matter, could collect sufficient evidence for the real existence of atoms and therefore convinced more and more scientists. However, even around the year 1900, some well-reputed chemists, such as Wilhelm Ostwald ( ), and physicists, e.g., Ernst Mach ( ), still doubted the real existence of atoms. They regarded the atomic model as only a working hypothesis that could better explain many macroscopic phenomena, but should not be taken as reality. In this chapter we will therefore discuss, after a short historical survey, the most important experimental proofs for the real existence of atoms. Furthermore, some measurements are explained that allow the quantitative determination of all atomic characteristics, such as their size, mass, charge distribution and internal structure. These experiments prove without doubt that atoms do exist, even though nobody has ever seen them directly because of their small size. 2.1 Historical Development Historically, the first concept of the atomic structure of matter was developed by the Greek philosopher Leucippus (around 440 B.C.) and his disciple Democritus ( B.C.) (Fig. 2.1), who both taught that all natural bodies consist of infinitely small particles that completely fill the volume of the bodies and are not Fig Democritus ( BC) (from K. Faßmann: Die Großen, BD I/2, Kindler-Verlag, Munich) further divisible. They called these particles atoms (from the Greek word atomos = indivisible). Outside the atoms there is only the empty space (a vacuum). Different atoms differ in size and shape and the characteristic properties of matter are, according to this model, due to different arrangements of equal or of differing atoms. All observable changes in the macroscopic world are caused by corresponding changes in atomic composition. Atom movements and collisions between atoms create and modify matter. We meet here for the first time the idea that the properties of macroscopic bodies can be explained by the characteristics of their constituents. This hypothesis, which comes close to our modern concept of atomic physics, had been an extension and refinement W. Demtröder, Atoms, Molecules and Photons, 2nd ed., Graduate Texts in Physics, DOI / _2, c Springer-Verlag Berlin Heidelberg 2010 8 2. The Concept of the Atom of former ideas by Empedocles ( B.C.), who believed that everything is composed of the four elemental constituents: fire, water, air and soil. The concept of Democritus represents in a way a symbiosis of the different doctrines of pre-socratic philosophers. First, the static hypothesis of Parmenides (around 480 B.C.) about the never-changing eternal existence of the world and secondly the dynamical doctrine of Heraclitus (around 480 B.C.), which stresses as the most important point the evolution instead of the static nature of things, since everything changes with time (nobody can submerge twice into the same river as the same man, because the river, as well as the man, is changing in time). According to Democritus, atoms represent static nature while their movements and their changing composition explain the diversity of matter and its time evolution. The famous Greek philosopher Plato ( B.C.) pushed the abstraction of the concept further. He used the hypothesis of the four elements fire, water, air, and soil but attributed to these elements four regular three-dimensional geometric structures, which are formed by symmetric triangles or squares (Fig. 2.2). Fire is related to the tetrahedron (four equilateral triangles), air to the octahedron (eight equilateral triangles), water to the icosahedron (20 equilateral triangles), and the soil, particularly important to mankind, to the cube (six squares or 12 isosceles triangles). Plato s ideas therefore reduced the atoms to mathematical structures that are not necessarily based on the real existence of matter. These mathematical atoms can change their characteristics by changing the arrangement of the elemental triangles. This is, according to Plato, equivalent to the observable evolution of matter. Aristoteles ( B.C.), a student of Plato, did not accept this concept of atoms since it contradicted his idea of a continuous space filled with matter. He Fig The platonic bodies also did not believe in the existence of empty space between the atoms. His influence was so great that Democritus hypothesis was almost abandoned and nearly forgotten until it was revived and modified later by Epicurus ( B.C.), who attributed atoms not only size but also a mass to explain why bodies fell down. After Epicurus the atomic theory was forgotten for many centuries. This was due to the influence of the Christian church, which did not accept the materialistic view that everything, even human beings, should be composed of atoms, because this seemed to be in contradiction to the belief in God as the creator of bodies and soul. There had occasionally been attempts to revive the atomic idea, partly induced by Arabic scientists, but they did not succeed against church suppression. One example was the Prior Nikolaus of Autrecourt in France, who was forced in 1348 to withdraw his newly developed atomic concept. The large shortcoming of all these philosophical hypotheses was the lack of experimental guidance and proof. They were more speculative. The real breakthrough of modern atomic physics was achieved by chemists in the 18th century. They found for many chemical reactions, by accurately weighing the masses of reactants and reaction products, that their results could be best explained by the hypothesis that all reactants consist of atoms or molecules that can recombine into other molecules (see below). Besides this increasing amount of experimental evidence for the existence of atoms, the atomic hypothesis won a powerful ally from theoretical physics when James Prescott Joule ( ), Rudolf Julius Clausius ( ), James Clark Maxwell ( ), and Ludwig Boltzmann ( ) developed the kinetic theory of gases, which could derive all macroscopic quantities of gases, such as pressure, temperature, specific heat, viscosity, etc., from the assumption that the gas consists of atoms that collide with each other and with the walls of the container. The temperature is a measure of the average kinetic energy of the atoms and the pressure represents the mean momentum the atoms transfer to the wall per second per unit wall area. The first ideas about the kinetic theory of gases and its relation to heat were published by John Herapath ( ) in a book, which was known to Joule. 2.2. Experimental and Theoretical Proofs for the Existence of Atoms 9 Quantitative information about the size of atoms and their internal structure, i.e., mass and charge distribution inside the atoms was only obtained in the 20th century. The complete theoretical description was possible after the development of quantum theory around 1930 (see Chaps. 3 and 4). In the Chronological Table at the end of this book one finds a compilation of historical landmarks in the development of atomic physics. For more detailed information on the history of atomic and molecular physics the reader is referred to the literature [2.1 6]. 2.2 Experimental and Theoretical Proofs for the Existence of Atoms Before we discuss the different experimental techniques developed for the proof of atoms, a general remark may first be useful. The objects of atomic physics are not directly visible since they are much smaller than the wavelength of visible light, unlike bodies in the macroscopic world. Therefore, indirect method for their investigation are required. The results of such experiments need careful interpretation in order to allow correct conclusions about the investigated objects. This interpretation is based on assumptions that are derived from other experiments or from theoretical models. Since it is not always clear whether these assumptions are valid, the gain of information in atomic physics is generally an iterative process. Based on the results of a specific experiment, a model of the investigated object is developed. This model often allows predictions about the results of other experiments. These new experiments either confirm the model or they lead to its refinement or even modification Dalton s Law of Constant Proportions The first basic experimental investigations that have lead to a more concrete atomic model, beyond the more speculative hypothesis of the Greek philosophers, were performed by chemists. They determined the mass ratios of the reactants and reaction products for chemical reactions. The basic ideas had already been prepared by investigations of Daniel Bernoulli ( ), who explained the experimental results of the Boyle Marriotte Law: p V = const at constant temperature by the movements of tiny particles in a gas with volume V which exert the pressure p onto the walls around V through collisions with the wall. These ideas laid the foundations of the kinetic gas theory, which was later more rigorously developed by Clausius, Maxwell, and Boltzmann. Following the more qualitative findings of Joseph Louis Proust ( ) on mass ratios of reactants and reaction products in chemical reactions, the English chemist John Dalton ( ) (Fig. 2.3) recognized, after many experiments of quantitative analyses and syntheses for various chemical compounds, that the mass ratios of reactants forming a chemical compound, are always the same for the same reaction, but may differ for different reactions. In this way, through collaboration between experimentalists and theoreticians, a successively refined and correct model can be established that reflects the reality as accurately as possible. This means that it allows correct predictions for all future experimental results. This will be illustrated by the successive development of more and more refined models of the atom, which will be discussed in the following sections and in Chap. 3. Fig John Dalton ( ) 10 2. The Concept of the Atom EXAMPLES g of water are always formed out of 11.1 g of hydrogen and 88.9 g of oxygen. The mass ratio of the reactants is then 1 : g of copper oxide CuO contains g Cu and g oxygen with a mass ratio of about 4 : Some reactants can combine in different mass ratios to form different products. For example, there are five different manganese oxides where 100 g of manganese combines either with g, g, g, g or g of oxygen. The different amounts of oxygen represent the mass ratios 2:3:4:6:7. From these experimental results Dalton developed his atomic hypothesis in 1803, which stated that the essential feature of any chemical reaction is the recombination or separation of atoms. He published his ideas in the paper A New System of Chemical Philosophy [2.7], which contains the three important postulates: All chemical elements consist of very small particles (atoms), which can not be further divided by chemical techniques. All atoms of the same chemical element have equal size, mass and quality, but they differ from the atoms of other elements. This means that the properties of a chemical element are determined by those of its atoms. When a chemical element A reacts with an element B to form a compound AB n (n = 1, 2,...) each atom of A recombines with one or several atoms of B and therefore the number ratio N B /N A is always a small integer. Dalton s atomic hypothesis can immediately explain the experimental results given in the above examples: 1. Two hydrogen atoms H recombine with one oxygen atom O to form the molecule H 2 O (Fig. 2.4). The observed mass ratio 11.1/88.9 is determined by the masses of the atoms H and O. From the mass ratio m(h)/m(o) = 1/16 (see Sects and 2.7), the measured mass ratio of the reactants follows as m(h 2 )/m(o) = 2/16 = 11.1/88.9. mh m H 2H AMU + mo H 2O 16 AMU 18 AMU Fig Reaction of hydrogen and oxygen to form water molecules as an example of Dalton s atomic hypothesis 2. For the reaction Cu + O CuO the mass ratio of the reactants corresponds to the relative masses m(cu)/m(o) = 64/16 = 4:1. 3. The different manganese oxides are MnO, Mn 2 O 3, MnO 2,MnO 3, and Mn 2 O 7. Therefore, the number of O atoms that combine with two Mn atoms have the ratios 2 : 3 : 4 : 6 : 7 for the different compounds, which is exactly what had been found experimentally. Since Dalton s laws only deal with mass ratios and not with absolute atomic masses, the reference mass can be chosen arbitrarily. Dalton related all atomic masses to that of the H atom as the lightest element. He named these relative masses atomic weights. Note: Atomic weights are not real weights but dimensionless quantities since they represent the ratio m(x)/ m(h) of the atomic masses of an atom X to the hydrogen atom H. Jörg Jakob Berzelius ( ) started to accurately determine the atomic weights of most elements in Nowadays this historic definition of atomic weight is no longer used. Instead of the H atom the 12 C atom is defined as reference. The atomic weight has been replaced by the atomic mass unit (AMU) 1AMU=(1/12) m( 12 C)= kg. All relative atomic masses are given in these units. EXAMPLES The mass of a Na atom is m(na) = 23 AMU, that of Uranium 238 is m(u) = 238 AMU and that of the nitrogen molecule N 2 is 2 14 = 28 AMU. 2.2. Experimental and Theoretical Proofs for the Existence of Atoms The Law of Gay-Lussac and the Definition of the Mole Joseph Louis Gay-Lussac ( ) and Alexander von Humboldt ( ) (Fig. 2.5) discovered in 1805 that the volume ratio of oxygen gas and hydrogen gas at equal pressures was always 1 : 2 when the two gases recombined completely to form water vapor. Further detailed experiments with other gases lead to the following conclusion: When two or more different gases completely recombine to form a gaseous chemical compound, the ratio of the volumes of reactands and reaction products at equal pressure and temperature is always given by the ratio of small integer numbers. EXAMPLES 1. 2 dm 3 hydrogen gas H 2 and 1 dm 3 oxygen gas O 2 recombine to form 2 dm 3 water vapor H 2 O (not 3dm 3 H 2 O as might be naively expected!) dm 3 H 2 and 1 dm 3 Cl 2 form 2 dm 3 HCl gas. Amadeo Avogadro ( ) (Fig. 2.6) explained these results by introducing the definition of molecules: A molecule is the smallest particle of a substance that determines the properties of this substance. It is composed of two or more atoms. Referring to the experimental results of Gay-Lussac, Avogadro concluded: At equal pressures and temperatures, the same volume of different gases always contains the same number of molecules. With this hypothesis the two preceding examples are described by the reaction equations: 2H 2 + O 2 2H 2 O, H 2 + Cl 2 2HCl. The total mass M of a gas with volume V containing N molecules with mass m is then: M = N m. (2.1) Fig Alexander von Humboldt ( ) (with kind permission from the Alexander von Humboldt foundation, Bonn) Fig Amadeo Avogadro ( ) with kind permission from the Deutsche Museum, Munich 12 2. The Concept of the Atom The mass ratio M 1 /M 2 of equal volumes of different gases at equal pressure and temperature therefore equals the mass ratios m 1 /m 2 of the corresponding molecules, since the number N of molecules is the same for both gases. It is convenient to introduce a specific reference quantity of molecules, called one mole [1mol]. The volume occupied by one mole of a gas is called the mole volume V M. The definition of a mole is as follows: 1 mol is the quantity of a substance that contains the same number of particles (atoms or molecules) as kg of carbon 12 C. This definition is equivalent to: 1 mol of atoms or molecules with atomic mass number X AMU has a mass of X grams. EXAMPLES 1. 1 mol helium He ˆ= 4 g helium 2. 1 mol oxygen O 2 ˆ= 2 16 g = 32 g oxygen 3. 1 mol water H 2 O ˆ= ( ) g = 18 g water 4. 1 mol iron oxide Fe 2 O 3 ˆ= ( ) g=160 g iron oxide The number N A of atoms or molecules contained in 1 mol is the Avogadro constant. Its experimental value is N A = (10) mol 1. From the hypothesis of Avogadro the statement follows: Under standard conditions (p = 1000 hpa, T = 0 C) 1 mol of an arbitrary gas always occupies the same volume V M, called the mole volume: V M = (39) dm 3 mol 1. Note: The value of V M depends on the definition of the standard conditions. There are two sets of these conditions in use: The old one uses a) T = 273,15K (0 0 C) and p = 1atm ( hPa) V M = 22, dm 3 mol 1 The new one, adapted in the CODATA list and NIST reference tables uses: b) T = 273,15K (0 0 C) and p = 1bar (1000hPa) V M = dm 3 mol Experimental Methods for the Determination of Avogadro s Constant Since the Avogadro constant N A is a fundamental quantity that enters many basic physical equations, several experimental methods have been developed for the accurate measurement of N A [2.8]. We will only present some of them here. a) Determination of N A from the general equation of gases From the kinetic theory of gases the general equation p V = N k T (2.2) can be derived for the volume V of an ideal gas under the pressure p at a temperature T, which contains N molecules. Here k is the Boltzmann constant. For 1 mol of a gas with volume V M, N becomes N A and (2.2) converts to p V M = N A k T = R T. (2.3) The gas constant R = N A k (2.4) is the product of Avogadro s and Boltzmann s constants. It can be determined from (2.3) when p, V M and T are measured. If the Boltzmann constant k and the gas constant R can be measured independently, the Avogadro constant N A can be determined from (2.4). α) Measurements of the gas constant R The gas constant R can be obtained from measurements of the specific heat. The internal energy of 1 mol is U = f 12 kt N A = 1 2 f R T, (2.5) where f is the number of degrees of freedom of the atoms or molecules of the substance. For example 2.2. Experimental and Theoretical Proofs for the Existence of Atoms 13 f = 3 for atoms, f = = 5 for diatomic molecules at low temperatures where the vibrations are not excited and f = 7 at higher temperatures. The molar specific heat C v for a constant mole volume of a gas is ( ) U C v = = 1 T v 2 f R. (2.6) This is the energy that increases the temperature of 1 mol of a gas by 1 K and can therefore be readily measured, giving the value of R, if the number of degrees of freedom f is known. Another way to measure the gas constant R is based on the difference To pump S Argon Gas inlet P Mi T R = C p C v (2.7) of the molar specific heats C p at constant pressure and C v at constant volume. The most accurate determination of R uses the measurement of the velocity of sound waves v s in an acoustic resonator (Fig. 2.7). A spherical volume is filled with argon at a pressure p and temperature T. A small loudspeaker S produces sound waves that lead to resonant standing waves if the sound frequency matches one of the radial eigenfrequencies f 0,n v s /λ 0,n with λ 0,n = r 0 /n of the spherical acoustic resonator with radius r 0. These resonantly enhanced sound waves are detected by a microphone Mi. The frequencies f 0,n of different resonances are measured. As is outlined in the solution of Problem 2.6, the gas constant is related to the measurable acoustic eigenfrequencies f 0,n, the sound velocity v s, the molar specific heats C p and C v, the temperature T and the volume V by R = M v2 s T κ = M T f 2 0,n r 2 0 κ n 2, (2.8) where M is the molar mass and κ = C p /C v [2.9]. Since the acoustic losses of the spherical resonator are low, the resonances are very sharp and the resonance frequencies f 0,n can be determined with high accuracy.

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