Đang tải... (xem toàn văn)
The Wave Nature of Matter Causes Quantization tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tấ...
Nguồn: http://www3.interscience.wiley.com/cgi-bin/fulltext/114129695/PDFSTART Biên dịch và giới thiệu bởi: Nguyễn Hoàng Việt www.luyenkim.net Bản chất vật lý độ bền vật liệu ** By Zhe-Feng Zhang* and Jürgen Eckert Bảng 1. Sự so sánh về độ bền phá hủy, phá hủy trượt do kéo và độ bền ruêng của các HKVĐH khác nhau từ các nguồn tham khảo đã có. Độ bền phá hủy và góc phá hủy trượt do kéo quan sát được bởi các tác giả đối với mỗi hệ HKVĐH khác nhau, sau đó 2 biến độ bền riêng và tỷ số của nó được tính toán theo tiêu chuẩn phá hủy đồng nhất. Tác giả Thành phần θ θθ θ T = F max /A 0 (GPa) θ θθ θ T (degree) τ ττ τ 0 (GPa) σ σσ σ 0 (GPa) a = τ ττ τ 0 / σ σσ σ 0 He et al. 5 Zr 52.5 Ni 14.6 Al 10 Cu 17.9 Ti 5 1.66 -55 0.96 1.91 0.504 Inoue et al. 6 Cu 60 Zr 30 Ti 10 2.00 ~54 1.14 2.36 0.485 Lewandowski et al. 7 Zr 40 Ti 12 Ni 9.4 Cu 12.2 Be 22 1.98 -51.6 1.11 2.44 0.455 Liu et al. 8 Zr 52.5 Ni 14.6 Al 10 Cu 17.9 Ti 5 1.65 -54 0.94 1.95 0.485 Mukai et al. 9 Pd 40 Ni 40 P 20 1.65 -56 0.97 1.85 0.522 Noskova et al. 10 Co 70 Si 15 B 10 Fe 5 1.48 -60 0.91 1.57 0.577 Xiao et al. 11 Zr 52.5 Ni 10 Al 10 Cu 15 Be 12.5 1.75 -55 1.01 1.96 0.504 Zhang et al. 12 Zr 52.5 Ni 14.6 Al 10 Cu 17.9 Ti5 1.66 -56 0.97 1.86 0.522 Zhang et al. 13 Zr 59 Cu 20 Al 10 Ni 8 Ti 3 1.58 -54 0.90 1.86 0.485 Zielinski et al. 14 Ni 75 Si 8 B 17 1.59 -53 0.90 1.93 0.464 Độ bền của vật liệu được đánh giá thường thường qua phương pháp xác định dộ bền kéo. Đối với một vật liệu nhất định diện tích mặt cắt ban đầu A 0 , nếu áp dụng lực kéo lớn nhất dẫn đến phá hủy mẫu là F max , độ bền phá hủy sẽ được tính toán là: max max o F A σ = được mô tả trong nhiều sách giáo khoa [1,2] . Đối với mẫu vật liệu hợp kim vô định hình (HKVĐH-metallic glass), có xảy ra sự phá hủy ở chế độ trượt (shear mode), xem hình 1, và bề mặt trượt phá hủy tạo ra 1 góc 56 o T θ = tương ứng với trục kéo. Hành vi phá hủy trượt này xảy ra rất phổ biến đối với HKVĐH, như được mô tả trong bảng 1 [5-14] . Theo định nghĩa trong sách giáo khoa [1-2] , độ bền phá hủy khi kéo của HKVĐH sẽ là max max o F A σ = . Tuy vậy, diện tích thực của bề mặt phá hủy sẽ là ( ) sin o T A θ và lực kéo pháp tuyến của mặt sẽ là ( ) max cos T F θ ⋅ . Điều này dẫn đến kết quả là ứng suất phá hủy là Nguồn: http://www3.interscience.wiley.com/cgi-bin/fulltext/114129695/PDFSTART Biên dịch và giới thiệu bởi: Nguyễn Hoàng Việt www.luyenkim.net ( ) ( ) max sin cos T T o F A θ θ ⋅ ⋅ có sự khác biệt so với max o F A như được định nghãi trong sách giáo khoa. Như vậy, nó sẽ làm nảy sinh một số câu hỏi thú vị và có ý nghĩa. Độ bền kéo thực sự của HKVĐH là max o F A hay ( ) ( ) max sin cos T T o F A θ θ ⋅ ⋅ ? Tại sao dưới tác dụng ứng suất kéo, HKVĐH lại không bị phá hủy theo mặt ứng suất pháp tuyến lớn nhất ( 90 o T θ = ) hoặc không phá hủy theo trục ứng suất trượt lớn nhất ( 45 o T θ = )? Bản chất vật lý về độ bền vật liệu là gì? Hình. 1. Hình thái vĩ mô phá hủy trượt do kéo của mẫu HKVĐH Zr 52.5 Ni 14.6 Al 10 Cu 17.9 Ti 5 . Độ bền phá hủy kéo pháp tuyến của mẫu vào khoảng 1.58 GPa và mặt phá hủy trượt tạo thành một góc 56° so với phương ứng suất kéo . Hình. 2. (a) Minh họa vật thể bị tác động tổ hợp trạng thái ứng suất ( ) , n n σ τ và phá hủy tại hai biến độ bền riêng ( ) , o o σ τ Đối với vật liệu chịu một lực kéo F, luôn luôn có tổ hợp ứng suất ( ) , n n σ τ trên bất kì mặt phẳng nào, như minh họa ở hình 2 (a). Để có hiểu biết rõ hơn về bản chất vật lý độ bền vật liệu và trả lời các câu hỏi thú vị ở trên., chúng tôi giả sử rằng đối với vật liệu đẳng hướng chỉ có 2 giá trị độ bền riêng o σ và o τ , như minh họa ở hình 2b, o σ định nghĩa là độ bền tới hạn của vật liệu ở Mode phá hủy I; o τ là độ bền tới hạn của vật liệu ở Mode phá hủy II. Nếu mặt phẳng bất kì của vật liệu chịu tổ hợp ứng suất ( ) , n n σ τ , độ bền phá hủy sẽ được biểu diễn theo tiêu chuẩn sau: ( ) ( ) 2 2 / / 1 n o n o σ σ τ τ + = (1) Trong khi đ ó, tr ạ ng thái ứ ng su ấ t ( ) , n n σ τ trên m ặ t tr ượ t b ấ t kì tuân theo ph ươ ng trình Mohr, ngh ĩ a là ( ) ( ) ( ) 2 The Wave Nature of Matter Causes Quantization The Wave Nature of Matter Causes Quantization Bởi: OpenStaxCollege After visiting some of the applications of different aspects of atomic physics, we now return to the basic theory that was built upon Bohr’s atom Einstein once said it was important to keep asking the questions we eventually teach children not to ask Why is angular momentum quantized? You already know the answer Electrons have wavelike properties, as de Broglie later proposed They can exist only where they interfere constructively, and only certain orbits meet proper conditions, as we shall see in the next module Following Bohr’s initial work on the hydrogen atom, a decade was to pass before de Broglie proposed that matter has wave properties The wave-like properties of matter were subsequently confirmed by observations of electron interference when scattered from crystals Electrons can exist only in locations where they interfere constructively How does this affect electrons in atomic orbits? When an electron is bound to an atom, its wavelength must fit into a small space, something like a standing wave on a string (See [link].) Allowed orbits are those orbits in which an electron constructively interferes with itself Not all orbits produce constructive interference Thus only certain orbits are allowed—the orbits are quantized 1/5 The Wave Nature of Matter Causes Quantization (a) Waves on a string have a wavelength related to the length of the string, allowing them to interfere constructively (b) If we imagine the string bent into a closed circle, we get a rough idea of how electrons in circular orbits can interfere constructively (c) If the wavelength does not fit into the circumference, the electron interferes destructively; it cannot exist in such an orbit For a circular orbit, constructive interference occurs when the electron’s wavelength fits neatly into the circumference, so that wave crests always align with crests and wave troughs align with troughs, as shown in [link] (b) More precisely, when an integral multiple of the electron’s wavelength equals the circumference of the orbit, constructive interference is obtained In equation form, the condition for constructive interference and an allowed electron orbit is nλn = 2πrn(n = 1, 2, ), where λn is the electron’s wavelength and rn is the radius of that circular orbit The de Broglie wavelength is λ = h / p = h / mv, and so here λ = h / mev Substituting this into the previous condition for constructive interference produces an interesting result: nh me v = 2πrn Rearranging terms, and noting that L = mvr for a circular orbit, we obtain the quantization of angular momentum as the condition for allowed orbits: h L = mevrn = n 2π (n = 1, 2, ) This is what Bohr was forced to hypothesize as the rule for allowed orbits, as stated earlier We now realize that it is the condition for constructive interference of an electron in a circular orbit [link] illustrates this for n = and n = Waves and Quantization The wave nature of matter is responsible for the quantization of energy levels in bound systems Only those states where matter interferes constructively exist, or are “allowed.” Since there is a lowest orbit where this is possible in an atom, the electron cannot spiral into the nucleus It cannot exist closer to or inside the nucleus The wave nature of matter is what prevents matter from collapsing and gives atoms their sizes 2/5 The Wave Nature of Matter Causes Quantization The third and fourth allowed circular orbits have three and four wavelengths, respectively, in their circumferences Because of the wave character of matter, the idea of well-defined orbits gives way to a model in which there is a cloud of probability, consistent with Heisenberg’s uncertainty principle [link] shows how this applies to the ground state of hydrogen If you try to follow the electron in some well-defined orbit using a probe that has a small enough wavelength to get some details, you will instead knock the electron out of its orbit Each measurement of the electron’s position will find it to be in a definite location somewhere near the nucleus Repeated measurements reveal a cloud of probability like that in the figure, with each speck the location determined by a single measurement There is not a well-defined, circular-orbit type of distribution Nature again proves to be different on a small scale than on a macroscopic scale The ground state of a hydrogen atom has a probability cloud describing the position of its electron The probability of finding the electron is proportional to the darkness of the cloud The electron can be closer or farther than the Bohr radius, but it is very unlikely to be a great distance from the nucleus There are many examples in which the wave nature of matter causes quantization in bound systems such as the atom Whenever a particle is confined or bound to a small space, its allowed wavelengths are those which fit into that space For example, ...The Proof is in the Pudding A Look at the Changing Nature of Mathematical Proof Steven G. Krantz July 25, 2007 To Jerry Lyons, mentor and friend. Table of Contents Preface ix 0 What is a Proof and Why? 3 0.1 What is a Mathematician? . . . . . . . . . . . . . . . . . . . . 4 0.2 The Concept of Proof . . . . . . . . . . . . . . . . . . . . . . . 7 0.3 The Foundations of Logic . . . . . . . . . . . . . . . . . . . . 14 0.3.1 The Law of the Excluded Middle . . . . . . . . . . . . 16 0.3.2 Modus Ponendo Ponens and Friends . . . . . . . . . . 17 0.4 What Does a Proof Consist Of? . . . . . . . . . . . . . . . . . 21 0.5 The Purpose of Proof . . . . . . . . . . . . . . . . . . . . . . . 22 0.6 The Logical Basis for Mathematics . . . . . . . . . . . . . . . 27 0.7 The Experimental Nature of Mathematics . . . . . . . . . . . 29 0.8 The Role of Conjectures . . . . . . . . . . . . . . . . . . . . . 30 0.8.1 Applied Mathematics . . . . . . . . . . . . . . . . . . . 32 0.9 Mathematical Uncertainty . . . . . . . . . . . . . . . . . . . . 36 0.10 The Publication of Mathematics . . . . . . . . . . . . . . . . . 40 0.11 Closing Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . 42 1 The Ancients 45 1.1 Eudoxus and the Concept of Theorem . . . . . . . . . . . . . 46 1.2 Euclid the Geometer . . . . . . . . . . . . . . . . . . . . . . . 47 1.2.1 Euclid the Number Theorist . . . . . . . . . . . . . . . 51 1.3 Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2 The Middle Ages and Calculation 59 2.1 The Arabs and Algebra . . . . . . . . . . . . . . . . . . . . . . 60 2.2 The Development of Algebra . . . . . . . . . . . . . . . . . . . 60 iii iv 2.2.1 Al-Khwarizmi and the Basics of Algebra . . . . . . . . 60 2.2.2 The Life of Al-Khwarizmi . . . . . . . . . . . . . . . . 62 2.2.3 The Ideas of Al-Khwarizmi . . . . . . . . . . . . . . . . 66 2.2.4 Concluding Thoughts about the Arabs . . . . . . . . . 70 2.3 Investigations of Zero . . . . . . . . . . . . . . . . . . . . . . . 71 2.4 The Idea of Infinity . . . . . . . . . . . . . . . . . . . . . . . . 73 3 The Dawn of the Modern Age 75 3.1 Euler and the Profundity of Intuition . . . . . . . . . . . . . . 76 3.2 Dirichlet and Heuristics . . . . . . . . . . . . . . . . . . . . . . 77 3.3 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . 81 3.4 The Golden Age of the Nineteenth Century . . . . . . . . . . . 82 4 Hilbert and the Twentieth Century 85 4.1 DavidHilbert 86 4.2 Birkhoff, Wiener, and American Mathematics . . . . . . . . . 87 4.3 L. E. J. Brouwer and Proof by Contradiction . . . . . . . . . . 96 4.4 The Generalized Ham-Sandwich Theorem . . . . . . . . . . . 107 4.4.1 Classical Ham Sandwiches . . . . . . . . . . . . . . . . 107 4.4.2 Generalized Ham Sandwiches . . . . . . . . . . . . . . 109 4.5 Much Ado About Proofs by Contradiction . . . . . . . . . . . 111 4.6 Errett Bishop and Constructive Analysis . . . . . . . . . . . . 116 4.7 Nicolas Bourbaki . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.8 Perplexities and Paradoxes . . . . . . . . . . . . . . . . . . . . 129 4.8.1 Bertrand’s Paradox . . . . . . . . . . . . . . . . . . . . 130 4.8.2 The Banach-Tarski Paradox . . . . . . . . . . . . . . . 134 4.8.3 The Monty Hall Problem . . . . . . . . . . . . . . . . . 136 5 The Four-Color Theorem 141 5.1 Humble Beginnings . . . . . . . . . . . . . . . . . . . . . . . . 142 6 Computer-Generated Proofs 153 6.1 A Brief History of Computing . . . . . . i-01-LBK-861765 8/13/04 9:38 PM Page i Roine Magnusson/Stone i-01-LBK-861765 8/13/04 9:38 PM Page ii The Nature of Matter This pancake ice has formed on a river in Sweden Pancake ice forms when surface slush, arising from snow falling on water that is already at the freezing temperature, freezes The surface slush collects into rounded floating pads that collide and separate Copyright © 2005 by The McGraw-Hill Companies, Inc All rights reserved Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior written permission of the publisher The National Geographic features were designed and developed by the National Geographic Society’s Education Division Copyright © National Geographic Society.The name “National Geographic Society” and the Yellow Border Rectangle are trademarks of the Society, and their use, without prior written permission, is strictly prohibited The “Science and Society” and the “Science and History” features that appear in this book were designed and developed by TIME School Publishing, a division of TIME Magazine.TIME and the red border are trademarks of Time Inc All rights reserved Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 0-07-861765-0 Printed in the United States of America 10 027/111 09 08 07 06 05 04 Roine Magnusson/Stone i-01-LBK-861765 8/13/04 9:38 PM Page iii Authors Eric Werwa, PhD Department of Physics and Astronomy Otterbein College Westerville, OH Education Division Washington, D.C Patricia Horton Dinah Zike Mathematics and Science Teacher Summit Intermediate School Etiwanda, CA Educational Consultant Dinah-Might Activities, Inc San Antonio, TX Thomas McCarthy, PhD Science Department Chair St Edward’s School Vero Beach, FL Series Consultants CONSULTANTS READING ACTIVITY TESTERS Jack Cooper Barry Barto Nerma Coats Henderson Ennis High School Ennis, TX Special Education Teacher John F Kennedy Elementary Manistee, MI Pickerington Lakeview Jr High School Pickerington, OH SAFETY Mary Helen Mariscal-Cholka Aileen Duc, PhD William D Slider Middle School El Paso, TX Science Teacher Hendrick Middle School, Plano ISD Plano, TX Science Kit and Boreal Laboratories Linda McGaw Science Program Coordinator Advanced Placement Strategies, Inc Dallas, TX MATH Michael Hopper, DEng Manager of Aircraft Certification L-3 Communications Greenville, TX Tonawanda, NY Sandra West, PhD Department of Biology Texas State University-San Marcos San Marcos, TX Series Reviewers Sharla Adams Sandra Everhart Michelle Mazeika-Simmons IPC Teacher Allen High School Allen, TX Dauphin/Enterprise Jr High Schools Enterprise, AL Whiting Middle School Whiting, IN Anthony J DiSipio, Jr Great Bridge Middle School Chesapeake Public Schools Chesapeake, VA 8th Grade Science Octorana Middle School Atglen, PA George Gabb K ◆ iii i-01-LBK-861765 8/13/04 9:38 PM Page iv Why I need my science book? Have you ever been in class and not understood all of what was presented? Or, you understood everything in class, but at home, got stuck on how to answer a question? Maybe you just wondered when you were ever going to use this stuff? These next few pages are designed to help you understand everything your science book can be used for besides a paperweight! Before You Read ● Chapter Opener Science is occurring all around you, and the opening photo of each chapter will preview the science you will be learning about The Chapter Preview will give you an idea of what you will be learning about, and you can try the Launch Lab to help Chapter A Very Brief History of Chemistry Multiple Choice Questions Which of the following is the logical progression of elements formed in a star? a b c d e Hydrogen Helium Argon Carbon Hydrogen Helium Carbon Argon Helium Hydrogen Argon Carbon Helium Hydrogen Carbon Argon Argon Carbon Helium Hydrogen Answer: b Section 0.2 Difficulty Level: medium Why is iron the heaviest element formed in a star prior to a super nova? a The formation of iron in a star starts a cooling process of the star, ending nucleosynthesis b Iron reacts with the hydrogen in stars to cause a violet explosion which leads to a super nova c Iron is the heaviest element that is stable at high temperatures, all others are radioactive d The formation of iron in a star causes a reaction with helium that causes nucleosynthesis to end e When iron is formed in the outer layers of a star is has enough kinetic energy to leave the gravity of the star and therefore is able to remove itself from the star Answer: a Section 0.2 Difficulty Level: medium 0-1 One would expect a fairly even distribution of elements on earth; instead we see an uneven distribution of elements through the earth Which of the following best describes why we observe this? a The nebula that formed the earth had elements that were not evenly distributed b Winds on the surface of the earth have moved around the heavy and light elements into bands c The early earth liquefied, resulting in heavier elements migrating towards the core, and lighter elements towards the surface This migration to the surface was largely by lava flows, which were inconsistent d Some elements were soluble in water washed them into pockets on the surface e The magnetic properties of the core caused the metals to pool into certain areas between the poles on earth Answer: c Section 0.2 Difficulty Level: medium The relative number of atoms of each element in a particular compound a b c d e is always l:l is the same as the density ratio is the same as the weight ratio is definite and constant cannot be determined experimentally Answer: d Section 0.4 Difficulty Level: easy 0-2 Which of the following postulates from Dalton’s atomic theory are now considered incorrect? I All the atoms of a given element are identical II Matter consists of very small particles known as atoms III Atoms are indestructible and also indivisible a b c d e III only II only I only I and II I and III Answer: e Section 0.4 Difficulty Level: medium Which of the following statements is/are consistent with Dalton’s atomic theory? I The atoms in a given sample of an element not share any common properties II Matter consists of particles called atoms III In chemical reactions, atoms merely rearrange, but not disintegrate a b c d e III only II only I only II and III I and III Answer: d Section 0.4 Difficulty Level: medium 0-3 Which of the following statements is/are NOT consistent with Dalton’s atomic theory? I The atoms in a given sample of an element not share any common properties II Matter consists of tiny particles called molecular substances III In chemical reactions, atoms merely rearrange, but not disintegrate a b c d e III only II only I only II and III I and II Answer: e Section 0.4 Difficulty Level: medium Which of the following postulates from Dalton’s atomic theory is incorrectly stated? a b c d The atoms in a given sample of an element are identical Matter consists of tiny particles called atoms In chemical reactions, atoms merely rearrange, but not disintegrate In a given chemical compound, the atoms can be present in various numerical ratios e In a given chemical compound, the atoms are always present in the same fixed numerical ratio Answer: d Section 0.4 Difficulty Level: medium 0-4 Which of the following is consistent with the postulates from Dalton’s atomic theory? a The atoms in a given sample of an element are not necessarily identical b Matter consists of tiny particles called ions c In chemical reactions, atoms Chapter 2: The Components of Matter Kaolinite, a clay mineral with the formula Al4Si4O10(OH)8, is used as a filler in slickpaper for magazines and as a raw material for ceramics Analysis shows that 14.35 g of kaolinite contains 8.009 g of oxygen Calculate the mass percent of oxygen in kaolinite A) 1.792 mass % D) 34.12 mass % B) 24.80 mass % E) 55.81 mass % C) 30.81 mass % Ans: E Difficulty: M Compound has a composition of 46.7 mass % of element A and 53.3 mass % of element B A and B also form a second binary compound (compound 2) If the compositions of the two compounds are consistent with the law of multiple proportions, which of the following compositions could be that of compound 2? A) 23.4 mass % A 76.6 mass % B D) 53.3 mass % A 46.7 mass % B B) 30.4 mass % A 69.6 mass % B E) 73.3 mass % A 26.7 mass % B C) 33.3 mass % A 66.7 mass % B Ans: B Difficulty: M What are the approximate carbon:hydrogen mass ratios in methane (CH4) and ethyne (C2H2)? A) 1:4 and 1:1 D) 3:2 and 12:1 B) 3:2 and 6:1 E) 3:1 and 6:1 C) 3:1 and 12:1 Ans: C Difficulty: M J J Thomson studied cathode ray particles (electrons) and was able to measure the mass/charge ratio His results showed that A) the mass/charge ratio varied with as the cathode material was changed B) the charge was always a whole-number multiple of some minimum charge C) matter included particles much smaller than the atom D) atoms contained dense areas of positive charge E) atoms are largely empty space Ans: C Difficulty: E Who is credited with measuring the mass/charge ratio of the electron? A) Dalton B) Gay-Lussac C) Thomson D) Millikan E) Rutherford Ans: C Difficulty: E Who is credited with first measuring the charge of the electron? A) Dalton B) Gay-Lussac C) Thomson D) Millikan E) Rutherford Ans: D Difficulty: E Page 14 Chapter 2: The Components of Matter Millikan's oil-drop experiment A) established the charge on an electron B) showed that all oil drops carried the same charge C) provided support for the nuclear model of the atom D) suggested that some oil drops carried fractional numbers of electrons E) suggested the presence of a neutral particle in the atom Ans: A Difficulty: E In a Millikan oil-drop experiment, the charges on several different oil drops were as follows: –5.92; –4.44; –2.96; –8.88 The units are arbitrary What is the likely value of the electronic charge in these arbitrary units? A) –1.11 B) –1.48 C) –2.22 D) –2.96 E) –5.55 Ans: B Difficulty: M Who is credited with discovering the atomic nucleus? A) Dalton B) Gay-Lussac C) Thomson D) Millikan Ans: E Difficulty: E E) Rutherford 10 Rutherford bombarded gold foil with alpha () particles and found that a small percentage of the particles were deflected Which of the following was not accounted for by the model he proposed for the structure of atoms? A) the small size of the nucleus B) the charge on the nucleus C) the total mass of the atom D) the existence of protons E) the presence of electrons outside the nucleus Ans: C Difficulty: M 11 Which one of the following statements about atoms and subatomic particles is correct? A) Rutherford discovered the atomic nucleus by bombarding gold foil with electrons B) The proton and the neutron have identical masses C) The neutron's mass is equal to that of a proton plus an electron D) A neutral atom contains equal numbers of protons and electrons E) An atomic nucleus contains equal numbers of protons and neutrons Ans: D Difficulty: M 12 The chemical symbol for potassium is A) P B) Po C) Pt D) Pm E) K Ans: E Difficulty: E 13 Which of the following symbols does not represent an element? A) O2 B) Co C) HF D) Cs E) Xe Ans: C Difficulty: E Page 15 Chapter 2: The Components of Matter 14 When an atom is represented by the symbol ZA X , the value of A is the A) number of neutrons in the atom B) number of protons in the atom C) atomic mass of the element D) total number of electrons and neutrons in the atom E) total number of protons and neutrons in the atom Ans: E Difficulty: E 15 An .. .The Wave Nature of Matter Causes Quantization (a) Waves on a string have a wavelength related to the length of the string, allowing them to interfere constructively (b) If we imagine the. .. The Wave Nature of Matter Causes Quantization The third and fourth allowed circular orbits have three and four wavelengths, respectively, in their circumferences Because of the wave character of. .. the electron cannot spiral into the nucleus It cannot exist closer to or inside the nucleus The wave nature of matter is what prevents matter from collapsing and gives atoms their sizes 2/5 The