What is the expanded form of the number 4,582,903?

4,000,000 + 500,000 + 80,000 + 2,000 + 900 + 3

4,000,000 + 500,000 + 80,000 + 20,000 + 3

4,500,000 + 80,000 + 2,000 + 900 + 3

4,500,000 + 500,000 + 80,000 + 2,000 + 3

Which of the following represents the number 704,062 written in word form?

seven hundred four thousand, sixty-two

seven hundred four thousand, six hundred two

seven hundred forty thousand, sixty-two

seven hundred forty thousand, six hundred

Which of the following represents the number 40,603 in word form?

forty thousand, six hundred three

four thousand, six hundred three

four hundred thousand, sixty-three

4.NS.1 | Quizalize

Understanding Quantum Theory of Electrons in Atoms The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is a precursor to chemical bonding. As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels. The energy levels are labeled with an n value, where n = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of n. This number, n, is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure 9.4.1). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has. This quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom. The following equation summarizes these relationships and is based on the hydrogen atom: The values nf and ni are the final and initial energy states of the electron. The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital, which is distinct from an orbit, is a general region in an atom within which an electron is most probable to reside. The quantum mechanical model specifies the probability of finding an electron in the three-dimensional space around the nucleus and is based on solutions of the Schrödinger equation. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in a multi-electron atoms and ions are located. Another quantum number is l, the angular momentum quantum number. It is an integer that defines the shape of the orbital, and takes on the values, l = 0, 1, 2, …, n – 1. This means that an orbital with n = 1 can have only one value of l, l = 0, whereas n = 2 permits l = 0 and l = 1, and so on. The principal quantum number defines the general size and energy of the orbital. The l value specifies the shape of the orbital. Orbitals with the same value of l form a subshell. In addition, the greater the angular momentum quantum number, the greater is the angular momentum of an electron at this orbital. Orbitals with l = 0 are called s orbitals (or the s subshells). The value l = 1 corresponds to the p orbitals. For a given n, p orbitals constitute a p subshell (e.g., 3p if n = 3). The orbitals with l = 2 are called the d orbitals, followed by the f-, g-, and h-orbitals for l = 3, 4, 5, and there are higher values we will not consider. There are certain distances from the nucleus at which the probability density of finding an electron located at a particular orbital is zero. In other words, the value of the wavefunction ψ is zero at this distance for this orbital. Such a value of radius r is called a radial node. The number of radial nodes in an orbital is n – l – 1. Consider the examples in Figure 9.4.2. The orbitals depicted are of the s type, thus l = 0 for all of them. It can be seen from the graphs of the probability densities that there are 1 – 0 – 1 = 0 places where the density is zero (nodes) for 1s (n = 1), 2 – 0 – 1 = 1 node for 2s, and 3 – 0 – 1 = 2 nodes for the 3s orbitals. The s subshell electron density distribution is spherical and the p subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found. Principal quantum number (n) & Orbital angular momentum (l): The Orbital Subshell: https://youtu.be/ms7WR149fAY If an electron has an angular momentum (l ≠ 0), then this vector can point in different directions. In addition, the z component of the angular momentum can have more than one value. This means that if a magnetic field is applied in the z direction, orbitals with different values of the z component of the angular momentum will have different energies resulting from interacting with the field. The magnetic quantum number, called ml, specifies the z component of the angular momentum for a particular orbital. For example, for an s orbital, l = 0, and the only value of ml is zero. For p orbitals, l = 1, and ml can be equal to –1, 0, or +1. Generally speaking, ml can be equal to –l, –(l – 1), …, –1, 0, +1, …, (l – 1), l. The total number of possible orbitals with the same value of l (a subshell) is 2l + 1. Thus, there is one s-orbital for ml = 0, there are three p-orbitals for ml = 1, five d-orbitals for ml = 2, seven f-orbitals for ml = 3, and so forth. The principal quantum number defines the general value of the electronic energy. The angular momentum quantum number determines the shape of the orbital. And the magnetic quantum number specifies orientation of the orbital in space, as can be seen in Figure 9.4.3. Figure 9.4.4 illustrates the energy levels for various orbitals. The number before the orbital name (such as 2s, 3p, and so forth) stands for the principal quantum number, n. The letter in the orbital name defines the subshell with a specific angular momentum quantum number l = 0 for s orbitals, 1 for p orbitals, 2 for d orbitals. Finally, there are more than one possible orbitals for l ≥ 1, each corresponding to a specific value of ml. In the case of a hydrogen atom or a one-electron ion (such as He+, Li2+, and so on), energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n, are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and orbitals that belong to different subshells have different energies. Orbitals within the same subshell (for example ns, np, nd, nf, such as 2p, 3s) are still degenerate and have the same energy. While the three quantum numbers discussed in the previous paragraphs work well for describing electron orbitals, some experiments showed that they were not sufficient to explain all observed results. It was demonstrated in the 1920s that when hydrogen-line spectra are examined at extremely high resolution, some lines are actually not single peaks but, rather, pairs of closely spaced lines. This is the so-called fine structure of the spectrum, and it implies that there are additional small differences in energies of electrons even when they are located in the same orbital. These observations led Samuel Goudsmit and George Uhlenbeck to propose that electrons have a fourth quantum number. They called this the spin quantum number, or ms. The other three quantum numbers, n, l, and ml, are properties of specific atomic orbitals that also define in what part of the space an electron is most likely to be located. Orbitals are a result of solving the Schrödinger equation for electrons in atoms. The electron spin is a different kind of property. It is a completely quantum phenomenon with no analogues in the classical realm. In addition, it cannot be derived from solving the Schrödinger equation and is not related to the normal spatial coordinates (such as the Cartesian x, y, and z). Electron spin describes an intrinsic electron “rotation” or “spinning.” Each electron acts as a tiny magnet or a tiny rotating object with an angular momentum, even though this rotation cannot be observed in terms of the spatial coordinates. The magnitude of the overall electron spin can only have one value, and an electron can only “spin” in one of two quantized states. One is termed the α state, with the z component of the spin being in the positive direction of the z axis. This corresponds to the spin quantum number ms=12. The other is called the β state, with the z component of the spin being negative and ms=−12. Any electron, regardless of the atomic orbital it is located in, can only have one of those two values of the spin quantum number. The energies of electrons having ms=−12 and ms=12 are different if an external magnetic field is applied. Figure 9.4.5 illustrates this phenomenon. An electron acts like a tiny magnet. Its moment is directed up (in the positive direction of the z axis) for the 12 spin quantum number and down (in the negative z direction) for the spin quantum number of −12. A magnet has a lower energy if its magnetic moment is aligned with the external magnetic field (the left electron) and a higher energy for the magnetic moment being opposite to the applied field. This is why an electron with ms=12 has a slightly lower energy in an external field in the positive z direction, and an electron with ms=−12 has a slightly higher energy in the same field. This is true even for an electron occupying the same orbital in an atom. A spectral line corresponding to a transition for electrons from the same orbital but with different spin quantum numbers has two possible values of energy; thus, the line in the spectrum will show a fine structure splitting. The Pauli Exclusion Principle An electron in an atom is completely described by four quantum numbers: n, l, ml, and ms. The first three quantum numbers define the orbital and the fourth quantum number describes the intrinsic electron property called spin. An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same set of all the four quantum numbers. What this means is that electrons can share the same orbital (the same set of the quantum numbers n, l, and ml), but only if their spin quantum numbers ms have different values. Since the spin quantum number can only have two values (±12), no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons. The properties and meaning of the quantum numbers of electrons in atoms are briefly

Eff..rs of ott.-PoFllat i What woLrld hoppen ro our colnrry i, it is ovetsp.pulored? When our counrry is ov€.-populdted, re @ €xp€ri.nce rh€ foll.wirg: Food is our bdsic h@d. Wh€n th€.Cs an ih.re.se ir populdtion it neans thar hore ,@d is iealed. It rheds ho .naJgh food, rrtrple irll srruggle wirh eddr oth€r in ordeLro €!'r- As o l!fllr, lhde rill be o f@d -- , ond ou, now]nert of on ihdiyiduol fron d c..tair - the move$eni o, on individudl our of o cerrain pla.e which help r€duce ihe populotion of th6t fr Arcih€. b.sic ned is w.ra. Wde. shorroge ocu.s when there is on ircreare of hu,nber of p@ple ro be $pptied. rn owr-popur.t d ore.s, woler is rdior€d, Ir rEB rhoi supplies like ti,tWSS ond ,IWSI can'i $pply enoish worer. Do you hdve enough supply of sai.. in your oreo? Aside f.om food alld worer, shelier is olso ohe o, our inportant heeds. As the populoiion ihcre.!e!, building n.w hoLr!€s or rhelt€r is limit.i. To find solulion to this prcbl€n, some goverihent og.ncies dnd orhs non{ov€Ihrehl offi.iofs (N6O) .onvefied sot@ ti.elields, du,np site. dnd nountcirlr inlo flbdivisions dnd relidentiols. Sut whot uould be ths effect o{ coMrtiig .i@fields to .6id€nri6l uits in our food supply? z , 2 Z Z :'", becouse there ore no enough space for prcpex garbage dkposol. ^s o r€sulr, sore peoPle lend to ihrow'their gorbdge onywh.f€. oorbdge baones brc{niry ond rursing ground of iEecrs and onidols ihot @se horm ro pe.ple. Dec.yiry garboge olso produces r,hpleaiant odor ard ehen burn if pmduces pois.nour qds @lled nelhohe As ihe populdtion incr€a3*, the 9d6.9e dso incraes. nris is T't ,,8 T H Wha you de living in on oa-populdi.d pla@, you moy oqaiae halrh prcblerns. Ir is be@@. the woi.r srpply is limit.d ihct will l..d you to poor hygi.ni. hobirs. In plo.4 like rhis, the surrouhdiigs naybe uniidy. o focrorthoi @uld oko cfFe.t your h4l'th. The common oilments rhot yd @uld oc$rire in ovesfDpllar€d ploces ore bEnchil is, o5l hnq. diqrrha and rube.culosis. 7,\ ,\\ \1" 6. Lnck of Herlrh sarvice llosi Pelple in 6n oM-populci€d 6ra 90 ro rubli. heilrh @trtas ond governhent hospirols be6u.e ii prcvides fr@ @Eulrorion oid los @sr rEdicdrions. A3 a ..suli, lh€s€ gow.nnenr dg€rciB b.@ne itud.4$re in mcetiig ihe n eds b..ou!€ df ihsrffici€nr funds. Lock of medicol personnel ,o odmaiisi€I is also s problen in mosr hosptols ev€n rhere or. od.audtc supply of hedicire!. 7_ Do you how wlry rhe crim€ roi€ hexs ih becdur€ fiDre pe.ple o.e fnJrrct€d d@ ro sLffici€.i naE io supp.rr their forniliG. ouf country inclY{ses? If is uh.mploym€ni dnd hdve no arinet .re u$dv gr€{rer ia dn dq-popltdled ra whq. tl, , a, v, tlr I E. Air ard Wat€r Pollutioh How dir be.o'nes pollut€d? I11€ dir b@'n€s p.llurn be.4ne of rhe hormfolgoees thot ser. produ.4 by the fdciori€s and vehicles. Itete {octories ond whi.l6 @ fuel ro run nochiB ond .JBin6. In ,h€ prc.ess, they give our Cdrboh Dioxide ond other ho.6ful gars.r such 6 Nittugei Oxide, Corbon l oioxide dnd Le.d iiio the oir. Do you know whot .ontdbute io ihe incr€asing number of whides qnd foctories? It is ihe inc.6e o, populdtion. As whdt I hove dis.!sse!, wirh a lihired sra.e 9@bd9e disposalie one of the problens thot .o!ld ise i, dh o!er-pop!,.t€d ploce. exn,jple ot thie orc rhos€ pelpl€ livi,rg oh the raverside teid 'ro ,hrou, lheir gEr&ge Hde you seen 'th. P6si9 river or the Tulyahan river? Did you {ind it Whdr do you think i! ihe eff€.t of ihis ih the.re4iures sho lives ih Ahothd f6do.s thal could.on rlbule to wdtd pollutioh dre oil s?ills, gorbqg€ fro,n boa, or ships ahd som€ ihdust.iol wosre. 9. Ite l@96f p4.enroge group. Individuols who orc this grclp. of olr popllarioh is compos.n of the working @pobla of s'rpporting ,heir fomilies nok !-up Though rhas group hol& the lojgeei percenroge of d. populaiion, rhis olso becomer one o{ oveFpopulored probl€]ns b€4use there ore rc jobs awildble fo. oll of iha10. Erergy Shortdge ltere will be on energy shortdge iJ ihe populdtion incre63"l be.dise rhe d.,nand i. €le.iriciry is high. Why is thai wh.n th. PoPqldion inclE.g, rhe d4ord in el4tricity is high? Ir B be.ouse there $,ould be 8to.e hdsat dnd blildirys to lighr ond nore el?riric oPPliohces ro run. rt.6rcznho!3.Ef+ed Whor is rhe grernho@ eff€.r? In whoi say il c.uld offect c2 6re.hhG. effed is rhe wdrniltg of rhe drltlosphee. lvhen the 5un worft rhe.nrrh s1jrf.@, sone of rhe h@r go€J bo.k ro rhe ornos?herc. Air an the dtnDsphere which is C@boi Dioxid. ,rops ihe heot 6hd it mok6 the a.th very worm. As ihe populdtion coniinuou!|,l gtol4 , the gt@rl$use etfe.t b@res no.e visible. Ir is becaosu ,hera ore mo.e focrories snd whicl.s iha, produce wdst€s ond fuma5 which cduses more C{.bo. Diodde ir the ormosphere. As a rcsutt, ,herc eiould be nore h4, ,rop in the ornosphere uhich osk6 th. @ih nuch wornerIf this will hoppen continuously, ,h€ fish ih th€ ocah *ill di€, ricerields/f@mlands will dry too due to lh€ wcm clitnole 12. Destruction of rhe Ozone Loyer A5 whot you hove l@med lrheh you de in v5-6, rhot the qzore ldver is 'the proiecrive loy€. of the olnosPhd€. ft protects us {rom the homful effects of ultrdviolei rays of the su. Do you khow ,hot our Ozore lol€t q4. dQ4tt\!ci.d? Il olreadY hod holes lhai dllow the ulrroviolet rdys to .4dt ihe @rrh. How do6 this hdpPei? Does th. in rc$e of poPuldioh h@€ sonething 'to do tr,lh ir? Yes, rhe I6i grov/irts PoPuldiion .odribuied o lot be@use 6 th' populotion incre3es, rhe u5e of refrigerd'tors, d€rosol lProvs 6nd pl4srics 6bo ihcre&s6. The sid producls coiiojn chemicol called Chlorofluorocdrbons (CFCS) which is mix wafh ihe dir in ihe ormosPher€. As o resulr. ihe hcrmfirl chernicol r€oches the Prolectiw ldver dnd lhrowh. hole in {hid ult@iolzi cahders aid cai4.ct3 ,F.*Y.iis hi!586$q€9.7,- Ho$ doas dcid rdin form? Is cid roii hdmful ro rEn? In the prcvious dis.ussions, yodt€ t.on€d rhd more vel .1e3 dnd fdctori€s or€ necded fo het the iii:.e.siry number o{ P@Pla. Lefs now fihd af hd f@tot.i€s dnd vehicle! .ontribure in the forrEtion of ocid When foctories 6nd whi.ler give off woste gd..3 ,hot will ,nix on lhe noisture i. rhe oir, it will ihen Produ.e sulPhu.i. ocld dnd Nitri. o.id. 'Ihe clol,Jd folb will ,h€h obsorb rhese ccids ond ehei ihe clold f.lls os .oin, ih. ccid is ahady Pdrr of itU/ha d.id ftin falls oh lok"!, ,46 or ocan ih€ fish sill die d.d if h fdlls oh fopnlonds,lhe pldni. together oith the soil B desrroyed. When you inhole dir with Niiric acid, your blood will los. irs @pobilily io fonspori Oxyg€h to your diff€.ai bodY Po.i3. ScieniisB include other rorns oJ dcidic pr€cipiigrion. Thes€ drc nisi, Do you krcw ihot Nuclerr power slotionr Use .adiodctive ,ndie.ials in producirE fuels, yet, rhey do and those .odioactiw rndlqlotE gi\e otf radio'ting en.rgy thoi is harmrul 'to livirq thilEs. wlren rodiotion enlert ihe body ot living things it {ill srq rhere for o lorg ,eriod of ri'ne. Exonple fhe rodiqtion vG srilled to the c.m. Then rhe @rn will be aie by rhe chicken, the .odiotion o the c.rn 'rill also 'tronsf€r to the chi.k€n. Wha on individuol als ihe nat of the chickeh sith mdiarion, helshe rill .ko oblorb ihe rodi@.tirc mtaid that will destrcy hB/her .€lls ond ruket hnn/hd si.r. Over-populoiion .on leld to food shoridg€, wdter shorroqe, housiB probl€ms, qdrbog€ probl€rs, lock of halrh sdi.e. tisa ol clit@ rote, oir ond woi€r pollution, uhanpl6ynat, eiergy 5horr69e, grenhoq3€ efreci, desrruction o( th. ozo@ lat/e?, rci.l roi. olld e.l€d. watta

NS.4

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6.NS.B.4

6.NS.B.4 Finding Common Factors and Multiples

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