Energy needed to vaporize one mole of Ca (s) is 192 kJ. I have found all information that i believe are needed to solve this question .. yet i dont know how to solve the question! Answer (1 of 2): CaF2 is not soluble in water due to extremely high lattice energy.CaCl2 is fairly soluble in water as it's lattice energy is lower compared to CaF2. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. For example, Cl exists as Cl2 in its elemental state. In this simple view, appropriate number of cations and anions come together to form a solid. Dissociation : Cl 2(g) 2Cl (g) H 3 = +242.8 KJ mol-1 . The application of these new equation are, however, still quite niche and the improvements not as significant. In this bond order calculator we will show you how to find the bond order using Lewis structures, or a formula derived from the molecular orbital theory. plz tell me what are the steps to solve this question. Now the metal and nonmetal will be combined to form the ionic solid. How do you find the lattice energy of CaCl2? (1) M a L b ( s) a M b + ( g) + b X a ( g) This quantity cannot be experimentally determined directly, but it can be estimated using a Hess Law approach in the form of Born-Haber cycle. \(Z\) is the number of charges of the ions, (e.g., 1 for NaCl). For these reasons they have not been included in the present lattice energy calculator. There are however difficulties in getting reliable energetic readings. Reference for second table (theoretical and BHFC values) is. Hence, the chlorine atom will accept the electrons donated by the calcium atom. Express your answer in terms of x. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. The above discussion is valid only for the sodium chloride (also called rock salt) structure type. It is an inorganic compound of white color without any odor. A highly concentrated solution of calcium chloride on the road prevents the formation of the dust owing to its hygroscopic nature and forming a liquid layer on the surface of the road. This reaction results in an appreciable increase in temperature i.e., exothermic reaction, and hence, dissolution of calcium chloride in water has a high enthalpy of solvation. The bond energy of Cl2 is 242.6kJ/mol of CI-Cl bonds. Jenkins, H. Donald B. 3 Answers C5H5N in water > C5H5NH+ & OH- Kb = [C5H5NH+] [OH-] / [C5H5N] 1.5e-9 = [x] 1. --------------------------------------------------------------------------------------------------------------------------------------------. Your email address will not be published. Atoms can come together in many different ways, and this lattice energy calculator is concerned with the energy stored when cations and anions ionically bond as a part of a larger, uniform structure. Let us initiate this discussion with the basics of chemical bonds i.e., Ionic and Covalent Bond. Housecroft, Catherine E. and Alan G. Sharpe. Completa las oraciones con la forma correcta del presente de subjuntivo de los verbos entre parntesis.? Before we get to grips with finding the lattice energy, it's important to know the lattice energy definition as it is quite peculiar. Your email address will not be published. This definition causes the value for the lattice energy to always be positive, since this will always be an endothermic reaction. The lattice energy is the total potential energy of the crystal. Which one of the following has the greatest Lattice Energy? Now, the calcium cation and the chlorine anions are attracted by the electrostatic force of attraction and hence, there is the formation of an ionic bond between these two atoms. Rearrangement to solve for lattice energy gives the equation: Lattice energy= Heat of formation- Heat of atomization- Dissociation energy- (sum of Ionization energies)- (sum of Electron Affinities). Chemists, for various reasons, like to have exact and sometimes unintuitive definitions, but they do serve a purpose, we assure you. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Welcome to Techiescientist.com. However, it is not true in the case of the calcium atom i.e., alkaline earth metals. In one definition, the lattice energy is the energy required to break apart an ionic solid and convert its component atoms into gaseous ions. Permalink is the link containing your input data. as compared to the calcium atom. It can also be calculated from the electrostatic consideration of its crystal structure. Evaluate the energy of crystallization, Ecryst for CaO. But for simplicity, let us consider the ionic solids as a collection of positive and negative ions. Calculate the pH of a solution of 0.157 M pyridine. Alternatively, it can be defined as the energy that must be supplied to one mole of an ionic crystal in order to separate it into gaseous ions in a vacuum via an endothermic process . It is, however, still an approximation, and improvements to the repulsion term have since been made. DeltaHf (Na2O (s))=-414kJ/mol Delta H sublimation(Na)=109 kJ/mol I.E. The size of the lattice energy is connected to many other physical properties including solubility, hardness, and volatility. Which one of the following has the largest lattice energy? Seattle, Washington(WA), 98106. THANKS! Constructing the born-haber cycle and getting the relevant data like formation enthalpy of cacl2 , atomisation of ca, first and second ionisation energies of ca , atomisation of cl and then first electron affinity of cl. Step-by-step solution Step 1 of 5 The standard enthalpy of formation of an ionic solid from its constituent elements is given by Hess's Law: .. (1) In the equation, = the standard enthalpy of formation of the solid compound However, the crystalline structure allows each ion to interact with multiple oppositely charge ions, which causes a highly favorable change in the enthalpy of the system. KF MgS RbI Is there a formula or some simple way of knowing which has the greatest energy? Skill:Explain the trend of lattice energy. The energy to break 1 mole of a lattice at the standard temperature and pressure is defined as the standard lattice energy. Corrundum Al2O3 has some covalent character in the solid as well as the higher charge of the ions. Answer and Explanation: 1 Become a. This definition causes the value for the lattice energy to always be positive, since this will always be an endothermic reaction. These compounds have an additional stability due to the lattice energy of the solid structure. After the removal of an electron, the electronic configuration of the unipositive calcium ion becomes [Ar] 4s1. *This is a common error due to confusion caused by the definition of electron affinity, so be careful when doing this calculation. Therefore, option (c) is correct. 2. This is a geometrical factor, depending on the arrangement of ions in the solid. A chemical bond is an Ionic bond if the electronegativity difference of the chemical bond is greater than 2.0 and it will be a covalent bond if the electronegativity difference is less than 2.0 on the Pauling scale. Me molesta que mis padres no ______ (cuidar) su alimentacin.. 3. Cl2(g)--> 2Cl(g) dH = 242.6 kJ EA of Cl(g) = -349 (Electron affinity of Cl) The electron affinity of Cl is-348kJ/mol. Dr. Leon L. Combs. In this case, \rho is a factor representing the compressibility of the lattice, and letting this term equal 30pm30\ \text{pm}30pm is sufficient for most alkali metal halides. Ca(s) --> Ca(g) dH = +192 kJ This kind of construction is known as a Born-Haber cycle. However, this sharing may be equal or unequal depending on the electronegativity of the constituent atoms of the molecules. Electron affinity : Cl (g) + e- Cl-(g) H 3 = -355 KJ mol-1 Because there is actually some element of repulsion between the anion and cation, the hard-sphere model tends to over-estimate the lattice energy. [References) CHEMWORK The lattice energy of CaCl, is-2247 kJ/mol, and the enthalpy of hydration of one mole of gaseous Ca2+ and two moles of gaseous Cl" ions is -2293 kJ/mol. The nearest neighbors of Na+ are 6 Cl- ions at a distance 1r, 12 Na+ ions at a distance 2r, 8 Cl- at 3r, 6 Na+ at 4r, 24 Na+ at 5r, and so on. You're probably well aware of how ubiquitous ionic lattices are - you'll find them in your food, medicine, and maybe even in the walls of your house - but by learning what lattice energy is, the lattice energy formula, and the lattice energy trend, your appreciation for chemistry will surely increase. (O) = -141 kJ/mol E.A. Hf of NaCl = -411 (Enthalpy of formation). (Na)=495 kJ/mol Bond Energy (O2) = 495 kJ/mol E.A. Cheetham, A.K. Calculate the lattice energy of CaCl 2 using a Born-Haber cycle and data from Appendices F and L and Table 7.5. It is this that causes ionic solids to have such high melting and boiling points. 5 Answers There is no word like addressal. This module will introduce the idea of lattice energy, as well as one process that allows us to calculate it: the Born-Haber Cycle. They are: 1. The Kb of pyridine, C5H5N, is 1.5 x 10-9. Values of lattice energies for various solids have been given in literature, especially for some common solids. Calculate the lattice energy of Cacl2 given that the heat of sublimation of Ca=121 kJ/mol and Hf Cacl2=-795 first ionization of ca = 589.5 second ionization of ca = 1145 the electron affinity of Cl = 349 the cl2 bond energy is 242.7 I have found all information that i believe are needed to If we then add together all of the various enthalpies (if you don't remember the concept, visit our enthalpy calculator), the result must be the energy gap between the lattice and the ions. The ionic and covalent bonds can be differentiated on the basis of the electronegativity difference of the bond between atoms. The atomic mass calculator takes the number of protons and neutrons in an atom and displays the atomic mass in atomic mass units and kilograms. Still, if you would like us to add some more, please feel free to write to us . The covalent bond results when there is a sharing of electron pair between atoms of the molecule. Lattice energy is often used to estimate the strength of an ionic bond. Diaphragm _____ 3. So, regardless of if you've been asked to find the lattice energy of CaO\text{CaO}CaO for a test, or want to work out the lattice energy of NaCl\text{NaCl}NaCl to aid in dinner conversation, learning how to calculate lattice energy will aid in your understanding of the physical world. Journal of Chemical Education. However, when plugging in a value, determine whether energy is released (exothermic reaction) or absorbed (endothermic reaction) for each electron affinity. Unfortunately, some of the factors for both the Born-Land and Born-Mayer equations require either careful computation or detailed structural knowledge of the crystal, which are not always easily available to us. For calcium the first ionization energy is 589.5kJ/mol and the second ionization energy is 1146kJ/mol. The starting point for such a model is the potential energy between two gaseous ions: Two alterations are necessary to make the above equation suitable for a mole of a lattice. The chemical bond is formed by the participation of only valence electrons, electrons of the outermost shell, of the atom. This energy is known as Lattice Energy (U) and its value depends upon the strength of the Ionic bond. Hsub of Na = 108 kJ/mol (Heat of sublimation) The other definition says that lattice energy is the reverse process, meaning it is the energy released when gaseous ions bind to form an ionic solid. These additional reactions change the total energy in the system, making finding what is the lattice energy directly difficult. Ca(s) + Cl2(g) ==> CaCl2(s) dHformation = -795 kJ. As there is the formation of cations and anion, there will be the electrostatic force of attraction between them. It is a two-step process i.e., loss of one electron per step. For example, we can find the lattice energy of CaO\text{CaO}CaO using the following information: Since we can find all of these energies experimentally, this is a surefire way of answering "What is the lattice energy of CaO\text{CaO}CaO?". Ea for F = -328 E11 forCs = +375.7 E12 for Cs = +2422 Heat of sublimation for Cs =, Na+(g) + Br-(g) --> NaBr (s) Hint: use the Born-Haber cycle Note: the reference state for Br2 is Br2(l) not Br2(g) Data (in kJ/mol): enthalpy of sublimation of Na(s): 107 standard enthalpy, I have to place the following in order of decreasing magnitude of lattice energy, but I don't know how to do it. ), https://calculla.com/crystal_lattice_energy, chem.libretexts.org: lattice energy: the Born-Haber cycle, wiredchemist.com: alternative table with lattice energy values, crystallography.net: public database with crystal structures, youtube.com: video about einstein solid model (crystal), Lattice energy (measured in Born-Haber-Fajan cycle) [kJ/mol], Equivalently, lattice energy can be defined as the. Lime, CaO, is know to have the same structure as NaCl and the edge length of the unit cell for CaO is 481 pm. Calculate the lattice energy of CaCl2 from the given data Ca A ( s) + Cl A 2 A ( g) CaCl A 2 A ( s) H f H f 0 = 795 kJ mol 1 Sublimation: (g) Ca A ( s) Ca (g) H H 1 0 = + 121 kJ mol 1 Ionisation: Ca A ( g) Ca A 2 + A ( g) + 2 e A H H 2 0 = + 2422 kJ mol 1 Dissociation: Cl A 2 A ( g) 2 Cl A ( g) H H 3 0 = + 242.8 kJ mol 1 Lattice energy i s the heat of formation when one mole of crystalline ionic solid is generated from its constituent ions in their gaseous state. I also have, The enthalpy of formation of MX is Hf = 457 kJ/mol. Study with Quizlet and memorize flashcards containing terms like Beryllium forms a compound with chlorine that has the empirical formula BeCl2. However, many nonmetals will exist as polyatomic species. Calcium chloride is crystalline in nature and can be easily dissolved in water. The calcium atom forms a positive calcium ion by losing two electrons and the chlorine atom forms a negative chlorine ion by accepting one electron. and hence, the calcium atom will lose its electron easily. Combs, Leon. How Ionic Bond formation takes place in Calcium Chloride, Differentiation between Ionic and Covalent bond. The lattice energy of CaO is 3460 kJ/mol. The Kb of pyridine, C5H5N, is 1.5 x 10-9. Calculate the lattice energy of Cacl2? The value calculated for U depends on the data used. How would you determine whether it is an ionic compound?, An ionic bond is formed between a cation A+ and an anion B. Legal. Can I use this word like this: The addressal by the C.E.O. _____ 1. Check out 4 similar electrochemistry calculators , What is lattice energy? Lattice energy can be defined as the energy required to convert one mole of an ionic solid into gaseous ionic constituents. In chemistry, the lattice energy is the energy change upon formation of one mole of a crystalline ionic compound from its constituent ions, which are assumed to initially be in the gaseous state.It is a measure of the cohesive forces that bind ionic solids. Did you know that NaCl is actually table salt! Chichester: Horwood, 1999. Some require such high temperatures that they decompose before they can reach a melting and/or boiling point. Lattice Energy: The Born-Haber cycle is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. Discussion Even though this is a type of potential energy, you can't use the standard potential energy formula here. 11 Uses of Platinum Laboratory, Commercial, and Miscellaneous, CH3Br Lewis Structure, Geometry, Hybridization, and Polarity. Ionization : Ca (g) Ca 2+ (g) + 2e-H 2 = + 242.8 KJ mol-1 . Size of the ion On comparing the four given compounds we can see that, M gO and CaO have higher charges (+2 and -2 on the cations and anions respectively) than N aCl and KCl (+1 and -1 on the cations and anions respectively). Therefore, the hard-sphere equation for lattice energy is: U=\frac {N_ {\text {A}}z^+z^-e^2 M} {4\pi\varepsilon_0r_0} U = 40r0N Az +z e2M where: N_ {\text {A}} N A Avogadro's number; and M M Madelung constant. Legal. (O -) = 878 kJ/mol Use this information to calculate the lattice energy of Na2O (s). \(e\) is the charge of an electron (\(1.6022 \times 10^{-19}\; C\)). When an atom forms a chemical bond, it tends to achieve the nearest noble gas configuration. *Note: In this general equation, the electron affinity is added. The lattice energy of Calcium chloride CaCl 2 is - 2195. Table 8.2 Figure 7.9 First, he found that **in most cases was equal to 0.345pm0.345\ \text{pm}0.345pm, and so replaced it by ddd, equal to 3.451011m3.45\times10^{11}\ \text{m}3.451011m. Next, he replaced the measured distance between ions, r0r_0r0, with merely the sum of the two ionic radii, r++rr^++r^-r++r. The trick is to chart a path through the different states of the compound and its constituent elements, starting at the lattice and ending at the gaseous ions. Evaluate the energy of crystallization, Ecryst . The Born-Haber cycle allows us to understand and determine the lattice energies of ionic solids. In one definition, the lattice energy is the energy required to break apart an ionic solid and convert its component atoms into gaseous ions. Both the salts have common anion Cl-. The hydration enthalpy is the enthalpy change when 1 mole of gaseous ions dissolve in sufficient water to give an infinitely dilute solution. 2nd Ionization Energy of Ca = 1145 $ kJ $ Bond Energy of Cl2 = 242.7 $ kJ $ Electron Affinity for chlorine = 2 $ \times $ (349 $ kJ $ ) Standard Enthalpy of Formation of CaCl2 (s) = 795 $ kJ $ We can substitute these values in the equation to find out the lattice energy Find more about crystallography with our cubic cell calculator! For calcium, the first IE = 589.5 kJ mol-1, the second IE = 1146 kJ mol-1. No creo que Susana _____ (seguir) sobre los consejos de su mdico. Hess' Law states that the overall energy of a reaction may be determined by breaking down the process into several steps, then adding together the changes in energy of each step. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. My aim is to uncover unknown scientific facts and sharing my findings with everyone who has an interest in Science. Lattice Energy is Related to Crystal Structure, status page at https://status.libretexts.org. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Spanish Help The lattice energy is the total potential energy of the crystal. The Born-Haber cycle to evaluate Elattice is shown below: Ecryst = -411-(108+496+244/2)-(-349) kJ/mol Lattice energy is influenced by a number of factors: 787.3 kJ/mol is the lattice energy of NaCl. How do we get to know that calcium chloride is an Ionic compound? Using the values giving in the discussion above, the estimation is given by Equation \ref{6.13.3a}: \[ \begin{align*} E_cryst&= \dfrac{(6.022 \times 10^{23} /mol (1.6022 \times 10 ^{-19})^2 (1.747558)}{ 4\pi \, (8.854 \times 10^{-12} C^2/m ) (282 \times 10^{-12}\; m} \left( 1 - \dfrac{1}{9.1} \right) \\[4pt] &= - 766 kJ/mol \end{align*}\]. The Madelung constant depends on the structure type, and its values for several structural types are given in Table 6.13.1. Metals have very low ionization energy (I.E.) As the ionic radii of either the cation or anion increase, the lattice energies decrease. How is lattice energy estimated using Born-Haber cycle? Chichester: Horwood, 1994. The Born-Haber Cycle requires that the elements involved in the reaction are in their gaseous forms. Calcium chloride is highly soluble in water owing to its ionic nature. Coventry, West Midlands, UK: University of Warwick, 2005. An Hinglish word (Hindi/English). This energy is known as Lattice Energy (U) and its value depends upon the strength of the Ionic bond. Ladd, Mark. The lattice energy (U) of the calcium chloride is -2195 kJ/mol. The Lattice energy, \(U\), is the amount of energy required to separate a mole of the solid (s) into a gas (g) of its ions. The ground state electronic configuration of the calcium atom is [Ar] 4s2. Now, one calcium ion will combine with two chlorine ions, and hence, ionic bond formation takes place between them by releasing energy. This will cause a release of energy, which is called the lattice energy. 4. To do this, the ionization energy of the metal will be added to the value from Step 3. Lattice energy is defined as the energy released during the conversion of gaseous ions into their corresponding solid ions. The lattice energies for NaCl most often quoted in other texts is about 765 kJ/mol. Born-Haber Cycle Add the changes in enthalpy to turn one of the elements into its gaseous state, and then do the same for the other element. This quantity cannot be experimentally determined directly, but it can be estimated using a Hess Law approach in the form of Born-Haber cycle. Experts are tested by Chegg as specialists in their subject area. A is the number of anions coordinated to cation and C is the numbers of cations coordinated to anion. 82, P. 950-952. Its values are usually expressed with the units kJ/mol. CaCl2 is an ionic compound owing to the large electronegativity difference between the calcium atom and chlorine atom, which is greater than 2.0. In summary, calcium chloride is an ionic compound owing to the large electronegativity difference of the Ca-Cl bond in calcium chloride, which is greater than 2.0. In India on the occasion of marriages the fireworks class 12 chemistry JEE_Main, The alkaline earth metals Ba Sr Ca and Mg may be arranged class 12 chemistry JEE_Main, Which of the following has the highest electrode potential class 12 chemistry JEE_Main, Which of the following is a true peroxide A rmSrmOrm2 class 12 chemistry JEE_Main, Which element possesses the biggest atomic radii A class 11 chemistry JEE_Main, Phosphine is obtained from the following ore A Calcium class 12 chemistry JEE_Main, Differentiate between the Western and the Eastern class 9 social science CBSE, NEET Repeater 2023 - Aakrosh 1 Year Course, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. This constant varies from lattice structure to lattice structure, and the most common are present in the lattice energy calculator. The chemical bond is the force of attraction between the atoms of the molecule, which brings atoms together in their chemical structure. Calculate the enthalpy of solution per mole of solid CaCl2. Calculate the lattice enegry of CaCl 2 from the given data. Hence, calcium chloride is an ionic compound. It is subtracted because it is a release of energy associated with the addition of an electron. (Elements in their natural state have an energy level of zero.) The lattice energy is usually given in kilojules per mole (kJ/mol). Subtract from this the heat of formation of the ionic solid that would be formed from combining these elements in the appropriate ration. Hence, a covalent bond may be polar or nonpolar. 2*Cl(g) + 2e -->2*Cl^-(g) dH =2*-348 kJ Calcium chloride is prepared by reaction of calcium hydroxide with hydrogen chloride. { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Lattice_Energy : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Lattice_Energy:_The_Born-Haber_cycle" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Lattice_Enthalpies_and_Born_Haber_Cycles : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "The_Born-Lande\'_equation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { Inorganic_Solids : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Lattice_Basics : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Lattice_Defects : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Metal_Lattices : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Solids : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Thermodynamics_of_Lattices : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "Born-Haber cycle", "Lattice Energy", "showtoc:no", "license:ccbyncsa", "energy of crystallization", "licenseversion:40" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FInorganic_Chemistry%2FSupplemental_Modules_and_Websites_(Inorganic_Chemistry)%2FCrystal_Lattices%2FThermodynamics_of_Lattices%2FLattice_Energy, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\).
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