The intermediate carbocation is the tertiary carbocation, (rather than the primary carbocation that would be produced by addition to the =CH2 end of the double bond). Alkynes demonstrate two pi bonds, which are extremely electron rich. Unlock more options the more you use StudyPug. Alkenes that have a higher number of alkylated carbon atoms are more stable due to +R (resonance of a positive charge) effect. Because water is a reactant, increasing the concentration of water shifts the equilibrium position towards the addition product while lowering the water concentration favors the elimination reaction. When alkanes larger than ethane are halogenated, isomeric products are formed. The reducing agent in the reaction is zinc: How to name alkenes using IUPAC organic nomenclature. For example permanganate (MnO4) and osmium tetroxide (OsO4), both of which contain transition metals in high-oxidation states, can accomplish this transformation (). This reaction is not only regiospecific,but it is also stereospecific. Alkenes contain at least one carbon-carbon double bond. (Recall that entropy is associated with the number of possible arrangements of the system. Why is double bond more reactive than single? Recall that G = H TS. If the null hypothesis is never really true, is there a point to using a statistical test without a priori power analysis? The more bonds formed, the more energy released. This is the reason why Alkanes are unreactive. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Alkenes contain at least one carbon-carbon double bond. Show transcribed image text Expert Answer. What is the mechanism of reaction of this alkene? These carbon atoms are linked through a double bond. Alkenes are known as Olefins because ethylene, which is the first member in the series of alkene also known as ethene was found to yield oily products when they were made to react with chlorine and bromine. This page explains what they are and discusses their physical properties. This means alkenes are very useful for making polymers, which are very long chains of hydrocarbons made by a repeating unit. Want to create or adapt books like this? With ethene, this reaction has the equation: This is an important test for a double bond because. The chapter then focuses on one specific reactionthe addition of hydrogen halides to alkenesto raise a number of important concepts, including carbocation stability and the Hammond postulate. The sigma bond is formed by end-to-endoverlap of sp2 hybrid orbitals, and the pi bond by side-to-side overlap ofthe p orbitals. An alkane is added to one, an alkene to the other. Chapter 3: Conformations and Configurations: the consequences of the three- dimensional nature of carbon compounds. Stability: The rule for alkenes is that the more highly substituted the alkene, the more stable it will be (all other things being equal). Why did DOS-based Windows require HIMEM.SYS to boot? Which benzene is more reactive? This, the first of two chapters devoted to the chemistry of alkenes, describes how certain alkenes occur naturally, then shows the industrial importance of ethylene and propylene (the simplest members of the alkene family). Homework problems? Again, the addition is trans, but now an incoming nucleophile (H2O) will attack the carbon that is most carbocation-like, that is it is the most stabilized, as shown here [latex]\rightarrow[/latex]. the reason is alkenes contain double bond and pi electrons so addition reactions can take place. It's not a question of more bonds, but type of bonds. The result is that both of the carbons in the original double bond end up linked to the sameO atom. Alkenes and alkynes are more reactive than alkanes. Our extensive help & practice library have got you covered. Accessibility StatementFor more information contact us atinfo@libretexts.org. Organic Chemistry Practice Multiple Choice Question Set 1. They are more likely to participate in a variety of reactions, including combustion, addition, hydrogenation, and halogenation reactions. In cyclopropane, they are 60. Alkenes Alkenes are unsaturated compounds with a C=C double bond. Acidity of Terminal Alkynes: One alkyne-specific reaction involves the acidity of protons attached to sp hybridized carbons. The hydrogen then migrates to the adsorbed alkene and adds across the double bond. The change in entropy also influences the thermodynamic favorability of a reaction. Protonation of the alkene to generate the more stable carbocation. But alkenes contain single and double chemical bonds. Why are alkanes very reactive? The reason for the ordering is that tertiary radicals have a lower energy (and are thus easier to form) than secondary radicals, which are in turn easier to form than primary radicals. A typical energy diagram is shown below. Why alkenes are called olefins? You don't always break both pi bonds and a sigma bond in alkyne reactions. As one might expect (by analogy with alkene reduction), the cis hydrogenated product is formed (). The experimental values of Ea have been compared to those calculated by the bondenergy-bondorder method using a simple threemass . By contrast, there little tendency for a double or a triple bond to react with a electron rich substance i.e. Alkenes are relatively stable compounds, but are more reactive than alkanes because of the reactivity of the carbon-carbon -bond. Rotation around a double bond requires breaking the overlap of the pi bond and its subsequent reformation. Molecular Orbitals The common factor in these reagents is that they are able to add oxygen in various ways to the C=C bond. Chapter 7: Nucleophilic attack at the carbonyl carbon: When a carbon is bonded to one or more electronegative atoms, it takes on a partial positive charge and it is electrophilic. In general,underSN2 conditions the ring opening is also stereospecificthat is the nucleophile will attack the least hindered carbon (). In particular, these molecules can participate in a variety of addition reactions and can be used in polymer formation. Alkanes on the other hand are called saturated compounds. Alkanes are $sp^3$ hybridized, and hence have $25$% $s$ orbital character and $75$% p character. Alkenes are more reactive than their related alkanes due to the relative instability of the double bond. The overall reaction appears to have added the elements of water in an anti-Markovinkov direction. : http://blc.arizona.edu/courses/181Lab/MoBiByMe/Tautomers.html. These react with HBr by abstracting a hydrogen, and leaving a bromine radical. We say that alkenes decolorize bromine water. Accounting & Finance; Business, Companies and Organisation, Activity; Case Studies; Economy & Economics; Marketing and Markets; People in Business 2022 - 2023 Times Mojo - All Rights Reserved As we move into more complex organic chemistry we will see that the ability to choose and predict outcomes is a major component of organic chemistry. Alkanes are saturated hydrocarbons which are formed by single bonding between the carbon atoms. Thanks for the A2A. It is worth noting that by controlling the reactionconditions, we can choose to produce either cisor trans diols. The halogenation of propane discloses an interesting feature of these reactions. To my knowledge, the relative unreactivity of alkanes compared to other hydrocarbons such as alkenes or alkynes is due to the high amount of saturation across the carbon atoms. While the heading for this section is called anti-Markovnikov addition, this does not mean that the reaction mechanism is actually different. Both alkenes and alkynes are unsaturated, which means that they contain double or triple carbon-carbon bonds. Alkenes are relatively stable compounds, but are more reactive than alkanes because of the reactivity of the carboncarbon -bond. Alkenes are more reactive due to the presence of a carbon carbon double bond (always important to state what the double bond is between or may not get the marks!). Questions for Exam 2 ONE. Examples of such reagents are shown (). Epoxides tend to be reactive and for this reason can be useful as synthetic intermediates. Addition of Sulfuric acid to Alkenes The carbon-carbon double bond in alkenes such as ethene react with concentrated sulfuric acid. Greater the $s$ character, 'smaller' the hybrid orbital. How do you know which double bond is more reactive? The reaction begins with an electrophilic attack by the double bond onto the reactant which produces a carbocation that then undergoes nucleophilic attack. CH 3CH=CH 2 . They are more reactive than single bonds because they are more electron rich. Alkynes are usually less reactive than alkenes in electrophilic addition reactions because the electrons are held more tightly in CC bonds then in C=C bonds. The product is still a Markovnikov product (seeabove) but is often formed more cleanly, that is, without unwanted alternatives. The carbonyl carbon in aldehydes generally has more partial positive charge than in ketones due to the electron-donating nature of alkyl groups. Cis-diols: Alkenes can be oxidized to produce cis-diols using a different type of reagent that adds atoms across the double bond via a cyclic intermediate. 3.3. So, for example, we see Markovikov addition across the triple bond with HBr (), the only differencebeing that if excess HBr ispresent, tworather than onebromine atom will be added; one to each of the originally triple-bonded carbons. Alkenes are more reactive than their related alkanes due to the relative instability of the double bond. Indeed we did! Typically, ozone cleaves the double bond and the reaction is treated with a mild reducing agent such as tin (Sn)[6], leading to the production of the corresponding aldehydes or ketones (). Alkenes ($sp^2$) and alkynes ($sp$) have 33.3% and 50% $s$ character respectively. Our worksheets cover all topics from GCSE, IGCSE and A Level courses. Fill the rings to completely master that section or mouse over the icon to see more details. Alkanes contain strong carbon-carbon single bonds and strong carbon-hydrogen bonds. bbc bitesize the chemistry of depression neurotransmitters and more the chemistry of things 10 basic concepts of chemistry the chemistry blog reagent the chemistry of fireworks teaching resources rsc education . formation (although we will see many more). In the case of unsymmetrical alkenes (where the groups attached to the double-bonded carbons are not exactly the same), the most stable carbocation is produced. Chemistry Practice MCQ Chemical Reactions Carbon. How can i dentify the nucleophile and the electrophile in #H-Br# + #HO^-)hArr Br^-#+#H_2O# Can you explain the mechanism of the reaction (step-by-step) by which the alkene + HBr is Why are electrophilic addition reactions the characteristic reactions of alkenes? However, alkanes are saturated hydrocarbons consisting of single bonds only whereas alkenes are unsaturated hydrocarbons which include a carbon-carbon double bond. From the course view you can easily see what topics have what and the progress you've made on them. This is the reason that fats contain more energy than carbohydrates both of these molecules have alkane backbones, but the basic idea is the same, since fats are less oxidized and therefore higher in potential energy. Acid-catalyzed addition of water across a double bond. Even though increase in $s$ character increases bond enthalpy, but it also increases the electron withdrawing capacity; and it is the latter which predominates. These are both strong bonds so these molecules will only react in the presence of strong source of energy, strong enough to break these bonds. Cyclohexane: Cyclohexane is comparatively stronger. In E-3-bromo-2-pentene, the CH3 and CH2CH3 groups are closer to one another than they are in Z-3-bromo-2-pentene; theresult is that they have different physical and chemical properties. Compounds with more than one double bond have the '-ene' suffix changed to show which carbon atoms in the chain the double bonds are found at, and a prefix to say how many double bonds there are. After you have completed Chapter 7, you should be able to. Business Studies. These structural isomers . Both a hydroxyl group and a methyl group make the aromatic ring more reactive compared to benzene; they are activating groups. The reaction with water results in a trans diol. Legal. This reagent adds across the double bond in the direction that you would expect, that is the electrophile (Lewis acid) boron adds to the least substituted carbon, but at the same time, a hydrogen adds to the most substituted carbon from the same side of the molecule. This bond is why the alkenes are more reactive than the alkanes . The result is that epoxidesare susceptible to nucleophilic attack at a ring carbon (). Another reaction which appears to violate what we have learned about the regiochemistry of addition across double bonds is the reaction of an alkenewith HBr in the presence of light or peroxides. This means that a more reactive metal will displace a less reactive metal from its compounds; Two examples are: Reacting a metal with a metal oxide (by heating) Reacting a metal with an aqueous solution of a metal compound; For example, it is possible to reduce copper(II) oxide by heating it with zinc. Get in touch with one of our tutor experts. Alkenes are unsaturated, meaning they contain a double bond . Like a sponge saturated by water, an alkane is saturated by bonds; it cant form any more bonds, but alkenes can so it is unsaturated./li>. Alkanes generally have the Carbon-Carbon single bond and Carbon-Hydrogen single bonds. This reaction is regioselective, that is, we can predict the orientation of reactant addition across the double bond. Triple bonds are made of sideways overlapping of p orbitals. In case of hydrogenation reactions, alkynes are more reactive than alkenes. These bonds can be broken through addition reactions and the molecule can become saturated. scientists and engineers from key institutions web revise the structures and reactions of organic chemistry including alkanes . Mechanism of removal of boron moiety from the double bond. Why are tertiary hydrogens more reactive? Such a reaction can be accomplished by reacting the alkene with dilute sulfuric acid at low temperatures. This means that it is easier to accept electrons, as now the effective strength of nucleus is more. A simple alkene contains a pair of carbons linked by a double bond; this double bond consists of a sigma bond and a pi bond. Question: Alkenes are more reactive than benzene and undergo addition reactions, such as decolourizing bromine water in reaction (A) below, in which the C=C double bond is lost. Whereas in cas. The first step is addition of a proton to produce the most stable carbocationwhich is then attacked by water (the nucleophile). Aldehydes only have one e donor group while ketones have two. The best answers are voted up and rise to the top, Not the answer you're looking for? When you break a triple bond you have to break all three bonds. 7805 views As the groups attached to each carbon get more complex, such nomenclature quickly becomes confusing. Why is ammonia less acidic than terminal alkynes? Suggest the name of chemical B. effective nuclear charge is more. Alkynes are traditionally known as acetylenes, although the name acetylene also refers specifically to C2H2, known formally as ethyne using IUPAC nomenclature. Note: when a mechanism involves single electrons moving (as in a homolytic bond cleavage, or any reaction of a radical species) we use what is called afishhook arrowwith only one head, rather than the typicalarrow that denotes movement of two electrons. You can see this from their general formulas. The three bonds consist of two pi and one sigma bond.. Pi are very easy to break and sigma is difficult.. The reaction occurs via a concerted (coordinated) movement of electrons. This is where the terms saturated and unsaturated come from. Radicals are species with unpaired electrons, and, as such, are very reactive. Thus propanone is less reactive than ethanal towards nucleophilic addition reactions. Halogenoalkanes are named using standard nomenclature rules. Why Do Cross Country Runners Have Skinny Legs? Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. After dealing with the question of cis-trans isomerism in alkenes, Chapter 7 introduces the reactivity of the carbon-carbon double bond. In alkanes all the carbon atoms have single bonds between them and are thus known as saturated hydrocarbons. Why aldehyde is more reactive than ketones? In fact ALL reactions are reversible in theory (this is called the principle of microscopic reversibility, https://en.wikipedia.org/wiki/Microscopic_reversibility . This means that electrons are more closely packed. After studying this section you should be able to: recognise the nature of cis-trans isomerism in alkenes. The resulting bromine radical now reacts with the alkene double bond to produce the most stable intermediate, which is (just as in the carbocations) the tertiary. { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Addition_of_Sulfuric_acid_to_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Addition_Reactions_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Addition_Reactions_of_Dienes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Catalytic_Hydrogenation : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Catalytic_Hydrogenation_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Catalytic_Hydrogenation_of_Alkenes_II : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Diazomethane_Carbenes_and_Cyclopropane_Synthesis : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Diels-Alder_Cycloaddition" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Electrophilic_Addition_of_Halogens_to_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Electrophilic_Addition_of_Hydrogen_Halides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Electrophilic_Addition_of_Hydrogen_Halides_II : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Free_Radical_Reactions_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Hydration_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Hydroboration-Oxidation_of_Alkenes" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Oxacyclopropane_Synthesis : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Oxidation_of_Alkenes_with_Potassium_Manganate : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Ozonolysis : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Ozonolysis_of_Alkenes_and_Alkynes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Polymerization_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Reactions_of_Alkenes_with_Halogens : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Stereoselectivity_in_Addition_Reactions_to_Double_Bonds : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Vicinal_Syn_Dihydroxylation : "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]()" }, { Naming_the_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Properties_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Reactivity_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Synthesis_of_Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { Acid_Halides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Alcohols : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Aldehydes_and_Ketones : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Alkanes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Alkenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Alkyl_Halides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Alkynes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Amides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Amines : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Anhydrides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Arenes : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Aryl_Halides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Azides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Carbohydrates : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Carboxylic_Acids : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Chirality : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Conjugation : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Esters : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Ethers : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Fundamentals : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Hydrocarbons : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Lipids : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Nitriles : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Organo-phosphorus_Compounds" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Phenols : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Phenylamine_and_Diazonium_Compounds : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Polymers : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Reactions : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Spectroscopy : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Thiols_and_Sulfides : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic-guide", "showtoc:no", "license:ccbyncsa", "licenseversion:40" ], https://chem.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FOrganic_Chemistry%2FSupplemental_Modules_(Organic_Chemistry)%2FAlkenes%2FReactivity_of_Alkenes, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\).
Fort Hood Soldier Killed,
Is Illinois Emissions Testing Extended?,
"pace Funding" Lawsuit,
Why Is It Illegal To Kill Snakes In Virginia,
Articles W