2n2 electrons

The way in which the exclusion principle determines the order of hydrogenlike energy-level occupation in many-electron atoms, is by dictating a unique set of quantum numbers, n, /, mi and spin ms, for each electron in the atom. Application of this rule shows that the sub-levels with l = 0, 1, 2 can accommodate no more than 2, 6, 10 electrons respectively. In particular, no more than two electrons with ms = , can share the same value of mi. Each principal level accommodates 2n2 electrons. 

Show that there are 2n2 electronic states available to electrons having principal quantum number n. 

The principal quantum number n may have positive integer values (1, 2, 3,. ..). n is a measure of the distance of an orbital from the nucleus, and orbitals with the same value of n are said to be in the same shell. This is analogous to the Bohr model of the atom. Each shell may contain up to 2n2 electrons. 

For example, if the first quantum number is 3 the second quantum number can take values of 2, 1, or 0. Each of these values of will generate a number of possible values of mt and each of these values will be multiplied by a factor of two since the fourth quantum number can adopt values of 1/2 or -1/2. As a result there will be a total of 2n2 or 18 electrons in the third shell. This scheme thus explains why there will be a maximum total of 2, 8, 18, 32, etc., electrons in successive shells as one moves further away from the nucleus. 

Note. The maximum number of electrons that any quantum level can accommodate is seen to be given by the formula 2n2 where n is the number of the quantum level, for example n = 3 the maximum number of electrons is therefore 18. 

The magnetic moments of the two electrons, rotating in the opposite directions, cancel each other out. That is why an orbital can hold at most two electrons with opposite spins. The number of orbitals in a shell is directly proportional to n2. Since one orbital can hold no more than two electrons, the maximum number of electrons in a shell can be calculated by 2n2. 

For n = 1, there are 2n2 = 2 l2 = 2 electrons in the first shell. These electrons belong to the 1 s orbital, and this orbital is denoted by 1 s2 where the superscript 2, shows the number of the electrons in the orbital. 

The reaction of methyllithium with the electron-poor [Re-Cl(CO)2N2(PPh ))2l followed by treatment with acid gives the carbene... 

N6 - 2N2 + N4, etc. Another possible isomer of Ng has a bicyclic D2h structure consisting of two five-membered rings fused together [35]. This Da, isomer would be very stable if synthesized because of the aromatic character with 1 Ort electrons in the bicyclic rings. 

These more recent studies support the earlier concept that Ti must be reduced, at least formally, to or below the oxidation state (II) before N2 in the complex can be reduced. The formally Ti(III) complexes, [(v5-C5H5)2TiR]2N2, have only one electron per Ti atom and so have the possibility of producing diazene (N2H2) from N2 and thus, by disproportionation, some N2H4 and/or NH3. This reaction apparently does not occur unless additional reducing equivalents are added (40). Only when two electrons per Ti or Zr atom are present are substantial yields of N2H4 or NH3 formed. 

T, T, CE The expression 2n2, where n is the principal energy level number, is used to determine the maximum number of electrons that can be held in a principal energy level. Remember to square the number first, then multiply by 2. 

Problem 1.3 According to Bohr s theory of the H atom [49], the energy (E ) at n levels of H is given as = -p/2n2 (in a.u.), where p is the reduced mass of H. The same Bohr s theory is applicable to Ps, with a reduced mass pPs of half mass of the electron or about half that of pH. Therefore, the ionization energy (I.E). for the ground state (n=l) of Ps is only about half that of H. 

There are always 2n2 possible combinations of quantum numbers. We divide these into orbitals. Orbitals are maps of the probability of the electron being located at a certain region in space. They are designated by their angular momentum quantum numbers. The values of magnetic and spin quantum numbers define the electrons within an orbital. 

The number of electrons present in each shell is governed by 2 rules. Rule 1—The nth (n is the number of shell) shell cannot have more than 2n2 n electrons. 

The Mo[HB C3H(CH3)2N2 3](NO)(CO)2 is a convenient starting material for preparing these stable, 16-electron compounds, and it may be obtained from Mo(CO)6 in good yield by the reaction sequence described below. It reacts with iodine to give Mo[HB C3H(CH3)2N2 3](NO)l2, the preparation of which has been reported earlier, but an improved synthesis is now described.1 The preparation of the ethoxy-1 and ethylamido-4 derivatives of this diiodide are also described to provide examples of its substitution reactions. Related derivatives may be prepared by similar methods. 

In general, for a particular value of the principal quantum number, there cannot be more than two s, six />, ten d and fourteen/ electrons and the total possible number of electrons for a given value of n is therefore equal to 2n2. Thus the introduction of the concept of spin leads to the doubling of the number of possible electronic states (equation 1.39). The possible distribution of electrons for a hydrogen like atom is shown in Table V and it is seen that the series 2, 8, 18, 32,. . . which has arisen from the application of the Pauli principle is in agreement with the numbers of elements occurring in the periodic table of MendeleefF. The electron shells with values for the principal quantum number 1, 2, 3, 4, etc are often referred to as the K, L, Af, W, etc shells. 

Certain features of the Bohr theory are still appropriate to the wave-niechanical picture in particular, the orbits can still be grouped in shells characterized by a principal quantum number n, and the maximum permissible number of electrons in any shell is still 2n2. Not all the electrons in one shell, however, are identical, for those in a shell of principal quantum number n are distributed over n sub-shells characterized by an azimuthal or subsidiary quantum number Z, which can assume any of the values o, 1,..., (n— 1). Electrons in sub-shells with l = o, 1, 2, 3 are commonly termed s,p, /electrons, respectively, and the state of an electron in respect of its principal and subsidiary quantum numbers is usually symbolized by a figure representing n followed by a letter representing Z. Thus is is an electron in the K shell with / = 0... 

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