# engineering mechanics, Exercises for Engineering

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VECTOR ANALYSIS

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Art Ian G. Bautista, ECE, ECT

 The type of bond allows us to explain a

material’s properties.

Example: Consider carbon which may exist as

both graphite and diamond. Whereas graphite

is relatively soft and has a ‘‘greasy’’ feel to it,

diamond is the hardest known material. This

dramatic disparity in properties is directly

attributable to a type of interatomic bonding

found in graphite that does not exist in

diamond.

1. Name the two atomic models cited, and note

the differences between them.

2. Describe the important quantum-mechanical

principle that relates to electron energies.

3. (a) Schematically plot attractive, repulsive,

and net energies versus interatomic

separation for two atoms or ions.

(b) Note on this plot the equilibrium

separation and the bonding energy.

4. (a) Briefly describe ionic, covalent, metallic,

hydrogen, and van der Waals bonds.

(b) Note what materials exhibit each of these

bonding types.

 Each atom consists of a very small nucleus

composed of protons and neutrons, which is

encircled by moving electrons.

 Both electrons and protons are electrically

charged, the charge magnitude being 1.60 x

10-19 C, which is negative in sign for electrons

and positive for protons; neutrons are

electrically neutral.

 Protons and neutrons have approximately the

same mass, 1.67 x 10-27 kg,

 Each chemical element is characterized by the

number of protons in the nucleus, or the

atomic number (Z).

 The atomic mass (A) of a specific atom may

be expressed as the sum of the masses of

protons and neutrons within the nucleus.

Although the number of protons is the same

for all atoms of a given element, the number of

neutrons (N) may be variable. Thus atoms of

some elements have two or more different

atomic masses, which are called isotopes.

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Bohr Atomic Model

 Electrons are assumed to revolve around the atomic nucleus in discrete orbitals, and the position of any particular electron is more or less well defined in terms of its orbital.

 Another important quantum-mechanical principle stipulates that the energies of electrons are quantized; that is, electrons are permitted to have only specific values of energy. An electron may change energy, but in doing so it must make a quantum jump either to an allowed higher energy (with absorption of energy) or to a lower energy (with emission of energy). Often, it is convenient to think of these allowed electron energies as being associated with energy levels or states. These states do not vary continuously with energy; that is, adjacent states are separated by finite energies.

Electron States

- values of energy that are permitted for electrons.

 Bohr model was eventually found to have some significant limitations because of its inability to explain several phenomena involving electrons. A resolution was reached with a wave- mechanical model, in which the electron is considered to exhibit both wavelike and particle-like characteristics. With this model, an electron is no longer treated as a particle moving in a discrete orbital; but rather, position is considered to be the probability of an electron’s being at various locations around the nucleus. In other words, position is described by a probability distribution or electron cloud.

Comparison of the (a)

Bohr and (b) wave

mechanical atom

models in terms of

electron.

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 Using wave mechanics, every electron in an atom is characterized by four parameters called quantum numbers. The size, shape, and spatial orientation of an electron’s probability density are specified by three of these quantum numbers. Furthermore, Bohr energy levels separate into electron subshells, and quantum numbers dictate the number of states within each subshell.

1. Principal Quantum Number, n - Signifies the shell

- these shells are designated by the letters K, L, M, N, O, and so on, which correspond, respectively, to n 1, 2, 3, 4, 5, . . . ,

- related to the distance of an electron from the nucleus, or its position.

2. Second Quantum Number, ℓ

- signifies the subshell

- denoted by a lowercase letter—an s, p, d, or f;

- it is related to the shape of the electron subshell.

- the number of these subshells is restricted by the magnitude of n.

3. Third Quantum Number, mℓ - determines the number of energy states for each subshell

4. Fourth Quantum Number, ms - spin moment of an electron

- two values are possible (+ 1

2 , −

1

2 )

one for each of the spin orientations.

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 This principle stipulates that each electron state can

hold no more than two electrons, which must have

opposite spins. Thus, s, p, d, and f subshells may

each accommodate, respectively, a total of 2, 6, 10,

and 14 electrons;

 Not all possible states in an atom are filled with

electrons. For most atoms, the electrons fill up the

lowest possible energy states in the electron shells

and subshells, two electrons (having opposite spins)

per state.

Schematic representation of the filled energy

states for a sodium atom.

Ground State

- When all the electrons occupy the lowest possible energies in accord with the foregoing restrictions.

 Structure of an atom

 represents the manner in which energy states are occupied.

 In the conventional notation, the number of electrons in each subshell is indicated by a superscript after the shell–subshell designation.

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Valence electrons

 electrons that occupy the outermost shell.

Stable Electron Configurations

– atoms whose states within the outermost or valence electron shell are completely filled. Normally this corresponds to the occupation of just the s and p states for the outermost shell by a total of eight electrons.

 Some atoms of the elements that have unfilled valence shells assume stable electron configurations by gaining or losing electrons to form charged ions, or by sharing electrons with other atoms. This is the basis for some chemical reactions, and also for atomic bonding in solids.

 elements have been classified according to electron configuration

 elements are situated, with increasing atomic number in seven horizontal rows called periods.

 The arrangement is such that all elements arrayed in a given column or group have similar valence electron structures, as well as chemical and physical properties. These properties change gradually, moving horizontally across each period and vertically down each column.

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Group 0

 inert gases, which have filled electron shells and stable electron configurations.

Inert Gases

Group VIIA

 elements are one electron deficient from having stable structures.

 termed as halogens.

 F, Cl, Br, I, and At

Halogens

Group VIA

 elements are two electron deficient from having stable structures.

Group IA

 having one electron in excess of stable structures.

 alkali

Group IIA

 having two electron in excess of stable structures.

 alkaline earth metals

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alkali and the

alkaline earth

metals Groups IIIB through IIB

which have partially filled d electron states and in some cases one or two electrons in the next higher energy shell.

 are termed the transition metals

transition

metals

Groups IIIA, IVA, and VA

 display characteristics that are intermediate between the metals and nonmetals by virtue of their valence electron structures.

 B, Si, Ge, As, etc.

Electropositive

 elements under metal classification.

 indicating that they are capable of giving up their few valence electrons to become positively charged ions.

Electronegative

 they readily accept electrons to form negatively charged ions, or sometimes they share electrons with other atoms.

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