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second lecture review of quantum mechanics

الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 4
أستاذ المادة عباس عبد علي دريع الصالحي       21/10/2019 18:10:25
university of babylon undergraduate studies
college of sciences
department of chemistry
course no. chsc. 424 physical chemistry
fourth-year - semester 1
credit hour: 3 hrs.

lectures of quantum mechanics
the scholar year 2019-2020
prof. dr abbas a-ali draea
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lecture no. two:
review of classical quantum mechanics
classical quantum mechanics.
experimental foundations of quantum mechanics.
ritz rule.
bohr theory of a hydrogen atom.
failed of bohr theory for a multi electronic system.

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2-1-classical quantum mechanics:

two fundamental concepts are dominated classical by quantum mechanics they are:
the concept of a particle, a discrete entity with definite position and momentum which moves in accordance with newton s laws of motion.
the concept of an electromagnetic wave.
an extended quantum mechanics entity with a presence at every point in space that is provided by electric and magnetic fields which change in accordance with maxwell s laws of electromagnetism. the classical world picture is neat and tidy ((أنيقة ومرتبه:
the laws of particle motion account for the material world that around us and the laws of electromagnetic fields account for the light waves which illuminate this world. this classical picture began to crumbleالانهيار in the 1900s when max planck published a theory of black-body radiation (a theory of thermal radiation in equilibrium with a perfectly absorbing body). planck provided an explanation of the observed properties of black-body radiation by assuming that atoms emit and absorb discrete quanta of radiation with energy. at the end of the 19th century, some men made as they called classical mechanics by their experiments that’s discovered new particles provided with negative charges.

2-2- experimental foundation of quantum mechanics:
experiments have been discovered by some scientist that’s proved the foundation of classical quantum mechanics.
2-2-1 – experiment of electric discharged tubes.
j. j. thomson 1897 was found out that if they constructed a glass tube with wires inserted in both ends, and pumped out as much of the air as they could, an electric charge passed across the tube from the wires would create a fluorescent glow. this cathode ray also became known as an ‘electron gun’.
thomson’s first cathode ray experiment
his first experiment was to build a cathode ray tube with a metal cylinder on the end. this cylinder had two slits in it, leading to electrometers, which could measure small electric charges. he found that by applying a magnetic field across the tube, there was no activity recorded by the electrometers and so the charge had been bent away by the magnet. this proved that the negative charge. that’s mean the ray carried a negative charge but were not sure whether the charge could be separated from the ray. they debated whether the rays were waves or particles, as they seemed to exhibit some of the properties of both.


figure 1. first experiment of the electric discharge tube.

he developed the second stage of the experiment, to prove that the rays carried a negative charge. to prove this hypothesis, he attempted to deflect them with an electric field. earlier experiments had failed to back this up, but thomson thought that the vacuum in the tube was not good enough, and he found ways to improve greatly the quality. for this, he constructed a slightly different cathode ray tube, with a fluorescent coating at one end and a near-perfect vacuum. halfway down the tube were two electric plates, producing a positive anode and a negative cathode, which he hoped would deflect the rays. as he expected, the rays were deflected by the electric charge, proving beyond doubt that the rays were made up of charged particles carrying a negative charge. this result was a major discovery in itself, but thomson resolved to understand more about the nature of these particles.
thomson s second experiment
he decided to try to work out the nature of the particles. they were too small to have their mass or charge calculated directly, but he attempted to deduce this from how much the particles were bent by electrical currents, of varying strengths.
thomson found out that the charge to mass ratio was so large that the particles either carried a huge charge or were a thousand times smaller than a hydrogen ion. he decided upon the latter and came up with the idea that the cathode rays were made of particles that emanated from within the atoms themselves.



figure 2. second experiment of the electric discharge tube.
the discharge tube is also called "croock tube". it is made of a glass tube which consists of two metallic plates. one plate is connected to a positive terminal of high voltage power supply and the other to the negative terminal. the plate connected to the positive terminal is called "anode" the other connected to a negative terminal is called "cathode". the tube is filled with any gas.
in the discharge tube experiment, at low pressure and at very high voltage, an electric current is passed. due to passage of electric current, a stream of rays is passed in the tube originating from the cathode. these rays are called "cathode rays".
properties of cathode rays
these rays originate from the cathode.
cathode rays travel in a straight line.
cathode rays carry a negative charge.
cathode rays are deflected by the electric field.
cathode rays are deflected by the magnetic field.
these rays consist of material particles.
the ratio e / m of these particles 1.76 x 108 coulomb / gram.
cathode rays consist of "electron".
the rays, upon striking glass or certain other materials, cause them to glow.

2-2- r. a. millikan experiment:
r. a. millikan is determined the charge of the electron by using the droping of oil. he obtained more precise results in 1910 with his famous oil-droping experiment in which he replaced water (which tended to evaporate too quickly) with oil.
the experiment entailed observing tiny charged dropinglets of oil between two horizontal metal electrodes. first, with zero applied electric field, the terminal velocity of a dropinglet was measured. at terminal velocity, the drag force equals the gravitational force, and these depend on the radius in different ways, so that the radius of the dropinglet, and therefore the mass and gravitational force, could be determined (using the known density of the oil). then an adjustable voltage was applied between the plates to induce an electric field, and the voltage was adjusted until the dropings were suspended in mechanical equilibrium, indicating that the electrical force and the gravitational force were balanced. now using the known electric field, millikan and fletcher could determine the charge on the oil dropinglet. by repeating the experiment for many dropinglets, they confirmed that the charges were all small integer multiples of a certain base value, which was found to be 1.5924×10?19 c, less than a 1% difference from the currently accepted value of 1.602176487×10?19 c. they proposed that this was the (negative of the) charge of a single electron.
initially, the oil dropings are allowed to fall between the plates with the electric field turned off. they very quickly reach a terminal velocity because of friction with the air in the chamber. the field is then turned on and, if it is large enough, some of the dropings (the charged ones) will start to rise. (this is because the upwards electric force fe is greater for them than the downwards gravitational force fg. a likely looking droping is selected and kept in the middle of the field of view by alternately switching off the voltage until all the other dropings have fallen. the experiment is then continued with this one droping. the droping is allowed to fall and its terminal velocity v1 in the absence of an electric field is calculated. the drag force acting on the droping can then be worked out using stokes law:


figure 2-3 a. diagram of oil droping experiment.


figure 2-3 b. apparatus of oil droping experiment.

where v1 is the terminal velocity (i.e. velocity in the absence of an electric field) of the falling droping, ? is the viscosity of the air, and r is the radius of the droping. the weight w is the volume d multiplied by the density ? and the acceleration due to gravity g. however, what is needed is the apparent weight. the apparent weight in air is the true weight minus the upthrust (which equals the weight of air displaced by the oil droping). for a perfectly spherical dropinglet the apparent weight can be written as:
w=4?(3.?.r3.d)
the mean value of oil droping fall between two plates is u=2gr2d/qn
at terminal velocity, the oil droping is not accelerating. therefore, the total force acting on it must be zero and the two forces f and w must cancel one another out (that is, f = w). this implies
once r is calculated, w can easily be worked out.
now the field is turned back on, and the electric force on the droping is where q is the charge on the oil droping and e is the electric field between the plates. for parallel plates where v is the potential difference and d is the distance between the plates. one conceivable way to work out q would be to adjust v until the oil droping remained steady. then we could equate fe with w. also, determining fe proves difficult because the mass of the oil droping w is difficult to determine without reverting to the use of stokes law. by dividing the free fall of oil droping into the resistance fall of oil droping by the electric field, the static charge of the can is found as in the following:
?1/?2= wg/(ee?wg)
a more practical approach is to turn v up slightly so that the oil droping rises with a new terminal velocity v2. then millikan repeated the experiment no. of times, each time varying the strength of x-rays ionizing the air. as a result no. of electrons attaching to the oil droping varied. then they obtained various values for q and is found to be a multiple of 1.6 x 10-19c.
by using avogadro s number to find the electronic charge for one particle
e charge =f/?n_a?^ =(2.8929*?10?^14 esu)/(6.0232*?10?^23 particle/mole)=4.8029*?10?^(-10)
at last electronic charge equal to 4.8029*10-10esu.
combination of the thomson experiment and millikan experiment the mass of electron been calculated as in the following:-
me=(4.8029*?10?^(-10) esu)/(5.2731*?10?^17 esu.?gm?^(-1) )=9.1083*?10?^(-31) kg

2-3- w. ritz rule:
the empirical rule, formulated by w. ritz in 1905, that sums and differences of the frequencies of spectral lines often equal other observed frequencies. the rule is an immediate consequence of the quantum-mechanical formula h? = ei ? ef relating the energy h? of an emitted photon to the initial energy ei and final energy ef of the radiating system h is planck s constant and ? is the frequency of the emitted light. for example, the illustration shows the photon energies h?3-2, h?3-1,
h?3-0 associated with transitions from level 3 to lower-lying levels, and so on. level 3 may radiate directly to the ground state 0, emitting ?3-0, or it may first make a transition to level 2, which subsequently radiates to the ground state, and so on. since the total energy emitted in these two alternative means of making transitions from 3 to 0 is exactly the same, namely e3 ? e0, it follows that h?3-0 = h?3-2 + h?2-0. similarly, h?3-0 = h?3-2 + h?2-1 + h?1-0, and so forth. the statement of w. ritz rule is:
the frequencies for a spectral line to the atomic spectral serious is represented by differences between two quantity the first is constant and the second is variable through the serious.
e=h?=r_h (1/(n^2 1)-1/(n^2 i)), since ni larger than n1
rh is rydberg constant of hydrogen atom equal to 109737.31cm-1
according to this rule, any species or atom cannot be absorbed or emitted energy as radiation in the continuous state but it occurs indefinite quantity that’s called quanta.
2-4-bohr theory of hydrogen atom:
niels bohr introduced the atomic hydrogen model in 1913. he has described it as a positively charged nucleus, comprised of protons and neutrons, surrounded by a negatively charged electronic cloud. in the model, electrons orbit is rounded the nucleus in atomic shells. the atom is held together by electrostatic forces between the positive nucleus and negative surroundings. the bohr model is used to describe the structure of hydrogen energy levels. the image below represents shell structure, where each shell is associated with principal quantum number n. the energy levels presented correspond with each shell. the amount of energy in each level is reported in ev, and the maximum energy is the ionization energy of 13.598ev.

figure 2-4: orbital shells of a hydrogen atom.
the movement of electrons between these energy levels produces a spectrum. the ballmer equation is used to describe the four different wavelengths of hydrogen which are present in the visible light spectrum. these wavelengths are at 656, 486, 434, and 410nm. these correspond to the emission of photons as an electron in excited state transitions down to energy level n=2. the rydberg formula, below, generalizes the ballmer series for all energy level transitions. to get the ballmer lines, the rydberg formula is used with nf of 2.
rydberg formula:
the rydberg formula explains the different energies of transition that occur between energy levels. when an electron moves from a higher energy level to a lower one, a photon is emitted. the hydrogen atom can emit different wavelengths of light depending on the initial and final energy levels of the transition. it emits a photon with energy equal to the difference of the square of the final (nf) and initial (ni) energy levels. energy =r_h (1/(n^2 f)-1/(n^2 i)) . the energy of a photon is equal to planck’s constant, h=6.626*10-34 m2.kg/s, times the speed of light in a vacuum, divided by the wavelength of emission. e=hc/? combining these two equations produces the rydberg formula
1/?=r_h (1/(n^2 f)-1/(n^2 i))
the rydberg constant (r) = 109737.316 m?1 or 1.097×107m?1.
i-postulates of bohr s theory
(1) every atom has consisted from the nucleus and the suitable number of electrons revolved around the nucleus in circular orbits.
(2) electrons revolved only in certain non-radiating orbits called stationery orbits for which the total angular momentum is an integral multiple of( h/2pi) where h is plank s constant.


figure 2-5: model of hydrogen atom.

l is the angular momentum of the revolving electrons
(3) radiation occurs when an electron jumps from one permitted orbit to another. it is emitted when the electron jumps from higher orbit to a lower orbit, i.e., e2 - e1 = h?, where ? is the frequency of radiation.

ii-derivation of bohr s radius
radii of orbits
according to bohr s second postulate, since
and
where m is mass of the electron, v is linear velocity, r is the radius of the orbit in which e revolves around the nucleus. now [because necessary centripetal force is provided by the electrostatic force of attraction between electron and nucleus] whose charge is ze where z is the atomic number of the atom.
substituting for v,

for hydrogen atom z = 1

i.e., r a n2 the stationary orbits are not equally spaced
on substituting the value h = 6.6x10-34 j. sec
n = 1, k = 9 x 109nm2/c2, m = 9.1 x 10-31kg, e = 1.6 x 10-19c,
we get r = 5.29 x 10-11m
the velocity of the electron in a stationary orbit substituting the expression for r in the equation.
we get
the resulting expression is
calculation shows that when n=1, velocity v of the electron is 1/137 time velocity of light is vacuum i.e.,

the total energy of the electron in stationary orbit:
the energy of electron revolving in a stationary orbit is of two types. kinetic energy due to velocity and potential energy due to the position of the electron.
now

(negative sign is for charge of an electron)
nowtotal energy = k.e + p.e


spectral series of hydrogenations.

on putting the value m, k, e, h, we get
=
for hydrogen atom

the negative sign implies that an electron is bound to the nucleus. as n increases, the total energy of the electron is more than that in the inner orbits.
2-5- failed of bohr theory for the multi electronic system:
the bohr model was an important step in the development of atomic theory. however, it has several limitations been occurring.
it is in violation of the heisenberg uncertainty principle. the bohr model considers electrons to have both a known radius and orbit, which is impossible according to heisenberg.
the bohr model is very limited in terms of size. poor spectral predictions are obtained when larger atoms are in question.
it cannot predict the relative intensities of spectral lines.
it does not explain the zeeman effect when the spectral line is split into several components in the presence of a magnetic field.
the bohr model does not account for the fact that accelerating electrons do not emit electromagnetic radiation.


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