Structure of matter Flashcards
Wave properties of particules
~Elementary particles possess (as do their systems: atoms, molecules) both corpuscular and wave properties (corpuscular-wave dualism)
wave property ← diffraction and interference experiments show that light is represented by waves
corpuscular property ← photoelectric effect demonstrates that light is a flux of energy in the form of photons.
~Wave theory of matter: Motion of a particle with mass m, momentum p, and energy E is related to the wavelength of de Broglie’s wave by
λ = h/p
And to the frequency by f = E/h
~The equation for wavelength suggests that wavelengths of elementary particles are very short (shorter than visible light). This is why the resolving power of an electron microscope is better than that of an optical microscope.
*The energy E of photon (J) is related to the frequency f of the wave and to its wavelength lamda (λ) by
E=hf = hc/ λ
where h = 6.63 x 10-34 J.s = 4.13x10-15 eV = Planks constant c=velocity of light
~The corpuscular-wave dualism has its consequences:
The Heisenberg Uncertainty Principle: it is impossible to determine with perfect accuracy the position and momentum of a particle simultaneously. If the position vector r is being measured, its momentum will change and vice versa.
Quantum numbers
The state (position and energy) of an electron can be described by a wave function composed of dimensionless parameters which equal the degrees of freedom.
~The degrees of freedom of an electron = 4
~Thus, atomic theory states that the any electron in an atom can be completely described by 4 quantum numbers.
With the exception of the last quantum number ms (spin), the numbers determine the geometry and symmetry of the electron cloud (orbital).
~The electron cloud (orbital) is a space around the nucleus in which the probability of finding an electron is high.
Pauli Exclusion Principle: no two electrons in a given atom can possess the same set of four quantum numbers. (i.e. each electron in a given atom has a unique set of quantum numbers and exist in the same quantum state)
- Principle Quantum Number (n)
n = any positive integer 1, 2, 3, …
n describes the electron’s total energy and shell in which it can be found
~n = 1, 2, 3, 4, 5, 6, 7 corresponds to shell K, L, M, N, O, P, Q
The greater the value of n, the higher the energy level and radius of the electron’s orbit
Maximum # of e- in energy level (shell) n = 2nsquared
*The difference in energy between two adjacent shells decreases with distance from the nucleus (1/n1squared - 1/n2squared) n= total energy electron M= 9.11x10 -31 E0= 8.854x10 -12 FM-1 e= 1.6x10 -19 coloumbs
- Orbital Quantum Number (L)
For any given n, L is a number from 0 → n-1
It is determined by the angular momentum L where the magnitude of Lis given by the equation 1.13
L describes the subshell of the electron
Subshells (s, p, d, f) correspond to l values of 0, 1, 2, 3
Determines the shape of the orbital (s is spherical, p is bilobed, etc)
Exp: s to equation 1.13 the value of orbital angular momentum is 0 = spherical shaped orbital.
The max # of e- that can exist within a subshell = 4l + 2 - Magnetic Quantum Number (ml)
Possible values are m1=0, +/- 1, +/-2, …, +/-L(orbital quantum number)
Chart for understanding:
Relationship between Quantum numbers Orbital Values Number of values for m S L=0, m= 0 1 P L=1, m=-1,0,+1 3 D L=2, m=-2, -1, 0, +1, +2 5 F L=3, m=-3,-2,-1,0,+1,+2,+3 7 G L=4, m=-4,-3,-2,-1,0-+1,+2,+3,+4 9
Simple: m represents the number of possible values for available energy levels of that subshell.
Specifies the particular orbital within the subshell where an electron can be found
Determines the spatial orientation of the orbital, describes loosely the direction of the angular momentum vector (L) in an external magnetic field
- Spin Quantum Number (ms)
+-1/2
Describes the spin of the electron due to it’s internal angular momentum (S) which does not depend onits orbital angular momentum
In the presence of an external magnetic field, electrons orient themselves in one of two possible orientations corresponding to +-1/2
Two electrons within the same orbital must have opposite values of spin (paired e-)
~Parallel e- are electrons in different orbitals that possess the same value of ms
- Electron transitions may be probable or improbable (allowed vs. forbidden)
~allowed: transitions in which l changes by +-1
~forbidden: transitions in which l changes by more than +-1
*During electron transitions n can vary arbitrarily
Ionisation and excitation
Excitation:
• Electrons with minimum energy are in the ground state, increase energy and becomes excited state
• Atom absorbs energy corresponding to difference between ground state and excited state
• Atom remains in excited state for short time (10-5-10-7s)
• Excitation energy emitted as one or more photons during deexcitation
• Metastable state of electron when it reaches an energy level from which transition to ground state is ‘forbidden’
• Deexcitation may be followed by radiation emission resulting in luminescence
Ionisation:
• Ionisation potential = binding energy of electron which is the energy required to remove the electron from the forces of the nucleus
• Always positive, equal in value to total energy of electron (with heavy atoms, other factors affect this)
• If electron gets energy greater than binding energy, some of this energy is used to remove electron from the system
• Results in formation of positive ion and increase in total energy of system
• Ionised atom is unstable and returns to ground state of minimum energy by emission of fluorescence radiation
Absorbed energy must be of order of eV for excitation or ionisation of outer electrons to take place, and keV for inner electrons. Excitation or ionisation of inner electrons may result in emission of UV light or Xrays
Structure of electron shells in atoms
Electron configuration is the pattern by which electrons fill an atom.
Electron filling is governed by two principles:
1. The arrangement is at minimal energy
2. No two electrons within the atom occupy the same quantum state
-electrons fill shells and subshells in order of increasing energy and each subshell is filled completely before the next subshell begins filling
-in the case of heavy atoms, higher shells can fill prior to lower ones if the total energy associated with the shell is lower in the higher shell.
-highest # of electrons in a shell is 2nsquared
Closed shell/subshell: one that is completely occupied by electrons
s (2 e-), p (6 e-), d (10e-), f (14 e-)
-Total orbital and spin angular moment =0 and distribution of the electrons‘ eff charge is symmetrical.
Hund’s Rule: within a given subshell, orbitals are filled so that there are a maximum number of half-filled orbitals with parallel spins.
-electrons prefer empty orbitals due to the repulsion of neg. charges that occurs in filled orbitals
-filled orbitals contain 2 e- with opposite spin
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6, 5s2, 4d10, 5p6, 6s2, 4f14, 5d10, 6p6, 7s2, 6d10
Atomic Nucleus
The atomic nucleus is formed by nucleons: protons and neutrons
~It can be described by:
Atomic # = Z, number of protons present in the nucleus
Mass # = A, total number of nucleons
Neutron # = N = A – Z, number of neutrons in the nucleus
! A= Z + N !
~The total electrical charge Z of a nucleus is 1.6 x 10-19 C
~Most of the atom‘s mass is found in the nucleus (nucleon mass is 2 x 103 times greater than electron mass)
~The mass of atoms is expressed in atomic mass units (AMU)
1 AMU = 1/12 the mass of a C-12 atom. 1 A.M.U.=1.66x10-27 kg = energy equivalent 931MeV
~the mass of an atom can be estimated with a mass spectrometer
Isotopes- nuclei of the same element with same proton #, different neutron # (same Z, diff A), 280 stable in nature, 1100 art. unstable Isobars- nuclei with equal # nucleons, # different protons (same A, diff Z) Isomers- same # of protons and neutrons but different energies of nuclei -not stable, e- fall to lower energy levels → emit radiation
Radius of proton = 1.23x10-15 m
Radius of heavy atoms can be calculated Ra= (1.23x10-15) x A1/3
*atomic radii decrease from left to right on the periodic table and increase down a group
Nuclear forces hold an atom together via the strong interaction
~Range is 10-15 m
~Strength is not dependent on nucleon charge
~The strong interaction is the strongest force in nature at this distance
Energy e- =0.51 MeV
Energy p =938MeV
Energy n =939MeV
Radius r =1.23x10-15 m
Binding energy in atomic nucleus
The binding energy of a nucleus determines its stability.
Mass Defect (∆m): the difference between the calculated and actual mass of a nucleus
~The mass of a nucleus is always less than the combined mass of it‘s protons and neutrons
~The difference is due to matter that has been converted to binding energy which holds the nucleons together.
*binding energy peaks at iron which is the most stable atom. In general, intermediate sized nuclei are the most stable.
Therefore a greater binding energy leads to a higher mass defect.
Potential barrier of atomic nucleus
~At short distances, strong interactions are stronger than electromagnetic interactions.
~A charged nucleus of Ze forms as electromagnetic force field around it with potential U(r) that is a function of the r = distance from nucleus.
~This creates a potential barrier around the nucleus for positively charged particles due to the electromagnetic interaction for positively charged particles (protons,dueterons,alpha-particles) that enter the nucleus.
*Therefore positively charged particles must overcome the potential barrier to enter the atom.
Physical Principles of Mass spectrometry
~Mass spectrometry can be used to determine the isotopic composition of a given sample.
~This method is based on the fact that the trajectory of charged particles moving in a magnetic field is dependent on their mass
Method:
- sample is ionized-(e- removed) to become positive ions of charge q
- the ions are accelerated in a longitudinal electric field by a potential difference U to E=qU (a beam of ions is formed)
- The ion beam is seperated into several beams depending on each ion’s specific charge q/M. The path/deflection of an ion in a magnetic field depends on its mass to charge ratio.
- Each beam is detected and its intensity is measured → intensity provides information about the relative amount of each isotope present in the sample (% composition)
E=qU = 1/2mv2
E= Energy, q= charge, U=potential difference
Scheme Fig. 17: Page 27
Physical principles of nuclear magnetic resonance, and MRI
- The magnetic properties of nuclei provide the basis for NMR
- Nucleons possess their own half-integer value of angular momentum.
- Magentic moment of proton is 2.8nm and neutron is 1.9nm (dependent on the mass of the particle)
- magnetic moment = how much spin (drehmoment) the elementary particle receives in an external magnetic field
- the magentic moments of nuclei with non-zero values of spin (and thus magnetic moments) can be utilized to provide information about the amount of particular nucleus within a given sample.
- if the number of both the protons and neutrons in a given nuclide is even, then there is no spin
- all elements have at least one isotope with a non-zero spin number in the nucleus
-The magnetic moments of such nuclei are oriented randomly under normal conditions and can align with an external magnetic field when one is present
~The potential energy of a nucleus/proton in the presence of an external magnetic field
B =μB where μ is the magnetic moment of the proton
~Protons can exist in two possible energy states within a magnetic field → +-μB depending on their spin number of +-1/2
In the higher energy state, the magnetic moments are oriented antiparallel to the field
In the lower energy state, the magnetic moments are aligned with the field
~If protons are exposed to electromagnetic radiation with energy that equals the difference between these two energy states, resonance exchange of energy occurs between the nuclei and incident wave.
E = hf = 2μB
w= angular frequency B= magnetic induction
~Resonance exchange of energy is called nuclear magnetic resonance
~This resonance exchange of energy occurs at Lamors frequency which is
f=w/2π
from w=2πf
- If a proton is in a lower energy state with energy -μB, a photon is absorbed and transition into a higher energy state with energy +μB occurs.
- If a proton is in a higher energy state with energy μB, then a photon is emitted and transition into lower energy state –μB occurs
The examples in the book (page 24-25) demonstrate that the values of resonance frequencies are very different for various nuclei.
Fact: of an oscilloscope - If the sample is located inside a coil, then on axis which lays perpendicular to the static field B, will be created an electromotoric force based on the procession motion of nuclear magnetisation M.
MRI (Magnetic Resonance Imaging)
- diagnostic application of NMR
- free protons in the hydrogen nuclei of water are utilized since almost all biological tissues contain water.
- The patient is placed into an external magnetic field which aligns the magnetic moments of hydrogen nuclei, pulses of radio waves are passed through the patient causing deflection of the magnetic moments. The magnetic moments then return to their lower energy aligned states with simultaneous emission of a photon by the nucleui. The strength, frequency, and time it takes for the nuclei to return to their pre-excited state produces a signal. The signal is analyzed by a computer and an image is produced in which the differences in tissue composition can be visualized.
- MRI produces 3 images in one: a distribution of proton density, and relaxation times T1 and T2.
T2 transverse relaxation time:
Time it takes for the probe to relax back after the deviation through short-time pulse.
Relaxation times are normally different from another except Water where they are basically identical. Exp: water: 3 secs, ice: T1: 360s , T2: 5 microseconds
- The frequency at which resonance exchange of energy occurs is dependent on the atom’s environment. Thus a compound can contain many nuclei that resonate at different frequencies producing a complex spectrum.
- magnetic moment is influenced by the shielding action of a nuclei’s own electrons as well as the electrons of other atoms in its vicinity.
Quantum properties of waves
Energy emitted from electromagnetic radiation comes in discrete bundles called quanta. The energy value of a quantum is
E=hf = hc/ λ
This suggests a particulate nature of electromagnetic radiation where each light particle (photon) carries an energy that is proportional to its frequency. High frequency (short wavelength) → high energy Low frequency (long wavelenth) → low energy
~The corpuscular-wave dualism has its consequences:
The Heisenberg Uncertainty Principle: it is impossible to determine with perfect accuracy the position and momentum of a particle simultaneously. If the position vector r is being measured, its momentum will change and vice versa.
Emission spectrum of the hydrogen atom
~Electrons within an atom can be excited to higher energy levels when heat or other forms of energy are applied. The excited state is not stable and short-lived. Thus the electron returns rapidly to its initial state simultaneously emitting energy in the form of a photon.
~The energy of the released photon equals the energy difference between the excited and initial states of the electron
~Since there are discrete values of electron energies only certain energies (frequencies, wavelengths) may be emitted by the atom.
~The different electrons within the atom will be excited to different energy levels and each will emit a photon characteristic of the energy transition it undergoes.
~Thus a line spectrum is produced with each line corresponding to a specific electron transition.
~Each element produces a unique emission spectrum that can serve as its fingerprint.
~The set of spectral lines observed during transitions from all higher levels into a certain energy level corresponding to the given n is called a series.
The atomic emission spectra of hydrogen is composed of several series
Lyman Series: lines corresponding to e- transition from higher energy levels into the ground state (n = 1)
I–observed in the ultraviolet region of light (high energy emission)
Balmer series: lines corresponding to e- transitions from higher energy levels into
n = 2
–observed in the region of visible light
Paschen series: lines corresponding to e- transitions from higher energy levels into
n = 3 (Paschen‘s series) and to higher values of n
–observed in the region of infrared light (low energy emission)
*The greatest energy emitted is in the Lyman series and corresponds to an electron falling from n = to n = 1
