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General Properties of Electromagnetic Radiation.
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The electromagnetic radiation is looked at as sinusoidal waves which are composed of a combination of two fields. An electric field (which we will use, in this course, to explain absorption and emission of radiation by analytes) and a magnetic field at right angle to the electric field (which will be used to explain phenomena like nuclear magnetic resonance in the course of special topics in analytical chemistry offered to Chemistry students only).
The classical wave model The classical wave model describes electromagnetic radiation as waves that have a wavelength, frequency, velocity, and amplitude. These properties of electromagnetic radiation can explain classical characteristics of electromagnetic radiation like reflection, refraction, diffraction, interference, etc. However, the wave model can not explain the phenomena of absorption and emission of radiation.
We will only deal with the electric field of the electromagnetic radiation and will thus refer to an electromagnetic wave as an electric field having the shape of a sinusoidal wave. The arrows in the figure below represent few electric vectors while the yellow solid sinusoidal wave is the magnetic field associated with the electric field of the wave.
Wave Parameters 1. Wavelength () The wavelength of a wave is the distance between two consecutive maxima or two consecutive minima on the wave. It can also be defined as the distance between two equivalent points on two successive maxima or minima. This can be seen on the figure below:
2. Amplitude (A) The amplitude of the wave is represented by the length of the electrical vector at a maximum or minimum in the wave. In the figure above, the amplitude is the length of any of the vertical arrows perpendicular to the direction of propagation of the wave.
3. Frequency The frequency of the wave is directly proportional to the energy of the wave and is defined as the number of wavelengths passing a fixed point in space in one second. 4. Period (p) The period of the wave is the time in seconds required for one wavelength to pass a fixed point in space.
5. Velocity (v) The velocity of a wave is defined as the multiplication of the frequency times the wavelength. This means: V = The velocity of light in vacuum is greater than its velocity in any other medium
Since the frequency of the wave is a constant and is a property of the source, the decrease in velocity of electromagnetic radiation in media other than vacuum should thus be attributed to a decrease in the wavelength of radiation upon passage through that medium.
6. Wavenumber () The reciprocal of wavelength in centimeters is called the wavenumber. This is an important property especially in the study of infrared spectroscopy. = k
Electromagnetic Spectrum The electromagnetic radiation covers a vast spectrum of frequencies and wavelengths. This includes the very energetic gamma-rays radiation with a wavelength range from 0.005 – 1.4 Ao to radiowaves in the wavelength range up to meters (exceedingly low energy). However, the region of interest to us in this course is rather a very limited range from 180-780 nm. This limited range covers both ultraviolet and visible radiation.
Mathematical Description of a Wave A sine wave can be mathematically represented by the equation: Y = A sin (t + ) Where y is the electric vector at time t, A is the amplitude of the wave, is the angular frequency, and is the phase angle of the wave. The angular frequency is related to the frequency of radiation by the relation: = 2 This makes the wave equation become: Y = A sin (2t+ )
Superposition of Waves When two or more waves traverse the same space, a resultant wave, which is the sum of all waves, results. Where the resultant wave can be written as: Y = A1 sin (2t+ ) + A2 sin (2t+ ) + ........ + An sin (2nt+ n)
Constructive Interference The resultant wave would has a greater amplitude than any of the individual waves which, in this case, is referred to as constructive interference. The opposite could also take place where lower amplitude is obtained.
The decrease in the intensity is a result of what is called a destructive interference. When the multiple waves have the same wavelength, maximum constructive interference takes place when 1 - 2 is equal to zero, 360 deg or multiple of 360 deg. Also maximum destructive interference is observed when 1 – 2 is equal to 180 deg, or 180 deg + multiples of 360 deg. A 100% constructive interference can be seen for interference of yellow and blue shaded waves resulting in a wave of greater amplitude, brown shaded.
The blue and yellow shaded waves interfere to give the brown shaded wave of less amplitude, a consequence of destructive interference of the two waves.
The Period of a Beat When two waves of the same amplitude but different frequencies interfere, the resulting wave exhibit a periodicity and is referred to as beat (see figure below). The period of the beat can be defined as the reciprocal of the frequency difference between the two waves: Pb = 1/()
Fourier Transform The resultant wave of multiple waves of different amplitudes and frequencies can be resolved back to its component waves by a mathematical process called Fourier transformation. This mathematical technique is the basis of several instrumental techniques like Fourier transform infrared, Fourier transform nuclear magnetic resonance, etc.
Diffraction of Radiation Diffraction is a characteristic of electromagnetic radiation. Diffraction is a process by which a parallel beam of radiation is bent when passing through a narrow opening or a pinhole. Therefore, diffraction of radiation demonstrate its wave nature. Diffraction is not clear when the opening is large.
Coherence of Radiation Two beams of radiation are said to be coherent if they satisfy the following conditions: 1. Both have the same frequency and wavelength or set of frequencies and wavelength. 2. Both have the same phase relationships with time. 3. Both are continuous.
Transmission of Radiation As mentioned before, the velocity of radiation in any medium is less than that in vacuum. The velocity of radiation is therefore a function of the refractive index of the medium in which it propagates. The velocity of radiation in any medium can be related to the speed of radiation in vacuum ( c ) by the relation: ni = c/vi Where, vi is the velocity of radiation in the medium I, and ni is the refractive index of medium i.
The decrease in radiation velocity upon propagation in transparent media is attributed to periodic polarization of atomic and molecular species making up the medium. By polarization we simply mean temporary induced deformation of the electronic clouds of atoms and molecules as a result of interaction with electric field of the waves.
Dispersion of Radiation If we look carefully at the equation ni = c/vi and remember that the speed of radiation in vacuum is constant and independent on wavelength, and since the velocity of radiation in medium I is dependent on wavelength, therefore the refractive index of a substance should be dependent on wavelength. The variation of the refractive index with wavelength is called dispersion.
Refraction of Radiation When a beam of radiation hits the interface between two transparent media that have different refractive indices, the beam suffers an abrupt change in direction or refraction. The degree of refraction is quantitatively shown by Snell's law where: n1 sin 1 = n2 sin 2
Reflection of Radiation An incident beam hitting transparent surfaces (at right angles) with a different refractive index will suffer successive reflections. This means that the intensity of emerging beam will always be less than the incident beam.
Scattering of Radiation When a beam of radiation hits a particle, molecule, or aggregates of particles or molecules, scattering occurs. The intensity of scattered radiation is directly proportional to particle size, concentration, the square of the polarizability of the molecule, as well as the fourth power of the frequency of incident beam. Scattered radiation can be divided into three categories:
Quantum Mechanical Description ofRadiation All the previously mentioned properties of radiation agrees with the wave model of radiation. However, some processes of interest to us, especially in this course, can not be explained using the mentioned wave properties of radiation. An example would be the absorption and emission of radiation by atomic and molecular species. Also, other phenomena could not be explained by the wave model and necessitated the suggestion that radiation have a particle nature. The familiar experiment by Heinrich Hertz in 1887 is the corner stone of the particle nature of radiation and is called the photoelectric effect.
The Photoelectric Effect When Millikan used an experimental setup like the one shown below to study the photoelectric effect, he observed that although the voltage difference between the cathode and the anode was insufficient to force a spark between the two electrodes, a spark occurs readily when the surface of the cathode was illuminated with light. Look carefully at the experimental setup:
It is noteworthy to observe the following points: 1. The cathode was connected to the positive terminal of the variable voltage source, where it is more difficult to release electrons from cathode surface. 2. The anode was connected to the negative terminal of the voltage source which makes it more difficult for the electron to collide with the anode for the current to pass. 3. The negative voltage was adjusted at a value insufficient for current to flow. The negative voltage at which the photocurrent is zero is called the stopping voltage.
At these conditions, no current flows through the circuit as no electrons are capable of completing the circuit by transfer from cathode to anode. However, upon illumination of the cathode by radiation of suitable frequency and intensity, an instantaneous flow of current takes place. If we look carefully at this phenomenon and try to explain it using the wave model of radiation, it would be obvious that none of the wave characteristics (reflection, refraction, interference, diffraction, polarization, etc. ) can be responsible for this type of behavior.
What actually happened during illumination is that radiation offered enough energy for electrons to overcome binding energy and thus be released. In addition, radiation offered released electrons enough kinetic energy to transfer to the anode surface and overcome repulsion forces with the negative anode. If the energy of the incident beam was calculated per surface area of an electron, this energy is infinitesimally small to be able to release electrons rather than giving electrons enough kinetic energy. When this experiment was repeated using different frequencies and cathode coatings the following observations were collected:
Conclusions 1. The photocurrent is directly proportional to the intensity of incident radiation. 2. The magnitude of the stopping voltage depends on both chemical composition of cathode surface and frequency of incident radiation. 3. The magnitude of the stopping voltage is independent on the intensity of incident radiation.
Energy States of Chemical Species The postulates of quantum theory as introduced by Max Planck in 1900, intended to explain emission by heated bodies, include the following: 1. Atoms, ions, and molecules can exist in certain discrete energy states only. When these species absorb or emit energy exactly equal to energy difference between two states; they transfer to the new state. Only certain energy states are allowed (energy is quantized).
2. The energy required for an atom, ion, or a molecule to transfer from a one energy state to another is related to the frequency of radiation absorbed or emitted by the relation: Efinal – Einitial = hn Therefore, we can generally state that: DE = hn
Types of Energy States Three types of energy states are usually identified and used for the explanation of atomic and molecular spectra: 1. Electronic Energy States: These are present in all chemical species as a consequence of rotation of electrons, in certain orbits, around the positively charged nucleus of each atom or ion. Atoms and ions exhibit this type of energy levels only.
2. Vibrational Energy Levels: These are associated with molecular species only and are a consequence of interatomic vibrations. Vibrational energies are also quantized, that is, only certain vibrations are allowed. 3. Rotational Energy Levels: These are associated with the rotations of molecules around their center of gravities and are quantized. Only molecules have vibrational and rotational energy levels.
The solid black lines represent electronic energy levels. Arrows pointing up represent electronic absorption and arrows pointing down represent electronic emission. Dotted arrows represent relaxation from higher excited levels to lower electronic levels. The figure to left represents atomic energy levels while that to the right represents molecular energy levels.
Line Versus Band Spectra Since atoms have electronic energy levels, absorption or emission involves transitions between discrete states with no other possibilities. Such transitions will only result in line spectra. However, since molecular species contain vibrational and rotational energy levels associated with electronic levels, transitions can occur from and to any of these levels. These unlimited numbers of transitions will give an absorption or emission continuum, which is called a band spectrum. Therefore, atoms and ions always give line spectra while molecular species give band spectra.
