CHM214 Light and Electromagnetic Radiations PDF

# Light and Other Forms of Electromagnetic Radiations Electromagnetic radiation is a radiation energy that exhibits wavelike behavior and travels through space at the speed of light in a vacuum. Electromagnetic waves are typically described by any of the following three physical properties: - freq...

# Light and Other Forms of Electromagnetic Radiations Electromagnetic radiation is a radiation energy that exhibits wavelike behavior and travels through space at the speed of light in a vacuum. Electromagnetic waves are typically described by any of the following three physical properties: - frequency (v) - wavelength (λ) - photon of energy (E) ## The Electromagnetic Spectrum The electromagnetic spectrum is the distribution of electromagnetic radiation according to frequency and wavelength. It is the range of all possible frequencies of electromagnetic radiations. The electromagnetic spectrum extends from the low frequencies used for modern radio communication to gamma (γ) radiation at short wavelength (high frequency) end. It includes: - radio waves - microwaves - infrared - visible light - X-rays - y-rays. These different forms of radiations all travel at the speed of light (c), but they differ in the frequencies (v) and wavelength (λ). Since speed of light (c) = v x λ, electromagnetic radiation that has a long wavelength has a low frequency and the radiation with high frequency has a short wavelength. ## Light Light is a form of electromagnetic radiation. Because it is a wave, light is bent when it enters a glass prism. When white light is focused on a glass prism, the light rays of different wavelength are bent by different amounts and the light is transformed into a spectrum of colors. Starting from the side of the spectrum where the light is bent by the smallest angle, the colors are: - red - orange - yellow - green - blue - violet. ## Lights and the Human Eye Lights contain the narrow band of frequencies and wavelength in the portion of the electromagnetic spectrum that the human eyes can detect. It includes radiations with wavelength between about 400 nm (violet) and 700 nm (red). The visible region is only a small portion of the total range of electromagnetic radiation. ## Diagram of the Electromagnetic Spectrum A description of the electromagnetic spectrum is shown in the diagram below: - The diagram shows the range of frequencies of electromagnetic radiation from low frequencies to high frequencies. - The diagram also shows the corresponding wavelengths of the radiation at each frequency. - It is shown that the visible light spectrum is only a small portion of the electromagnetic spectrum. ## Questions 1. Photosynthesis uses 660nm light to convert CO2 and H2O into glucose and O2. Calculate the energy and frequency of this light. 2. The brilliant red colors seen in fireworks are due to the emission of light with wavelength around 650nm when strontium salts such as Sr(NO3)2 and SrCO3 are heated. Calculate the frequency of red light of wavelength 6.50 x 10^2 nm. # Assignment 1. An Fm radio station broadcasts at 99.5MHz. Calculate the wavelength of the corresponding radio waves. # Atomic Spectra When an element is heated in a flame or a discharge of electricity is passed through a gas, colors of light characteristics of the element are seen, for example, sodium emit yellow light and mercury provides a greenish glow. This light when passed through a spectroscope produces a spectrum called emission spectrum. The emission spectrum thus obtained consists of a series of lines called line spectra. The line spectrum is also known as atomic spectrum since it originated in the atom of the element. Each element gives a unique pattern of spectra lines and each line has a characteristic wave length. # Brief explanation on the emission spectrum of hydrogen When the electric current is passed through the glass tube that contains hydrogen gas at low pressure, the molecules in the gas break into atoms. These atoms absorb energy from the electric current and the single electron present in the 1st energy level jump to higher vacant energy levels 2, 3, 4 etc., depending on the amount of energy absorbed by the atoms of the gas. When the electron reaches the higher energy level, becomes excited and unstable. Being unstable, it comes back to one of the lower energy levels or even the lowest (n=1). In this process, energy absorbed earlier is liberated by it in the form of a photon of light of specific frequency and wavelength which appears as spectral lines in the emission spectrum. Emission spectrum of hydrogen is obtained due to the jump of electrons from higher energy level to a lower energy level. # Notes on discoveries of more series of lines in the hydrogen spectrum Between 1906 and 1924, four more series of lines were discovered in the emission spectrum of hydrogen by searching the infrared spectrum at longer wavelength. Each of these lines fits the same general equation where n1 and n2 are integers; ## Equation for the lines in the hydrogen spectrum **Equation:** v = R_{H}(\frac {1}{n_{1}^{2}} - \frac {1}{n_{2}^{2}} ) , where R_{H} is the Rydberg constant. These five series of lines are referred to by their discoverers' names. | Series | Year of Discovery | Region | n₁ | n₂ | | ----------- | ------------------ | -------- | ---- | -------- | | Lyman | 1906 | ultraviolet | 1 | 2, 3, 4... | | Balmer | 1885 | visible | 2 | 3, 4, 5.. | | Paschen | 1908 | Infrared | 3 | 4, 5, 6... | | Brackett | 1922 | Infrared | 4 | 5, 6, 7.. | | Pfund | 1924 | Infrared | 5 | 6,7... | # Bohr Model of Atom In 1914, Niels Bohr proposed a theory of the hydrogen atom which explained the origin of its spectrum and which also led to an entirely new concept of atomic structure. The following postulations were made by Bohr in working out his theory: 1. The electron moves in an orbit around the central nucleus and only certain orbits are allowed. 2. The electron does not radiate energy when in these orbits but has associated with it a definite amount of energy. 3. When an electron moved from one orbit to another, it either radiated or absorbed energy. If it moved towards the nucleus, energy was radiated and if it moved away from the nucleus energy was absorbed. # Derivations of some important equations/Formula ## Equation for the calculation of atomic radius of hydrogen For an electron to remain in its orbit, the electrostatic attraction between the electron and the nucleus which tends to pull the electron towards the nucleus must be equal to the centrifugal force which tends to throw the electron out of its orbit. For an electron of mass m, moving with a velocity v in an orbit of radius r **Equation:** Centrifugal Force = mv^2/r If the charge on the electron is e, the number of charges on the nucleus is Z and the permittivity of a vacuum is ɛo; **Equation:** Coulombic attractive force = Ze^2/4πεo r^2 For an electron to remain in its orbit, **Equation:** mv^2/r = Ze^2/4πεo r^2 ........eqn. 1 **Equation:** v^2 = Ze^2/4πεo mr ........eqn. 2 According to Planck's quantum theory, energy is not continuous but is discrete. This means that energy occurs in packets called quanta of magnitude h/2π where h is Planck's constant. The energy of an electron in an orbit i.e. its angular momentum (mvr) must be equal to a whole number n of quanta **Equation:** mvr = nh/2π **Equation:** v = nh/2πmr Square both sides **Equation:** v^2 = n^2h^2/4π^2m^2r^2 Combining this with eqn. 2 **Equation:** Ze^2/4περmr = n^2h^2/4π^2m^2r^2 **Equation:** r = εon^2h^2/πme^2Z ........ eqn. 3 When n = 1 and Z = 1 then (all other variables are constant **Equation:** r = a_{H} i.e the radius of the first Bohr orbit or atomic radius of hydrogen. **Equation:** a_{H} = εoh^2/πme^2 = 0.529177 x 10^-10m or 52.9177pm or 0.0529nm r can also be written as follows: **Equation:** r= n^2 a_{H} /Z ........ eqn. 4 For hydrogen Z=1, when n= 1, r= 1^2 x 0.0529nm when n= 2, r = 2^2 x 0.0529nm when n = 3, r= 3^2 x 0.0529nm This gives the picture of the hydrogen atom where an electron moves in circular orbits of radius perpendicular to 1^2, 2^2, 3^2 and radiates energy only when the electron jumps from one orbit to another. ## Equation for the Calculation of Energy The total energy associated with an electron revolving in nth orbit is the sum of the kinetic energy and potential energy and is given by **Equation:** E_{n}= 1/2mv^2 - Ze^2/4πεo r ........ eqn.5 But centripetal force = centrifugal force **Equation:** Ze^2/4πεo r^2 = mv^2/r **Equation:** Ze^2/4πεo r = mv^2 or **Equation:** 1/2mv^2 = Ze^2/8πεo r ........eqn.6 Incorporating eqn. 6 into eqn. 5 we have **Equation:** E_{n}= Ze^2/8πεo r - Ze^2/4πεo r ........eqn.7 **Equation:** E_{n}= -Ze^2/8πεo r Substitute r in eqn. 3 into eqn. 7 Then, **Equation:** E_{N} = -Ze^2/8πεo x me^2Z^2/εo^2 n^2 h^2 ........eqn.8 **Equation:** E_{N} = -Z^2 e^4m/8εo^2 n^2h^2 ## Equation for the calculation of change in energy (ΔΕ) When an electron falls from a higher orbit n2 to lower orbit n1, the change in energy is equal to the difference of energies of the electron in n2 orbit and in n1 orbit and is denoted by ΔΕ. Thus ΔΕ = energy of the electron in n2 orbit - energy of the electron is n1 orbit. **Equation:** ΔΕ= E2-E1 **Equation:** ΔΕ= -Z^2 e^4m/8εo^2 n2^2 h^2 - [-Z^2 e^4m/8εo^2 n1^2 h^2 **Equation:** ΔΕ= -Z^2 e^4m/8εo^2 h^2 [1/n1^2 - 1/n2^2] ........eqn.9 But ΔΕ = hcv **Equation:** hcv = -Z^2 e^4m/8εo^2 h^2 [1/n1^2 - 1/n2^2] **Equation:** v = -Z^2 e^4m/8εo^2 hc [1/n1^2 - 1/n2^2] ........eqn.10 **Equation:** v = R_{H }Z^2 [1/n1^2 - 1/n2^2] ........eqn.11 Where R_{H} is the Rydberg constant which is me^4/8εo^2 hc ## Equation for the calculation of frequency **Equation:** ΔΕ = hv **Equation:** v = ΔΕ/h Substitute ΔΕ in eqn. 9 multiply by 1/h **Equation:** v = -Z^2 e^4m/8εo^2 hc [1/n1^2 - 1/n2^2] Where Z= 1 for hydrogen atom and R_{H} = me^4/8εo^2 hc **Equation:** v = R_{H}[1/n1^2 - 1/n2^2 ].........eqn.12 ## Equation for the calculation of ionization energy of hydrogen The ionization energy (I.E) or Ionization potential (I.P) of an atom or molecule is the energy needed to completely remove an electron from a gaseous atom or molecule in its ground state forming a positive ion. For hydrogen atom, the process is **Equation:** H(g) + I.P → H(g) + e Recall that for the hydrogen atom the change in energy is given by **Equation:** ΔΕ = me^4/8εo^2 h^2 [1/n1^2 - 1/n2^2] When an electron is excited from the ground state n=1 to an energy level in which the nucleus exerts no attraction on the electron (n= ∞), then the atom is ionized. **Equation:** Then I.E = ΔΕ = me^4/8εo^2 h^2 [1/1^2 - 1/∞^2] **Equation:** I.E = me^4/8εo^2 h^2 **Equation:** I.E= R_{H} c h or **Equation:** I.E_{H} = R_{H} c h Where R = Rydberg constant; c = speed of light; h = Planck's constant. # Questions 1. Given that the Bohr radius for the hydrogen atom equals 5.292 x10^-11 m, calculate (i) the radius of the first allowed Bohr orbit for Li^2+ an hydrogen like atom. (ii) the velocity of an electron in the first allowed Bohr orbit of Li^2+. **Solution** (i) Using eqn.4, **Equation:** r = n^2 a_{H}/Z where a_{H} is the atomic radius of hydrogen and the value is given as 5.292 x10^-11 m For Li^2+, n=2 and Z=3, **Equation:** r = 2^2 / 3 x 5.292 x 10^-11m **Equation:** r = 7.056 x 10^-11 m **Equation:** r ~ 7.06 x 10^-11 m (ii) Using this equation **Equation:** v = nh/2πmr n= 2, h = 6.626 x 10^-34 Js, m_{e} = 9.1085 x 10^-31 kg **Equation:** v = (2 x 6.626 x 10^-34 kgm^2 s^-2 s)/(2 x 3.14 (9.1085 x 10^-31kg) (7.06 x 10^-11 m)) **Equation:** v = 3.28 x 10^6 ms^-1 2. Calculate the energy required for the ionization of an electron from the ground state of hydrogen atom given that the Rydberg constant (R∞) equals 109737cm^-1, hence calculate the 3rd ionization potential of Lithium. **Solution** Using the equation **Equation:** I.E = R∞ ch **Equation:** I.E = (109737cm^-1) (3.0 x 10^10 cm s^-1) (6.626 x 10^-34 Js) **Equation:** I.E = 2.18 x 10^-18 J = 2.18 x 10^-21 kJ To calculate the 3rd I.P of lithium **Equation:** IP3 = Z^2 I.E_{H} **Equation:** IP3 = (3^2) (2.18 x 10^-21 kJ) **Equation:** IP3 of Li = 1.962 x 10^-20 kJ # Limitations of Bohr's Theory 1. Bohr proposed his theory based on some assumptions only to explain the discontinuous nature of the hydrogen spectrum, there was no theoretical basis. But modern wave mechanics provided its justification. 2. This theory is successfully applicable to hydrogen-like atom i.e, single electron systems e.g He^+ H, Li^2+, Be^3+, but totally fail when applied to multi-electron systems. 3. Bohr treated electrons as a particle, but modern concept accepts its wave-particle duality. 4. When spectral lines are viewed by spectroscope, each single line appears to have more single lines closely spaced together, called fine structure, indicating variation of electronic energy in the same quantum states. Bohr's theory fails to explain this. 5. Calculated ionization energy obtained using the I.E formula does not tally with experimental value. 6. Bohr considered electrons as particle revolving in an orbit and calculated its energy accurately. But modern development in physics showed that the position and momentum of an electron could not be calculated simultaneously with accuracy. In doing so, there-must be some error (uncertainty principle). # Mechanical Concept of Atomic Structure ## De-Broglie's Equation Einstein in 1905 suggested that light shows a dual character i.e. light behaves both as a material particle as well as a wave. De-Broglie in 1923, extended Einstein's view and said that all forms of matter like electrons, protons, neutrons, atoms, molecules, etc. also show a dual character. He derived this relationship: **Equation:** λ = h/mv Where λ is the wavelength of a moving particle with mass m moving with a velocity v and h is the Planck's constant. This equation allows us to calculate the wavelength for a particle. ## Heisenberg's Uncertainty Principle According to this principle, it is not possible to determine simultaneously and precisely both the position and momentum (or velocity) of a microscopic moving particle like electron, proton, etc. Heisenberg said that the uncertainty in position (represented by Δx) times the uncertainty in momentum (represented by Δp) must be greater than a constant number equal to Planck's constant(h) divided by 4π (π is a constant approximately equal to 3.14). Mathematically, this principle is expressed as **Equation:** Δx x Δp ≥ h/ 4π ........ eqn.1 Where Δx = Uncertainty involved in position Δp = Uncertainty in momentum h = Planck's constant Generally the relation eqn. 1 is written as **Equation:** Δx x Δp = h/4π ........eqn.2 On putting Δp= m x Δv (where m =mass of the particle and Δv= uncertainty in velocity) in eqn.2, we have **Equation:** Δx x (m x Δv) = h/4π .........eqn.3 ## Questions 1. A cricket ball weighs 200g. If the uncertainty in its position is 5 pm, what is the uncertainty in the velocity of the ball (Plancks's constant=6.626 x 10^-34 Js) **Solution** According to Heisenberg's uncertainty principle **Equation:** Δx x Δp = h/4π or **Equation:** Δx x (m x Δv) = h/4π **Equation:** (5pm) x (200g) x Δν = 6.626 x 10^-34Js/ 4 x 3.14 **Equation:** Δν = 6.626 x 10^-34 kgm^2s^-1 / 5 x 10^-12 m x 0.2 kg x 4 x 3.14 **Equation:** Δν = 6.626 x 10^-34 / 1.256 x 10^-11 ms^-1 **Equation:** Δν = 5.275478 x 10^-23 ms^-1 2. According to Bohr's theory of H-atom, the velocity of an electron in the 1st orbit is 2.183 x 10^6 ms^-1. If the uncertainty in the position of the electron is 5pm, what will be the uncertainty in its velocity? (mass of electron= 9.10939 10^-31kg; h = 6.626 x 10^-34 Js) **Equation:** Δx x (m x Δv) = h/4π **Equation:** (5pm) x (9.10939 x 10^-31 kg) x Δν = (6.626 x 10^-34 J s / 4 x 3.14 **Equation:** Δν = 6.626 x 10^-34 kgm^2s^-1 / 5 x 10^-12 m x 9.109 x 10^-31kg x 4 x 3.14 **Equation:** Δν = 6.626 x 10^-34 / 5.72069692 x 10^-41 ms^-1 **Equation:** Δν = 1.1582 x 10^7 ms^-1 # Schrodinger's Wave Equation According to de-Broglie, electron has a dual character i.e. the electron behaves as a material particle and as a wave. In 1926, Erwin Schrodinger (an Austrian Physicist) said that if an electron behaves as a wave, there must be a wave equation which should be able to describe the wave motion of the electron. He proposed that since an electron behaves as a wave, it should obey the same wave equation of motion which all other known types of wave obey. He derived an equation which describes the wave motion of an electron-wave propagating three dimension (x, y and z axes) in space. This wave equation is called the Schrodinger's wave equation. This equation is written in many forms, one of which is given below: **Equation:** (d^2ψ/dx^2) + (d^2ψ/dy^2) + (d^2ψ/dz^2) + (8π^2m/h^2)(E-V)ψ = 0 Where ψ is the wave function; m is the mass of the electron; E is the total energy of the electron; V is the potential energy of the electron h is the Planck's constant; x, y and z are Cartesian coordinates. # The Quantum Numbers The Schrodinger model assumes that the electron is a wave and tries to describe the region in space or orbitals where electrons are most likely to be found. Instead of trying to tell us where the electron is at any time, the Schrodinger model describes the probability that an electron can be found in a given region of space at a given time. There are four quantum numbers which specified the allowed energies and general behavior of the atomic electron. Three of these are obtained by solving the Schrodinger equation and the spin quantum number is obtained by experimental evidence. ## The Principal Quantum Number (n) This is primarily responsible for determining the overall energy of an atomic orbital. It also indicates the relative size of the orbital and the relative distance from the nucleus e.g. orbitals for which n=2 are larger than those for which n=1. The principal quantum number (n) has values which are integral and non zero i.e. n=1, 2, 3, 4...... ## The Azimuthal Quantum Number (l) This is also known as subsidiary quantum or angular quantum number. It describes the shape of the orbital and determines the angular momentum of the electron. It has integral values 0, 1, 2, 3.... (n-1) For a given value of n, the maximum permitted value of l is n-1 e.g. when n=3, l=0, 1 and 2. l values can be represented by code letters as follow | Values of l | Code letter | | ----------- | ----------- | | 0 | s | | 1 | p | | 2 | d | | 3 | f | The letters originate in the spectroscopic terms for the series sharp, principal, diffuse, fundamental - which are words used historically to describe aspects of line spectra. ## The Magnetic Quantum Number (ml) This is assigned to distinguish between orbitals of the same type in a given level. It may assume any integral value between +l and -l including zero. | l | ml | | --- | --- | | 0 | 0 | | 1 | +1, 0, -1 | | 2 | +2, +1, 0, -1, -2 | | 3 | +3, +2, +1, 0, -1, -2, -3 | The magnetic quantum number (ml) describes the orientation in space of a particular orbital. ## The Spin Quantum Number (ms) This determines the orientation of the orientation of the electron magnetic moment in a magnetic field, either in the direction of the field (+1/2) or opposed to it (-1/2) Together, the quantum numbers n, l, and ml define an atomic orbital, while the quantum no ms describes the electron Spin within the orbital. # Atomic Orbitals The region in space where the probability of finding an electron is high is called an orbital. There are different kinds of orbitals of different sizes and shapes and these are disposed about the nucleus in specific ways. The energy of an electron determines which kind of orbital it occupies. ## The s orbital An orbital with l=0 is spherically symmetrical around the nucleus and is called an S orbital. Note that there is an S orbital in every principal shell. The 2s orbital has the same shape with the 1s orbital except that it has a bigger radius than 1s. Similarly, 3s orbital has a bigger radius than 2s and energy of 2s orbital is higher than the energy of 1s orbital. ## The p orbital An orbital with l=1 has two regions of higher probability and is called a p orbital. It is sometimes called a dumbbell or polar shapes. After the 2s orbitals are the 3 orbitals of the same energy called the 2p orbitals. (Orbitals are degenerate when they have the same energy) it consists of 2 lobes with the atomic nucleus lying between them. There are 3p orbitals namely px, py, and pz.. ## The d orbitals Beginning with n=3, each principal shell have a set of 5d orbitals. An orbital with l=2 is called a d orbital. There are 5 possible ml values for l=2 i.e, +2, +1, 0, -1, -2. Thus an orbital can have any one of the different orientations. ## Shells and subshells of orbitals Orbitals that have the same value of the principal quantum number form a shell. Orbitals within a shell are divided into subshells that have the same angular quantum number. Chemist describes the shell and subshells in which an orbital belongs with a two character code such as 2p or 4f. The 1st character indicates the shell (n=2, or n=4). The second character identifies the subshells. The following letters are used to indicate different subshells s,p,d,f. | n | l | Subshell | No of orbital in the subshell | No of electrons needed to fill d subshell | Total | | ---- | ---------- | -------------- | ------------------------------ | --------------------------------------- | ----- | | 1 | 0 | 1s | 1 | 2 | 2 | | 2 | 0 | 2s | 1 | 2 | 8 | | | 1 | 2p | 3 | 6 | | | 3 | 0 | 3s | 1 | 2 | 18 | | | 1 | 3p | 3 | 6 | | | | 2 | 3d | 5 | 10 | | | 4 | 0 | 4s | 1 | 2 | 32 | | | 1 | 4p | 3 | 6 | | | | 2 | 4d | 5 | 10 | | | | 3 | 4f | 7 | 14 | | Notice that the number of subshell in a shell is equal to the principal quantum number for the shell e.g n=3 contain 3 subshells. ## The Relative Energies of Atomic Orbitals The most important factor influencing the *energy of the orbital* is its size and therefore the value of the principal quantum number *n*, within a given shell, the *subshells gradually have the lowest energy*. The energy of the subshells gradually becomes larger as the value of the angular quantum number becomes *larger i.e s<p<d<f*. The diagram below depicts the energy of atomic orbitals. "The order of increasing energy of atomic orbitals can be predicted by simply following the arrows in this diagram (see the diagram on attached sheet). This diagram can also be used as a general method of filling electrons into the orbital. The diagram predicts the following order of increasing energy for atomic orbitals: 1s < 2s < 2p <3s<3p<4s< 3d< 4p<5s<4d< 5p<6s<4f<5d<6p<7s<5f<6d<7p<8s ## Aufbau Principle To predict the electron configuration of an atom, we need to consider the Aufbau principle. This principle assumes that electrons are added to an atom one at a time, starting with the lowest energy orbital until all the electron have been placed in an appropriate orbital. E.g | Atom | Number of electrons | Configuration | | ----- | ------------------- | -------------- | | H | 1 | 1s^1 | | He | 2 | 1s^2 | | Li | 3 | 1s^2 2s^1 | | Be | 4 | 1s^2 2s^2 | | B | 5 | 1s^2 2s^2 2p^1 | Other rules that guide the filling of atomic orbitals are: - **Hund's rule:** states that for degenerate orbitals, electrons occupy each level singly *before pairing*. - **Pauli Exclusion Principle:** states that in an atom or molecule, no two electrons can *have the same four electronic quantum numbers*.