Quantum Theory and Structure of Atoms (Full lecture), Lecture notes of Inorganic Chemistry

Download Quantum Theory and Structure of Atoms (Full lecture) and more Lecture notes Inorganic Chemistry in PDF only on Docsity! Inorganic Chemistry CHEM 110 Western Mindanao State University (WMSU) Note about: Quantum Theory and Structure of Atoms Objectives: At the end of this session, you can: 1. Trace the development of quantum theory 2. Describe Quantum theory 3. Discuss the properties of waves 4. Tell what is electromagnetic radiation 5. Discuss the Plank’s theory 6. Describe photoelectric effect Historical development of the Quantum theory: Classical Model ▪ What is quantum? Quantum is the Latin word for amount and it means the smallest possible discrete unit of any physical property, such as energy or matter. ▪ 19th century early attempts by physicists to understand atoms were met with limited success ▪ they assume that molecules behave like rebounding balls, ▪ physicists were able to predict and explain some macroscopic phenomena, e.g., pressure exerted by gas. ▪ limitation of this model- did not account for the stability of molecules, i.e., could not explain the forces that hold atom together ▪ years to realize and accept- that the properties of atoms and molecules not governed by the same physical laws as larger objects. ▪ 1900 - new era in physics started. Planck while analyzing the data on radiation emitted by solids heated at various temperature, discovered that atoms and molecules emit energy only in certain discreet quantities or “quanta” and ▪ physicists had always assumed that energy is continuous an it any amount of energy could be released in a radiation process. Niels Bohr and Max Planck, are two of the founding fathers of Quantum Theory, each received a Nobel Prize in Physics for their work on quanta. Max Planck study the effect of radiation on a “blackbody” substance, and the quantum theory of modern physics is born. o Planck demonstrated that energy, in certain situations, can exhibit characteristics of physical matter. Bohr’s - the atomic model: It shows the atom as a small, positively charged nucleus surrounded by orbiting electrons. o theory of atomic structure in which the hydrogen atom (Bohr atom) is assumed to consist of a proton as nucleus with a single electron moving in distinct circular orbits around it. Each orbit corresponding to a specific quantized energy state: the theory was extended to other atoms. It is commonly known that light somehow consists both of light waves and also particle-like photons. The notion of these photons comes from quantum theory (and from Einstein directly, who first introduced them in 1905 as “light quanta”). o Einstein is considered the third founder of Quantum Theory because he described light as quanta in his theory of the photoelectric effect, for which he won the 1921 Nobel Prize. Quantum theory o 1905, Albert Einstein explained the photoelectric effect by postulating that light, or more generally all electromagnetic radiation(emr), can be divided into a finite number of “energy quanta” that are localized points in space. They are collectively known as the quantum theory. https://www.uib.no/en/hms- portalen/75292/electomagnetic-spectrrum = 5.75 × 1014 𝑠 or 5.75 × 1014𝐻𝑧 This means that 5.75 × 1014𝐻𝑧 waves passed a fixed point every second, which is in accordance to the speed of light. Figure 2 shows ▪ long radio waves are emitted by large antennas- broadcasting stations ▪ shorter visible light waves are produced by the motions of electrons within atoms and molecules. ▪ shortest waves, which also have the highest frequency, are associated with gamma (γ) rays, Figure 2 which result from changes within the nucleus of the atom. Note: the higher the frequency, the more energetic the radiation. ▪ UV radiation, X-rays and γ are high-energy radiation. Planck’s Quantum Theory Light bulb (tungsten) - emits bright white light. Electric heater - emits dull red. Measurement taken in the latter part of the 19th century, showed that the amount of the radiation energy emitted by an object at a certain temperature depends on its wavelength. But, dependence on its wavelength whether its short or long became a problem and they could not account and they surmise that a bask thing missing from the laws of classical physics. ▪ Planck solved the problem, where he said that atoms and molecules could emit (or absorb) energy only in discrete quantities (like small packages or bundles). ▪ named quantum to the smallest quantity of energy that can be emitted (or absorb) in the form of electromagnetic radiation. ▪ The energy 𝐸 of a single quantum of energy is expressed as: 𝐸 = ℎ𝑣 where: ℎ is the Planck's constant 6.63 × 𝑙𝑂−34𝐽. 𝑠, 𝑣 is the frequency of the radiation ▪ Based on quantum theory, energy is always emitted in multiples of ℎ𝑣. Example, 1ℎ𝑣, 2ℎ𝑣,3ℎ𝑣... but cannot be, for example, 1.67ℎ𝑣 or 4.98 ℎ𝑣. The Photoelectric effect what is it all about? 1905, Albert Einstein used this theory to solve another breakthrough in physics. ▪ A phenomenon in which electrons are ejected from the surface of certain metals exposed to light of at least a certain minimum frequency, called the threshold frequency. ▪ The # of electrons ejected was proportional to the intensity (or brightness) of the light, but the energies of the ejected electrons were not. ▪ Below the threshold frequency no electron were ejected no matter how intense the light is. ▪ The photoelectric effect could not be explained by the wave theory (light behaves like a wave) of light. ▪ Einstein, however, made an assumption, and suggested that a beam of light is really a stream of particles, and these particles of light is called photons. ▪ Using Planck theory of radiation, Einstein deduced that each photon must possess energy 𝐸, given by the equation 𝐸 = ℎ𝑣, where 𝑣 is the frequency of light. ▪ Consider a metal, if you shine a beam of light onto a metal surface, you can think of a shooting a beam or particles-photons — at the metal atoms. ▪ So if the 𝑣 of photons is such that hv is exactly equal to the energy that binds the electrons in the metal, then the light will have Just enough energy to knock the electron loose. ▪ If we use a light of higher frequency, then only the electrons will be knocked loose, but they will also acquire some 𝐾𝐸, which can be summarized as follows: ℎ𝑣 = 𝐾𝐸 + 𝐵𝐸 where 𝐾𝐸 is the kinetic energy of the ejected electron and 𝐵𝐸 is the binding energy of the electron in the metal. Rearranging the above equation: ▪ 𝐾𝐸 = ℎ𝑣— 𝐵𝐸 ▪ this equation tells you, that the more energetic the photon (i.e., the higher its frequency) the greater the 𝐾𝐸 of the ejected electron. ▪ Or the more intense the light, the greater the number of electrons is emitted by the target metal; the higher the frequency of the light, the greater the 𝐾𝐸 of the ejected electrons. Exercise: From the equation discussed above, solve the ff: ▪ Calculate and compare the energy in joules of a) a photon with a 𝜆 of 5.00 × 10−4𝑛𝑚 (Infrared region) and b) a photon with a 𝜆 5.00 × 10−2𝑛𝑚 (x-ray region). a) 𝐸 = ℎ𝑣 and 𝑣 = 𝑐/𝜆 𝐸 = ℎ × 𝑐/𝜆 = (6.63 × 10−34𝐽. 𝑠) × 3.00 × 108𝑚/𝑠 5.00 × 10−4𝑛𝑚 × 1.0 × 10−9𝑚/1𝑛𝑚 = 3.97 × 10−21𝐽 (Infrared) b) 𝐸 = ℎ𝑣 and 𝑣 = 𝑐/𝜆 𝐸 = ℎ × 𝑐/𝜆 = (6.63 × 10−34𝐽. 𝑠) × 3.00 × 108𝑚/𝑠 5.00 × 10−2𝑛𝑚 × 1.0 × 10−9𝑚/1𝑛𝑚 = 3.97 × 10−17𝐽 (x-ray) Bohr’s Theory of the Hydrogen Atom Objective/s: At the end of the discussion, you can: 1. describe the Bohr’s theory of Hydrogen atom 2. tell what line emission spectra are 3. discuss the emission spectrum of the hydrogen atom 4. Elucidate Bohr’s quantum theory In 1913, after Planck and Einstein, a theoretical explanation of the emission spectrum of the H atom was presented by Bohr. Recall the structure of an atom (protons, neutrons and electrons). In Bohr’s theory of the Hydrogen atom includes the following: 1. Electrons moves around the nucleus in circular orbits. 2. Electron’s motion in the permitted orbits must be fixed in value or quantized. 3. The emission of energy by an energized H atom to the electron dropping from a higher energy-orbit to a lower one and giving up of a quantum energy (a photon) in the form of light. The figure below shows the line emission spectra of hydrogen atom. Based on electrostatic attraction and Newton’s laws of motion, Bohr showed that the energies that the electron in the H atom can possess are given by: Eq. 5.1 𝐸𝑛 = −𝑅𝐻(𝑙/𝑛), where 𝑅𝐻, the Rydberg constant 2.18 × 10−18𝐽. 𝑛 is an integer called the principal quantum number, it has the values 𝑛 1,2,3… the negative sign is an arbitrary convention, signifying that the energy of electron in the atom is lower than energy of the free electrons — electrons that are infinitely far from the nucleus. ▪ Energy of free electrons is arbitrarily assigned 0 value of zero. 𝐸 = 0 ▪ As the electron gets closer to the nucleus (as 𝑛 decreases), 𝐸𝑛 becomes larger in absolute value, but also more negative. ▪ The most negative value, then, is reached when 𝑛=1, which correspond to the most stable energy State — ground state, which refers as the lowest energy state of a system (H). https:llwww.khanacademy.org/science/class11chemistryindia/xfbb6cb8fc2b d00c8:ininstructur-of-atom/xfbb6cb8fc2bd00c8:in- inbohrsmodelofhydrogenatom/a/absorptionemission-lines Dual Nature of atoms Objectives: At the end of the session, you can: 1. tell what is wave theory 2. describe how De Broglie explained the dual nature of electron. 3. Perform calculation involving wavelengths of a particle Recall: Bohr’s theory! Physicists were puzzled and question Bohr’s theory, that is, why is the electron in a H atom restricted to orbiting the nucleus at certain fixed distances? Or simply why the energies of the H atom are quantized? ▪ Bohr could not offer an explanation of his theory. ▪ 1924, de Brogue offered an explanation to this query, that is, ▪ If light waves can behave like a stream of particles (photons), then maybe particles like electrons can possess wave properties. Broglie avers that an electron bound to the nucleus behaves like standing waves (generated by plucking a guitar string) called as such because such waves do not travel along the string. ▪ Nodes1 points on the string do not move at all, that is the amplitude of the wave at this point is zero. (exists at each end or may be between the end). ▪ The greater the 𝑣 of vibration, the shorter the 𝜆 of the standing wave and the greater the number of nodes. ▪ de Broglie argued that if an electron does behave like a standing wave in the H atom, the length of the wave must fit the circumference of the orbit exactly, or otherwise the wave would partially cancel itself on each successive orbit. Eventually the amplitude of the wave would be reduced to zero, and the wave would not exist. ▪ He further said that the relation between the circumference of an allowed orbit (2𝜋𝑟) and the 𝜆 of the electron is expressed as: 2𝜋𝑟 = 𝑛 𝜆 where: r is the radius of the orbit 𝜆 is the wavelength of the electron wave 𝑛 is 1,2,3 ... ▪ Because n is an integer, it follows that r can hove only certain values as n increases from 1 to 2 to 3... ▪ Because the energy of the electrons depends on the size of the orbit (value of r), its value must be quantized. ▪ This reasoning led de Broglie to conclude that waves can behave like particles and particles can exhibit wavelike properties. ▪ de Brogue deduced that the particle and wave properties are related by the expression 𝜆 = ℎ/𝑚𝑢 Eqn. 5.3 where: 𝜆 = wavelengths 𝑚 = mass 𝑢 = velocity Eqn 5.3 tells us that a particle in motion can be treated as a wave, and a wave can exhibit the properties of a particle Answer this. Using Eqn. 5.3, calculate the 𝜆 of the " particle" in the following two (2) cases. a) the fastest serve in tennis s about 140 miles/hour, or 63m/s. Calculate the 𝜆 associated with a 6.0 𝑥10−2 𝑘𝑔 tennis ball travelling at this speed, b) Calculate the 𝜆 associated with an electron (9.1094 𝑥 1031𝑘𝑔) moving at 63m/s. Quantum Mechanics (or Wave Mechanics) Objectives: At the end of this topic, you can: 1. describe the loopholes In Bohr’s theory; 2. tell the event that led to the quantum mechanics; 3. define quantum mechanics; 4. Discuss Heisenberg Uncertainty principle. Weaknesses of Bohr’s theory 1. Did not account for the emission spectra of atoms containing more than one electron(e.g., He and Li) 2. Did not explain why extra lines appear in the H emission spectrum when a magnetic field is applied. 3. Another problem arise when the wave-like properties of electrons was discovered, that is the “position” of a wave be specified. Strength: He theorized that energy of an electron in an atom is quantized (which remains unchallenged) Heisenberg Uncertainty Principle To solve the problem of the position or location of electron that behaves like a wave- ▪ Werner Heisenberg formulated this idea known as the Heisenberg Uncertainty Principle that states: it is impossible to know simultaneously both the momentum 𝑝 (𝑚 𝑥 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦) and the position of a particle with certainty. Mathematically expressed as: 𝛥𝑝 𝛥𝑝 ≥ ℎ/4𝜋𝑟 Eqn.5.4 where: Δ x and Δp are the uncertainties in measuring the position and momentum. ▪ Eqn. 5.4 tells us that if we make Δp a small quantity- our knowledge of the position will become correspondingly less precise ( that is Δx will become larger). In like manner, if the position of the particle is known more precisely, then its momentum measurement must be less precise. ▪ If we apply Heisenberg uncertainty principle to H atom, it is observed in reality the electron does not orbit in the nucleus in a well-defined path, as what Bohr thought. ▪ If it did, we could determine both the position of the electron (from the radius of the orbit) and its momentum (from its k.e.) at the same time a violation of the uncertainty principle. Erwin Schrödinger ▪ 1926 Erwin Schrödinger using a complicated mathematical technique formulated an equation that describes the behavior and energies of submicroscopic particles in general, an equation analogous to Newton’s laws of motion for macroscopic objects. ▪ His equation incorporates both particle behavior, in terms of mass m, and wave behavior in terms of a wave function Ψ (psi), which depends on the location in space of the system (such an electron in an atom). In his mind, the wave function itself has no direct physical meaning ▪ However, in his equation the probability of finding the electron in a certain region in space is proportional to the square of the wave function Ψ2 Where did this idea come from? ▪ The idea relating Ψ2 to probability got from wave theory. ▪ Wave theory stipulates that intensity of light is proportional (α) to the square of the amplitude of the wave of Ψ2. What does this mean? ▪ this means that the most likely place to find a photon is where the intensity is greatest, that is, where the value of theΨ2 is greatest. ▪ Or you can associate Ψ2 with the likelihood of finding an electron in regions surrounding a nucleus. ▪ Schrödinger’s equation launched a new field, known as quantum mechanics. The Quantum Mechanical Description of the Hydrogen atom ▪ Based on quantum mechanics we cannot pinpoint an electron in an atom, it does define the region where the electron might be at a given time. ▪ Electron density gives the idea of the probability that an electron will be found in a particular region of an atom. ▪ the square of the wave function, 1412 defines the distribution of electron density in 3D space around the nucleus. Regions of high electron density represent high probability of locating an electron, whereas the opposite holds for regions of low electron density ▪ The quantum mechanical model of H atom (Bohr’s model) we refer to atomic orbital not an orbit. ▪ An atomic orbital is perceived to be as the wave function of an electron in an atom. ▪ When an electron is in a certain orbital, it means that the distribution of the electron density or the probability of finding the electron in space is described by the square of the wave function associated with that orbital. ▪ Therefore, an atomic orbital has a characteristic energy, as well as a characteristic distribution of electron density. Quantum Numbers Objectives: At the end of the topic discussion, you can: 1. tell what quantum numbers are 2. describe each quantum numbers 3. describe the shapes of s, p, d, f and 4. relate these shapes to quantum numbers. Quantum Numbers In quantum mechanics, there are three(3) quantum numbers are needed to describe the distribution of electrons in hydrogen and other atoms. ▪ These are derived from Schrödinger’s equation for the H atom. ▪ These are the: 1) principal quantum number (𝑛) 2) angular momentum quantum number (𝐼) 3) magnetic quantum numbers (𝑚𝑙) 1. The principal quantum number (𝑛) can have an integral value 1,2,3 ... ▪ It also relates to the mean distance of the electron from the nucleus in a particular orbital. ▪ The larger the 𝑛 is, the greater the average distance of an electron in the orbital from the nucleus and therefore the larger the orbital.