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8 -- Part 2: Electrons in Atoms

- A beam of slow electrons is diffracted by a crystal of nickel.
- The same year, G. P. Thomson directed a beam of electrons at a thin metal foil.
- He obtained the same pattern for the X-rays of the same wavelength as for the electrons by aluminum foil.

- Thomson and Davisson won the physics prize in 1937.
- J. J. Thomson's son was George P. Thomson.

- Thomson won the physics prize in 1906 for his discovery of the electron.
- Thomson the son showed that the electron is a wave, while Thomson the father showed that the electron is a particle.

- Wave-particle duality is important when it is comparable to atomic or nuclear dimensions.
- Baseballs and automobiles are too small to measure, so the concept has little meaning when applied to them.
- The laws of classical physics are adequate for these objects.

- The laws of classical physics allow us to make predictions.
- We can calculate the exact point at which a rocket will land.

- The more precisely we measure the variables that affect the rocket's trajectory, the more accurate our calculation will be.
- There is no limit to the accuracy we can achieve.
- Physical behavior can be predicted with certainty in classical physics.

- The wavelength is calculated using equation (8.10).
- To use it, we have to collect the electron mass, the electron velocity, and Planck's constant, and then adjust the units so that they are expressed in terms of kilogram, m, and s.

- The mass of the electron is expressed in kilograms.

- The electron velocities are as follows: U is 0.
- 100 * c is 0.
- 100 * 108 m s-1 is 3.00

- The constant is 6.626 and it is 10-34 kg m2 s-2.

- The wavelength in meters can be obtained by converting the unit J to kg m2 s-2.

- The behavior of particles can be determined through hypothet ical experiments.
- The position of the particle 1x2 is one of the variables that must be measured.

- An experiment designed to locate the position of a particle with great precision can't also measure the momentum of the particle.

- If we know how a particle is moving, we can tell you where it is, but we can't tell you where he won the prize.

- One way to rationalize this result is to think of a superposition of many matter waves of different de Broglie wave.
- The lengths are suggested by Figure 8-16.
- The interference pattern caused by the superposition of many waves of dif can be seen here.

- The uncertainty principle is not easy for most people to accept.

- A collection of waves can be combined into a packet.
- The uncertainty in the resulting momentum is greater because each wavelength corresponds to a different value.

- A small range of wavelength contributes to the wave packet if the momentum is known.
- A wave packet that is not highly local is given by the superposition of waves of similar wavelength.
- The less we know about the particle's position, the less we know about the wave packet.

- The concept of wave-particle duality and the Heisenberg uncertainty princi ple have a profound influence on how we should think of an electron.
- An electron is neither a wave nor a particle.
- The more certain we are about some aspect of an electron's behavior, the less electric potential difference we have.

- A 12 eV electron can be shown to have a speed of 106 m>s.

- To convert an uncertainty to a fraction, we have to divide it by 100%.
- The uncertainty of the velocity is obtained by taking the number and dividing it by the actual speed.

- About 10 atomic diameters is the uncertainty in the electron's position.
- The uncertainty in the electron's speed makes it difficult to pin down its position.

- Superman is traveling at one-fifth the speed of light because of his mass.

- The mass of an electron is 1/2000th of the mass of a protons.

- electrons are matter waves and should show wavelike properties The Heisenberg uncertainty principle limits the precision in determining an electron's position.

- Waves on the ocean's surface travel great distances because of the wind.

- The wave travels along the entire length of the rope.
- An alternative form of a wave can be seen in the strings of the guitar.

- The string experiences up-and-down displacements with time, and they vibrate between the limits set by the blue curves.

- The wave at the fixed ends of the string is zero.
- By plucking it.

- The allowed wavelength of standing waves is between the tized and the permitted wavelength.

- A standing wave can represent the plucked guitar string.
- It is not appropriate to describe the electron in a hydrogen atom with a model that combines the particle and wave nature of the electron.
- The correct model for the hydrogen atom is based on a three-dimensional treatment.

- These patterns are two-dimensional cross-sections of a three-dimensional wave.
- A standing wave is an acceptable representation.
- It has a number of waves about the nucleus.
- The pattern is not acceptable.
- The number of wavelengths is nonintegral, and successive waves tend to cancel each other; that is, the crest in one part of the wave overlaps a trough in another part of the wave, and there is no wave at all.

- The wave function should correspond to a standing wave within the boundary of the system being described.

- A quantum particle confined to a onedimensional box, a line, is the simplest system for which we can write a wave function.
- The matter waves of a particle are represented by the wave function that looks like a string guitar.

- The wave function is a function.

- The waves were zero.
- The a particle in a one wave function and the sine function both reach their maximum values at one-fourth the length of the box.
- At the center of the box, both boxes are zero; the wave function has a node.

- The first three wave functions length, both functions reach their minimum values, and at and their energies are shown the farther end of the box, both functions are again zero.

- The wave function has a sign.

- The equation that gives the form of the wave function and the boundaries within which the quantum mechanical particle is confined is the answer to how we arrived at equation (8.13).
- The equation below is a differential equation if youTrademarkiaTrademarkiaTrademarkiaTrademarkiaTrademarkiaTrademarkiaTrademarkia, the equation below is a differential equation if youTrademarkiaTrademarkia,Trademarkia, the equation below is a differential equation ifTrademarkiaTrademarkia,Trademarkia, the equation below is a differential equation ifTrademarkiaTrademarkia,Trademarkia, the It describes a one-dimensional standing wave for a particle in a box.
- The wave function is the solution to this equation.

- By differentiating the wave function twice, we can get the wave function times a constant.
- Functions satisfy this requirement.
- There are two trigonometric functions that have this property.

- The two expressions we have for d2c>dx2 can be compared.

- Again, we identify a number.

- When x is 0 and when x is, L e must have c.

- We used a boundary condition of the system to help us choose the correct form of the wave function.
- It is a common procedure to solve quantum mechanical problems.

- The solution for the particle in a box is unacceptable.
- The function sin goes to zero at the edges of the box, but cos does not.

- The standing wave conditions described earlier for the standing waves of a guitar string need the matter wave's wavelength to fit.

- According to the uncertainty principle, our knowledge of the momentum must decrease if we decrease the size of the box.
- The particle can't be at rest because the zero-point energy isn't zero.
- There is nothing uncertain about a particle at rest because the position and momentum must be uncertain.

- German physicist Max Born answered the question in the year 1926.

- Born argued that the value of c2 is more important than the value of c itself.

- The wave function has a probability density of 5.

- There is no need for three quantum numbers.
- We can now discuss the solution to the quantum mechanical problem at the points where c2 is 0.

- The answer is yes.
- Consider the wave function for the lowest energy level of a particle in a box.
- We established in Are You Wondering 8-3 that cn(x) is a sin.

- The particle must be between 0 and L.

- The total probability of finding the particle between x and x is the sum of all the other probabilities and must be equal to 1.

- We have finished the derivation of equation (8.13) by using the Born interpretation.

- We have a 100% chance of finding an electron in the n level.
- There are five maxima in c2 at 15 pm, 45 pm, 75 pm, 105 pm, and 135 pm for a one-dimensional box 150 pm long.

- The position at 30 pm is in the wave function and there are four of them.

- There are five peaks in the c2 function and the total area between 0 and 30 pm is 20% of the total probability.
- We have a 20% chance of finding the particle between 0 pm and 30 pm.

- The particle is in the box in the n state.

- When we make a measurement, we'll find the particle on one side of the structure.
- We have a 20% chance of finding the particle between 0 and 30 pm, and the maximum chance occurs at 15 pm.

- A particle is in a one-dimensional box.

- The ground and excited states correspond to the same number.
- We can calculate the wavelength of the photon from the relationship.

- The electron mass is 9.109, the length of the box is 1.00, and the constant h is 6.626.

- If we needed the photon in kJ mol-1, we would have to use 6.022 as the energy of the photon.

- The wavelength of the photon emitted when an electron in a box falls from the n to the n is calculated.

- An electron is excited by a photon in a one-dimensional box from the ground state to the first excited state.
- Take the length of the box into account.

- A conceptual model for understanding the hydrogen atom, a simple system consisting of a single electron interacting with just one nucleus, will be developed using ideas from Section 8-5.
- This simple model system is one of the most important models in chemistry because it provides the basis for understanding multielectron atoms, the organization of elements in the periodic table, and the physical and chemical properties of the elements and their compounds.
- Concepts and terminology used throughout chemistry will be introduced as we explore this model.

- Section 8-5 has a few key ideas.
- The energy of the particle is quantized if it is confined to a one-dimensional box.
- The particle can only have certain quantities of energy.
- An interesting correlation between the energy of each state and the number of nodes in the associated wave function is found for the particle in a box.

- The electron in a hydrogen atom is confined by its attraction to the nucleus.
- It should come as no surprise that the energy of the hydrogen atom is quantized if we accept the basic idea that the electron in a hydrogen atom is "Confined" by its attraction to the nucleus.
- The allowed energies won't be the same as for the particle in a box, but they will be restricted to certain values.
- We should expect that the state of the electron will be characterized by quantum numbers and a wave function that can be analyzed to reveal important features.
- By the end of the next section, we will see that all of these assertions are true.

- The wave nature of the electron was incorporated into the equation for the hydrogen atom proposed by Schrodinger in 1927.
- The process of solving the equation is complex.
- We will use ideas from earlier sections to describe and interpret the solutions.

- The basic postulate of quantum mechanics is the Schrodinger equation.

- It is not possible to derive it from other equations.
- The form of the equation can be justified.

- The wavelength of a matter wave is the next step.

- The equation of a free particle moving in one direction.

- The particle's strength varies as it moves from point to point.

- This is the equation that was obtained.

- The solutions can be found in Table 8.2.

- The constants are defined in Table 8.1.

- The spherical polar coordinates mathematical form of these orbitals is more complex than the particle in a and Cartesian coordinates box, but they can be interpreted in a straightforward way.

- The coordinates x, y, and Z Wave functions are easy to analyze in terms of the three variables that are needed to define a point with respect to the nucleus.

- The coordinate system could be used to solve the equation.

- Since the hydrogen atom is a three-dimensional system, each orbital has three quantum numbers to define it.
- The functional forms R(r) and Y(u, f) are most conveniently represented in graphical form by the particular set of quantum numbers.
- In Section 8-8, we will use graphical representations of orbitals to better understand the description of electrons in atoms.

- The permittivity of vacuum P0 is 8.854187817 The Committee on Data for Science and Technology is called CODATA.

- The three quantum numbers are related to the solution of the wave equation for the hydrogen atom.

- The rules expressed in equations (8.17), (8.18), and (8.19) must be determined if the given set of quantum numbers is allowed.

- The physical significance of the various quantum numbers as well as the rules interrelating their values is important.

- The relationships among the quantum numbers give a logical organization of orbitals into shells and subshells.

- The first and second principal shells have the same number of points.
- The other quantum numbers and the principal quantum number have the same physical characteristics.

- The number of subshells in a principal electronic shell is the same as the number of allowed values.
- There is a single subshell in the first principal shell.
- The second principal shell 1n is 22 with the allowed values of 0 and 1, and the third principal shell is 32 with the allowed values of 1 and 1.
- There are at least two subshells in the principal shell with n and so on.
- The value of the quantum number affects the name given to a subshell.

- The number of orbitals in a subshell is the same as the number of allowed values.
- The total number of orbitals in a subshell is 2/ + 1.
- The names of the subshells are the same as the names of the orbitals.

- A combination of a number and a letter is used to designate the principal shell in which a given subshell is found.

- The quantum numbers are n, 2, m and 0.

- The type of orbital is determined by the number.
- The designation is 4d because n is 4.

- We need to memorize the quantum number rules in order to solve this problem.
- In the later chapters, this information will be important.

- The quantum numbers n, m, and 1 are related to the orbital designation.

- Write all the combinations of quantum numbers that define hydrogen-atom orbitals with the same energy as the 3s.

- There are shells and subshells for the hydrogen atom.

- The subshells are made up of orbitals.

- All the subshells within a principal electronic shell have the same energy.

- The answer is no.
- As a result of the atom absorbing or emitting a photon, the state of the electron in the hydrogen atom may change.

- There are other rules that must be obeyed.
- The selection rules must also be obeyed.
- The selection rules are summarized.

- If the spectrum is measured in the presence of a magnetic field, the restriction for C/m/ applies.

- We won't try to justify the selection rules except to say that they arise from the fact that a photon carries a certain quantity of energy and one unit of momentum.
- When an atom absorbs or emits a photon, the energy of the atom changes, but the angular momentum of the atom also increases or decreases by one unit.
- The selection rules don't allow for a transition like this.

- Section 8-8 requires a fourth quantum number, ms, to describe an electron.
- The value of ms doesn't change when a photon is absorbed or emitted, according to the selection rule.

- The three-dimensional probability density distributions obtained for the various orbitals in the hydrogen atom will be our major undertaking in this section.
- The probability densities of the hydrogen atom's orbitals will be represented through the Born interpretation of wave functions.
- The shape of the probability density for each type of orbital will be shown.
- Even though we will provide some additional quantitative information about orbitals, your primary concern should be to acquire a broad qualitative understanding.
- You can apply this understanding in our discussion of how orbitals enter into a description of chemical bonding.

- In Chapter 11, we will matical solutions of the Schrodinger wave equation, remember that orbitals are wave functions.

- The square of the wave function is a basis for discussing quantity that is related to probabilities.
- Bonding between atoms is based on probability density distributions.

- Table 8.2 shows the forms of the radial wave function R1r2 and the angular wave function Y1u, f2 for a one-electron, hydrogen-like atom.

- All types 1s, p, d, f2 have the same behavior.
- The names given to the parts are related to their functional forms.
- The equations apply to any one-electron atom, that is, a hydrogen atom or a hydrogen-like ion.
- The term s is equal to 2Zr>na0 in the table.

- This distance is the lowest energy in the model.
- The name commemorates the work of a pioneer.

- To get the wave function for a particular state, we simply divide the radial part by the angular part.

- The radial function crosses the horizontal axis 1 times before decaying to a value of zero.

- The R(r) is the function of the hydrogen atom having n being 1, 2, or 3.
- The number of times R(r) crosses the horizontal axis is the same as the number of radial nodes for a given orbital.
- For s orbitals, R(r) has a maximum value of 0, whereas other orbitals have a maximum value of 0.

- The main features come from the math ematical forms of the radial functions.
- The radial functions decay to a value of zero because the exponential factor e-s>2 appears in all of them.
- The number of radial nodes and the value of the radial function at the nucleus are determined by factors.
- Each radial function crosses the horizontal axis this number of times, because of a polynomial of order n - / - 1 that crosses the horizontal axis up to n - / - 1 times.
- When it's 0, it's 0, except when it's 0.

- The main features of the radial functions have been rationalized.
- We need to consider the precise forms of these functions.

- The polar graphs will be used to plot the functions.
- The magnitude of the function at a particular value of the angles is given in a polar graph.
- The planes selected for the figure show the shapes of the functions.
- Let's take a closer look at the shapes of the wave functions.

- The function has the same value for both values.

- A sphere is the polar graph of this function.

- Although the mathematical forms of these functions are different, their polar graphs show that they are the same in shape and orientation.

- The pz orbital's function is proportional to cos U.
- The phase of the orbital is an important consideration when developing models for describing chemical bonding.

- The mathematical forms of the functions can be seen in Table 8.2.
- The number of nodes is the same as the value.

- Let's look at the function of the dx2y2 orbital.
- The function is proportional to sin2u cos 2f.
- The function cos 2f can be plotted as a polar graph.

- The phase is positive for two of the lobes and negative for the other two.
- We can either move clockwise or counterclockwise from one part to another.

- The functions of the s, p, and d are shown.
- The magnitude of the function for a given value is determined by the distance from the origin to a point on the curve.
- The colors blue and red are used to tell if the function has a positive or negative value.

- Four of them have the same shape, but they are different with respect to the axes.
- The dxy, dxz, and dz2 orbitals each have two nodal planes.

- The dz2 is a different shape and has two different types of nodes.
- The dz2 orbital has conical surfaces.

- We will not consider their shapes because they are not often seen.

- The complete wave function is given by the product of a radial function and an angular function.
- The radial function is shown in red while the angular function is shown in blue.

- We can project the three-dimensional surface onto a two-dimensional map.
- The points are joined by the circular lines.
- There is a large (positive) value for the contours close to the nucleus.
- The lower value is for the contours farther away.

- The highest density of points can be found in a graph with the largest values.

- The value of c is represented by the height above the xy plane, which is an arbitrary choice.
- The iso surface is called an iso surface for this reason.
- The density of points is highest when the magnitude of c is large.

- The surface shows the variation of probability density.
- Increasing distance from the nucleus decreases the probability density.

- High-energy standing waves are characterized by the fact that the number of nodes increases as the energy increases.

- Let's take a look at the wave function.

- When we consider multielectron atoms, this difference will be important.

- The xy plane is plotted as a distance above or below the value of c. The nucleus is thought to be at the beginning at x and y.
- The colors are used to show the regions with either a positive or a negative value.

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