Chem Ch 6
What are the possible values of mℓ for an electron in a d orbital?
-2,-1,0,1,2 Since the allowed values for mℓ range from −ℓ to +ℓ, once you know the value for ℓ you know the values for mℓ.
The electron configuration for Si is 1s22s22p63s23p2. From this method of writing the electron configuration, you cannot predict the number of unpaired electrons. You can write the electron configuration as an orbital diagram as (see picture)...
...where each orbital is denoted by a box and each electron is denoted by a half arrow.
What is the only possible value of mℓ for an electron in an s orbital?
0 Since the allowed values for mℓ range from −ℓ to +ℓ, once you know the value for ℓ you know the values for mℓ.
Identify which sets of quantum numbers are valid for an electron. Each set is ordered (n,ℓ,mℓ,ms).
1,0,0,1/2 4,3,2,1/2 2,1,1,1/2 3,2,2,1/2 2,1,-1,1/2
What is the de Broglie wavelength of a 1.22 × 106 g truck that is moving at 105 km/hour ?
1.86 × 10−38m
What is the maximum number of electrons that a d subshell can hold?
10 electrons
Give the complete ground-state electron configuration for silicon (Si).
1s^22s^22p^63s^23p^2
Cu has an anomalous electron configuration. Write the observed electron configuration of Cu.
1s^22s^22p^63s^23p^64s^13d^10
Give the actual ground-state electron configuration for copper (Cu) using the complete form.
1s^22s^22p^63s^23p^64s^13d^10 The expected ground-state electron configuration of copper is 1s22s22p63s23p64s23d9; however, the actual configuration is 1s22s22p63s23p64s13d10 because a full d subshell is particularly stable. There are 18 other anomalous elements for which the actual electron configuration is not what would be expected.
Give expected ground-state electron configurations for the following atoms. Zn (Z = 30)
1s^22s^22p^63s^23p^64s^23d^10
Give expected ground-state electron configurations for the following atoms. Sn (Z = 50)
1s^22s^22p^63s^23p^64s^23d^104p^65s^24d^105p^2
Give expected ground-state electron configurations for the following atoms. Pb (Z = 82)
1s^22s^22p^63s^23p^64s^23d^104p^65s^24d^105p^66s^24f^145d^106p^2
Give expected ground-state electron configurations for the following atoms. Ti (Z = 22)
1s^22s^22p^63s^23p^64s^23d^2
For the electronic transition from n = 3 to n = 8 in the hydrogen atom, calculate the energy.
2.08 × 10−19 J
With some manipulation, the Rydberg equation can be rewritten in the form E=constant×((1/nf²)−(1/ni²)) which allows you to calculate the energy of the emitted light. What is the value of the constant needed to complete this equation?
2.18×10−18 J
The biological effects of a given dose of electromagnetic energy generally become more serious as the energy of the radiation increases: Infrared radiation has a pleasant warming effect; ultraviolet radiation causes tanning and burning; and X rays can cause considerable tissue damage. What energies in kilojoules per mole are associated with the following wavelengths: X rays with λ = 4.73 nm ?
2.53×104 kJ/mol
How many photons of frequency 1.50 × 1014 s−1 are needed to give 30.1 J of energy?
3.03 × 10^20 photons
Which of the following set of quantum numbers (ordered n, ℓ, mℓ, ms) are possible for an electron in an atom?
4, 2, -1, -1/2 5, 2, 1, -1/2 4, 2, 1, -1/2 The allowed values for mℓ range from −ℓ to +ℓ and the allowed values for ℓ are integers between zero and n−1. Once you know the value for n, you can determine the acceptable ℓ values, and from there the acceptable mℓ values. The ms values are fixed at either 1/2 or −1/2.
The biological effects of a given dose of electromagnetic energy generally become more serious as the energy of the radiation increases: Infrared radiation has a pleasant warming effect; ultraviolet radiation causes tanning and burning; and X rays can cause considerable tissue damage. What energies in kilojoules per mole are associated with the following wavelengths: ultraviolet light with λ = 231 nm ?
518 kJ/mol
How many electrons can an n = 6 shell theoretically hold?
72 electrons
The biological effects of a given dose of electromagnetic energy generally become more serious as the energy of the radiation increases: Infrared radiation has a pleasant warming effect; ultraviolet radiation causes tanning and burning; and X rays can cause considerable tissue damage. What energies in kilojoules per mole are associated with the following wavelengths: infrared radiation with λ = 1.62×10−6 m ?
73.9 kJ/mol
How many orbitals are there in the third shell (n=3)?
9 Nine orbitals (one s, three p, and five d) can hold a maximum of 18 electrons.
The Heisenberg Uncertainty Principle
A student is examining a bacterium under the microscope. The E. coli bacterial cell has a mass of m = 1.60 fg (where a femtogram, fg, is 10−15g) and is swimming at a velocity of v = 8.00 μm/s , with an uncertainty in the velocity of 9.00 % . E. coli bacterial cells are around 1 μm ( 10−6 m) in length. The student is supposed to observe the bacterium and make a drawing. However, the student, having just learned about the Heisenberg uncertainty principle in physics class, complains that she cannot make the drawing. She claims that the uncertainty of the bacterium's position is greater than the microscope's viewing field, and the bacterium is thus impossible to locate.
The Rydberg Equation
An astrophysicist working at an observatory is interested in finding clouds of hydrogen in the galaxy. Usually hydrogen is detected by looking for the Balmer series of spectral lines in the visible spectrum. Unfortunately, the instrument that detects hydrogen emission spectra at this particular observatory is not working very well and only detects spectra in the infrared region of electromagnetic radiation. Therefore the astrophysicist decides to check for hydrogen by looking at the Paschen series, which produces spectral lines in the infrared part of the spectrum. The Paschen series describes the wavelengths of light emitted by the decay of electrons from higher orbits to the n=3 level.
Electron Configurations of Atoms
An atom consists of a small, positively charged nucleus, surrounded by negatively charged electrons. We organize the electrons in a logical manner. As the atomic number increases, electrons are added to the subshells according to their energy. Lower energy subshells fill before higher energy subshells. The order of filling is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. The periodic table can be used to help you remember the order.
Orbital-Filling Diagrams
An orbital-filling diagram shows the number of electrons in each orbital, which are shown in order of energy. The placement of electrons in orbitals follows a certain set of rules. Lower energy subshells fill before higher energy subshells. The order of filling is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. The periodic table can be used to help you remember this order. An orbital can hold up to two electrons, which must have opposite spins. Hund's rule states that if two or more orbitals with the same energy are available, one electron goes in each until all are half full. The electrons in the half-filled orbitals all have the same value of their spin quantum number.
Calculate in kilojoules per mole the energy necessary to completely remove an electron from the first shell of a hydrogen atom (R∞ = 1.097×10^−2nm^−1).
E = 1310 kJ/mol
What is the energy in kilojoules per mole of photons corresponding to the shortest-wavelength line in the series of the hydrogen spectrum when m=1 and n>1 ?
E = 1310 kJ/mol
Atomic Radii
Each element in the periodic table has a distinctive atomic radius.
The Photoelectric Effect
Electrons are emitted from the surface of a metal when it's exposed to light. This is called the photoelectric effect. Each metal has a certain threshold frequency of light, below which nothing happens. Right at this threshold frequency, an electron is emitted. Above this frequency, the electron is emitted and the extra energy is transferred to the electron. The equation for this phenomenon is KE=hν−hν0 where KE is the kinetic energy of the emitted electron, h=6.63×10−34J⋅s is Planck's constant, ν is the frequency of the light, and ν0 is the threshold frequency of the metal. Also, since E=hν, the equation can also be written as KE=E−E0 where E is the energy of the light and E0 is the threshold energy of the metal.
How much energy does the electron have initially in the n=4 excited state?
En = −1.37×10−19 J
Hospital X-ray generators emit X-rays with wavelength of about 15.0 nanometers (nm), where 1nm=10−9m. What is the energy of a photon of the X-rays?
Ephoton = 1.33×10−17 J
Green light has a frequency of about 6.00×1014s−1. What is the energy of a photon of green light?
Ephoton = 3.98×10−19 J
Quantum Numbers
Every electron in an atom is described by a unique set of four quantum numbers: n, ℓ, mℓ, and ms. The principal quantum number, n, identifies the shell in which the electron is found. The angular momentum quantum number, ℓ, indicates the kind of subshell. The magnetic quantum number, mℓ, distinguishes the orbitals within a subshell. The spin quantum number, ms, specifies the electron spin.
Identify the atom with the following ground-state electron configuration:
Fe
Which type of electromagnetic radiation has the greatest energy?
Gamma rays
What might the photon from Part C (slide above) be useful for?
Getting a suntan
What is the kinetic energy of the emitted electrons when cesium is exposed to UV rays of frequency 1.9×1015Hz?
KE = 6.37×10−19 J
In what period and group on the periodic table, respectively, is the element with the following electron configuration located and what is the element's identity? [Kr]5s 24d 105p 4
Period 5, group 6A, tellurium (Te)
The work function of iron metal is 451 kJ/mol. Will photons of violet light with = 390 nm cause electrons to be ejected from a sample of iron?
Photons with λ = 390 nm won't eject electrons from a sample.
Quantum Number Rules
Quantum numbers can be thought of as labels for an electron. Every electron in an atom has a unique set of four quantum numbers. The principal quantum number n corresponds to the shell in which the electron is located. Thus n can therefore be any integer. For example, an electron in the 2p subshell has a principal quantum number of n=2 because 2p is in the second shell. The azimuthal or angular momentum quantum number ℓ corresponds to the subshell in which the electron is located. s subshells are coded as 0, p subshells as 1, d as 2, and f as 3. For example, an electron in the 2p subshell has ℓ=1. As a rule, ℓ can have integer values ranging from 0 to n−1. The magnetic quantum number mℓ corresponds to the orbital in which the electron is located. Instead of 2px, 2py, and 2pz, the three 2p orbitals can be labeled −1, 0, and 1, but not necessarily respectively. As a rule, mℓ can have integer values ranging from −ℓ to +ℓ. The spin quantum number ms corresponds to the spin of the electron in the orbital. A value of 1/2 means an "up" spin, whereas −1/2 means a "down" spin.
Anomalous electron configurations
Some atoms, such as some transition metals and some elements in the lanthanide and actinide series, do not adhere strictly to Hund's rule and Pauli's principle. The reason the anomalies are observed is the unusual stability of both half-filled and completely filled subshells. This behavior can be explained with an example of the chromium atom. Using Hund's rule and Pauli's principle, you can write the expected electron configuration of the Cr atom that strictly follows these rules as 1s22s22p63s23p64s23d4 . However, by moving an electron from the 4s orbital to the 3d orbital you obtain a half-filled 3d orbital. This half-filled orbital is more stable than the combination of the filled 4s orbital and the partially filled 3d orbital. Thus, the observed electron configuration of the Cr atom is 1s22s22p63s23p64s13d5.
Identify the specific element that corresponds to the following electron configuration: [Kr]5s 24d 105p 4.
Te
Electron configurations which of the following elements are anomalous?
Th Mo Cu Ag U Pt
In the animation, you can see that the electrons occupy different orbitals according to the energy level of each orbital. A single box represents an orbital. The unpaired electron is represented assingle harpoon upwhereas the paired electrons in the same orbital are represented by two arrows pointing in opposite directions:single harpoon up and single harpoon down. Watch the animation and identify which of the following statements are correct.
The C atom has two unpaired electrons. In the Li atom, the 3s, 3p, and 3d orbitals have different energies. Electrons generally occupy the lowest energy orbital first. The arrangement of the orbitals in a multielectron atom is different from the arrangement in a single-electron atom owing to the electron-electron repulsions in a multielectron atom. In the case of a single-electron atom, the orbitals in a given principal shell have the same energy. However, in the case of a multielectron atom, the orbitals in a given principal shell have different energies. Electrons occupy the lowest energy orbital first. Each orbital can hold a maximum of two electrons of opposite spins. When more than one orbital of equal energy is available, electrons will first occupy these orbitals singly with parallel spins. Thus, the C atom has two unpaired electrons in its 2p subshell. The element that follows C is N. It has three unpaired electrons in the 2p subshell. In the N atom, all the three degenerate 2p orbitals are filled with single-electrons each. Thus, it has attained half-filled orbitals. For the next atom, oxygen, the pairing of electrons will occur. The filling of the electrons in the different orbitals of an atom determines the electron configuration of the atom and indicates the presence or absence of unpaired electrons in the atom.
Emission Line Energy
The Rydberg equation expresses the wavelength, λ, of emitted light based on the initial and final energy states (ni and nf) of an electron in a hydrogen atom:
Rules for writing electron configuration
The electron configuration of an atom describes how the electrons fill the orbitals within an atom. Two of the rules that explain how electrons fill orbitals are as follows: Hund's rule of maximum multiplicity states that when more than one orbital of equal energy is available, electrons will first occupy these orbitals singly with parallel spins. The pairing of electrons will start only after all the degenerate orbitals are singly occupied or are half-filled. Pauli's exclusion principle states that each orbital can hold a maximum of two electrons of opposite spins. For example, the electron configuration of a C atom with the atomic number 6 is (see picture) Here, a single box represents an orbital, and an electron is represented as a half arrow. Orbitals of equal energy are grouped together. According to Pauli's exclusion principle, each orbital can hold a maximum of two electrons of opposite spins. If you observe the electron configuration of the carbon atom, 1s and 2s orbitals hold two electrons of opposite spins. The fifth and sixth electrons enter the 2p orbital. Because the 2p subshell has three orbitals of equal energy, according to Hund's rule, the fifth and sixth electrons occupy 2p orbitals singly with parallel spins instead of pairing.
The Bohr Equation
The electron from a hydrogen atom drops from an excited state into the ground state. When an electron drops into a lower-energy orbital, energy is released in the form of electromagnetic radiation.
Outer electron configurations
The outer electron configuration of an element includes everything except the noble-gas core. For example, the element C has an electron configuration of [He]2s22p2 and an outer electron configuration of 2s22p2. Similarly, the element Pb has an electron configuration of [Xe]6s24f145d106p2 and an outer electron configuration of 6s24f145d106p2.
Relating Quantum Numbers and Electron Configurations to the Periodic Table
The periodic table lists all known elements arranged by atomic number. Atomic number is the nuclear charge, the number of protons in the nucleus of an an atom of a particular element. For a neutral atom, the number of protons is equal to the number of electrons. Each column of the table, called a group, contains elements with the same number of valence electrons that are in different quantum levels. Each row of the table, called a period, contains elements with differing numbers of valence electrons that are in the same principal quantum level. The four main blocks of the table (s, p, d, and f) contain elements whose highest energy electrons have the same azimuthal quantum number (ℓ).
By looking at the uncertainty of the bacterium's position, did the student have a valid point?
The student is wrong. The uncertainty of the bacterium's position is tiny compared to the size of the bacterium itself.
The electron configuration can also be represented by writing the symbol for the occupied subshell and adding a superscript to indicate the number of electrons in that subshell. For example, consider a carbon atom having an atomic number of 6. The total number of electrons in a neutral carbon atom is 6. The electron configuration of the carbon atom represented by the orbital diagram is (see picture)
This electron configuration can be written as 1s22s22p2 where 1s, 2s, and 2p are the occupied subshells, and the superscript "2" is the number of electrons in each of these subshells. Use the rules for determining electron configurations to write the electron configuration for Si. 1s^22s^22p^63s^23p²
Properties of Waves
To understand electromagnetic radiation and be able to perform calculations involving wavelength, frequency, and energy. Several properties are used to define waves. Every wave has a wavelength, which is the distance from peak to peak or trough to trough. Wavelength, typically given the symbol λ (lowercase Greek "lambda"), is usually measured in meters. Every wave also has a frequency, which is the number of wavelengths that pass a certain point during a given period of time. Frequency, given the symbol ν (lowercase Greek "nu"), is usually measured in inverse seconds (s−1). Hertz (Hz), another unit of frequency, is equivalent to inverse seconds. The product of wavelength and frequency is the speed in meters per second (m/s). For light waves, the speed is constant. The speed of light is symbolized by the letter c and is always equal to 2.998×108 m/s in a vacuum; that is, c=λν=2.998×108m/s Another term for "light" is electromagnetic radiation, which encompasses not only visible light but also gamma rays, X-rays, UV rays, infrared rays, microwaves, and radio waves. As you could probably guess, these different kinds of radiation are associated with different energy regimes. Gamma rays have the greatest energy, whereas radio waves have the least energy. The energy (measured in joules) of a photon for a particular kind of light wave is equal to its frequency times a constant called Planck's constant, symbolized h: Ephoton=hν where h=6.626×10−34J⋅s These two equations can be combined to give an equation that relates energy to wavelength: E=hcλ
How would the dx2−y2 orbital in the n=5 shell compare to the dx2−y2 orbital in the n=3 subshell? The contour of the orbital would extend further out along the x and y axes. The value of ℓ would increase by 2. The radial probability function would include two more nodes. The orientation of the orbital would be rotated 45∘ along the xy plane. The mℓ value would be the same.
True: A, C, E False: B, D The following representation of this orbital, 5dx2−y2, depicted when it is bisected by the xy plane, shows the effect of the radial nodes on the orbital contours.
Two of the types of ultraviolet light, UVA and UVB, are both components of sunlight. Their wavelengths range from 320 to 400 nm for UVA and from 290 to 320 nm for UVB. Compare the energy of microwaves, UVA, and UVB.
UVB radiation causes sunburn whereas UVA radiation does not. However, UVA, which causes tanning, is thought to be even more dangerous. The precise wavelengths of ultraviolet light that contribute to the formation of skin cancers still need to be determined by scientists.
Characteristics of an Atomic Orbital
Wave functions provide information about an electron's probable location in space. This can be represented by an electron-density distribution diagram, called an orbital. An orbital is characterized by a size, shape, and orientation in space.
Electron Configurations: Rules and Principles
When writing the ground-state electron configuration of a many-electron atom, three main rules must be followed: The aufbau principle: Electrons are added to the lowest energy orbitals available. The Pauli exclusion principle: No two electrons in an atom can have the same set of four quantum numbers (n, ℓ, mℓ, and ms). Hund's rule: For degenerate orbitals, the lowest energy state is attained when the number of electrons with the same spin is maximized. So for a degenerate set of orbitals, one electron goes into each orbital until all the orbitals of the subshell are half-filled. Once all the orbitals of the subshell are half-filled the pairing of electrons can take place. Note that aufbau is the German word for "building up."
Give the ground-state electron configuration for copper (Cu) using noble-gas shorthand.
[Ar]4s^13d^10
Give the ground-state electron configuration for silicon (Si) using noble-gas shorthand.
[Ne]3s^23p^2
What is the label for the orbital shown here that indicates the type of orbital and its orientation in space?
dxz
As an electron drops from the n=5 level to the n=2 level, ____.
light of one color is emitted
Compare the orbital shown in Parts A and B (the two previous slides) to the orbital shown here in size, shape, and orientation. Which quantum number(s) would be different for these two orbitals?
mℓ only The label for this orbital would be dx2−y2. The actual value of mℓ assigned to a given orientation is not arbitrary. It is determined based on how the hydrogen atom behaves in a magnetic field. This also accounts for the name given to this quantum number.
Give the possible combinations of quantum numbers for the following orbitals. A 2p orbital
n = 2, l = 1, ml= -1,0,1
Give the possible combinations of quantum numbers for the following orbitals. A 3s orbital
n = 3, l = 0, ml= 0
Give the possible combinations of quantum numbers for the following orbitals. A 4d orbital
n = 4, l = 2, ml= -2,-1,0,1,2
Give a possible combination of n and l quantum numbers for the following fourth-shell orbital:
n=4, l=2
A ray of red light has a wavelength of about 7.0×10−7 m. Will exposure to red light cause electrons to be emitted from cesium?
no
Is this wavelength longer or shorter than the diameter of an atom (approximately 200 pm)?
shorter
What is the change in energy, ΔE, in kilojoules per mole of hydrogen atoms for an electron transition from n=9 to n=2?
ΔE = -312 kJ/mol Each hydrogen atom emits energy in the form of a photon of light and hence the energy carries a negative sign. The emission spectra of various elements are used for lighting purposes as in sodium and mercury vapor lights as well as neon signs.
What is the change in energy if the electron from Part A now drops to the ground state?
ΔE = −2.05×10−18 J Energy was released in this transition, so we express ΔE as a negative number (it is a net loss of energy from the point of view of the system). However, you should use the absolute value of ΔE for the remaining calculations.
What is the uncertainty of the position of the bacterium?
Δx = 4.58×10^−11 m
What wavelength λ should the astrophysicist look for to detect a transition of an electron from the n=5 to the n=3 level?
λ = 1.28×10−6 m
A microwave oven operates at 2.30 GHz . What is the wavelength of the radiation produced by this appliance?
λ = 1.30×108 nm Some people lose their wireless Internet connection at home while their microwave oven is turned on because both happen to operate near 2.40 GHz.
What is the longest-wavelength line in nanometers in the infrared series for hydrogen where m = 3?
λ = 1875 nm
What is the de Broglie wavelength in meters of a small car with a mass of 1150 kg traveling at a velocity of 55.0 mi/h (24.6 m/s)?
λ = 2.34×10^−38 m
A radio station's channel, such as 100.7 FM or 92.3 FM, is actually its frequency in megahertz (MHz), where 1MHz=106Hz and 1Hz=1s−1. Calculate the broadcast wavelength of the radio station 93.10 FM.
λ = 3.220 m
What is the shortest-wavelength line in nanometers in the infrared series for hydrogen where m = 3?
λ = 820.4 nm
What is the wavelength λ of the photon that has been released in Part B?
λ = 9.70×10−8 m
What is the wavelength in meters of an FM radio wave with frequency ν = 149.0 MHz ?
λFM = 2.01 m
What is the wavelength of a medical X ray with ν = 5.55×1017 Hz ?
λXray = 5.41×10−10 m
What is the threshold frequency ν0 of cesium? Note that 1 eV (electron volt)=1.60×10−19 J.
ν0 = 9.39×1014 Hz
What is the frequency of a gamma ray with 5.17×10−11 m ?
νgamma = 5.80×1018 Hz
What is the frequency of a radar wave with 17.3 cm ?
νradar = 1.73×109 Hz
What is the azimuthal quantum number (also called the angular-momentum quantum number), ℓ, for the orbital shown here?
ℓ= 2 For the known elements, only s, p, d, and f orbitals are used. However, quantum theory predicts the existence of orbitals with values higher than ℓ=3. For example, an orbital with ℓ=4 would be given the letter designation of g.
