Quantum Numbers and Periodic Trends By Sarah C. Petitto using materials from Jack F. McKenna According to quantum mechanics, each electron in an atom is described by a set of four different quantum numbers. Three of the quantum numbers (n, l, and ml) specify the wave function that gives the probability of finding the electron at various points in space. The fourth quantum number (ms) refers to a magnetic property of electrons called "spin." The quantum numbers are derived from the solution of Schrödinger's equation for the hydrogen atom. A wave function for an electron in an atom is called an atomic orbital. An atomic orbital is pictured qualitatively by describing the region of space where there is a high probability of finding an electron. s orbitals In each energy level there is an s orbital which is spherically symmetric. p orbitals Starting in the second energy level, there are three p orbitals. P orbitals consist of two lobes oriented along each of the three axes with a planar node cutting through the nucleus. All of the p orbitals have the same energy and are said to be degenerate. d orbitals Starting in the third energy level, there are five d orbitals, which have more complicated shapes than s and p orbitals. Each of the five d orbitals have the same energy despite the fact that they do not all look alike. f orbitals Starting in the fourth energy level, there are seven f orbitals, which have even more complicated shapes than s, p, or d orbitals. Each of the seven f orbitals have the same energy. Electron configuration The arrangement of the electrons in the various atomic orbitals within an atom. The atomic number Z of an element is the number of protons in the nucleus of an atom. It is also the number of electrons in a neutral atom. NOTE: When asked for the electron configuration of an atom, the electron configuration of the ground state is normally understood. 1. Principal Quantum Number (n) The energy of an electron in an atom depends principally on the quantum number n. The smaller the value of n, the lower the energy. The value of n can have any positive integer value--1, 2, 3, ... The size of an orbital also depends on n. The larger the value of n is, the larger average distance the electron is from the nucleus. Orbitals of the same quantum state n are said to belong to the same shell. 2. Angular Momentum Quantum Number (or Azimuthal Quantum Number) (l) The angular momentum quantum number distinguishes orbitals of a given n having different shapes; it can have any integer value from 0 to n - 1. Within each shell of quantum number n, there are n different kinds of orbitals, each with a distinctive shape denoted by an l quantum number. Orbitals of the same n, but different l, are said to belong to different subshells (or sublevels) of a given shell. The different subshells are usually denoted by letters: l orbital 0 s 1 p 2 d 3 f 4 g … … The rather odd choice of letter symbols for angular momentum quantum numbers survives from an old spectroscopic terminology which described certain features of spectra as sharp, principal, diffuse, and fundamental. Following f, the orbital designations follow in alphabetical order. For n = 1 l=0 one kind of orbital (s). For n = 2 l = 0, 1 two kinds of orbitals (s and p). For n = 3 l = 0, 1, 2 three kinds of orbitals (s, p, and d). 3. Magnetic Quantum Number (ml) The magnetic quantum number distinguishes orbitals of a given n and l of a given energy and shape but having a different orientation in space; the allowed values are integers from -l to 0 to +l. NOTE: There are 2l + 1 orbitals in each subshell of quantum number l. For l = 0 (s subshell), the only allowed ml quantum number is 0 only one orbital in the s subshell. For l = 1 (p subshell), ml = -1, 0, and +1 three different orbitals in the p subshell. The orbitals have the same shape but different orientations in space. In addition, each orbital of a given p subshell has the same energy. For l = 2 (d subshell), ml = -2, -1, 0, +1, and +2 five different orbitals in the d subshell. The orbitals do not have the same shape or orientation in space. However, each orbital of a given d subshell has the same energy. For l = 3 (f subshell), ml = -3, -2, -1, 0, +1, +2, and +3 seven different orbitals in the f subshell. The orbitals do not have the same shape or orientation in space. However, each orbital of a f given subshell has the same energy. 4. Spin Quantum Number (ms) The spin quantum number refers to the two possible orientations of the spin axis of an electron; the possible values are +1/2 (spin up, ) and -1/2 (spin down, ). Pauli Exclusion Principle States that no two electrons in an atom can have the same four quantum numbers. If two electrons in an atom should have the same n, l, and ml values (that is, these two electrons are in the same orbital), then they must have different values of ms. In other words, only two electrons may occupy the same atomic orbital, and these electrons must have opposite spins. As a result of the Pauli exclusion principle, the configuration with both electrons having the same spin is ruled out as unacceptable. The Pauli exclusion principle can be tested by a simple observation. If the two electrons is the 1s orbital of a helium atom have the same, or parallel, spins ( ), their net magnetic fields would reinforce each other. Such an arrangement would make the helium atom paramagnetic. Paramagnetic substances are those that are attracted by a magnet. If the electron spins are paired, or antiparallel to each other (), the magnetic fields cancel each other out and the atom is diamagnetic. Diamagnetic substances are slightly repelled by a magnet. By experiment, we find that the helium atom is diamagnetic in the ground state. Electron Configuration The four quantum numbers n, l, ml, and ms enable us to label completely an electron in any orbital in an atom. In a sense, we can regard the four quantum numbers as the "address" of an electron in an atom in somewhat the same way that a street address, city, state, and postal zip code specify the address of an individual. The Building-Up Principle In writing the electron configurations of elements we use the aufbau principle. (The German word "aufbau" means "building up.") The aufbau principle dictates that as protons are added one by one to the nucleus to build up the elements, electrons are similarly added to the atomic orbitals. The order of filling orbitals is based on energy which is found to increase in the order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f ..... Hund's Rule To determine which electron configuration has the greatest stability, states that the most stable arrangement of electrons in sublevels is the one with the greatest number of parallel spins. Because electrons repel each other, it makes sense that they remain as far apart as possible. Clearly, they can remain farther apart and be lower in energy if they are in different spatial regions than if they are in the same orbital occupying the same region of space. Periodic Properties The periodic table is one of most important organizing principles of chemistry and its arrangement is based on the regular (or periodic) behavior of the elements. There are several periodic properties of the elements that are discernable based on the underlying organization of the periodic table. Ionic Charges When chemical nomenclature was introduced, you were simply told to learn that all Group IA elements had a charge of +1 and all Group IIA elements had a charge of +2. Once electronic configurations of the elements were introduced, the charge of Group IA ions was understandable because each element has an ns1 configuration for its valence shell. Group IIA elements have an electronic configuration of ns2 for the valence shell, which explains why they have a charge of +2. By the loss of the valence electrons, elements in Group IA and IIA become isoelectronic with the noble gas in the period above. Elements in Group VIIA have an electronic configuration of ns2 np5 for their valence shell. They could lose seven electrons and become isoelectronic with the noble gas in the period above or they could gain one electron and become isoelectronic with the noble gas in the same period. Clearly, based on what we observe in nature, elements in Group VIIA gain one electron and form ions with a -1 charge. Atomic Radius The size of an atom is difficult to define exactly because atoms are not like billiard balls or basketballs that have a fixed size. Because the electrons in the valence shell are in constant motion, the size of an atom is much more difficult to describe. In a similar way it is difficult to define where our atmosphere ends and "outer space" begins. Despite the difficulty in exactly determining the size of atoms, we can make some general statements about their size. When considered in light of their position on the periodic table, we can clearly observe regular or periodic changes. In general, the stronger the attraction between the nucleus and the valence shell electrons, the smaller the size. The strength of this attraction is measured in terms of the effective nuclear charge ( Zeff). If there is a large effective nuclear charge, the smaller the atomic radius. In going down a group, the size of an atom increases because the additional shells make the atom larger than the increasing nuclear charges makes the atom smaller. As a result, atomic radii increase from Li -> Na -> K -> Rb -> Cs. In going across a period, in general, the size of an atom gets smaller. Moving from left to right across the periodic table, the number of electrons in the inner shells remains constant while the nuclear charge increases. The electrons that are added to counterbalance the increasing nuclear size are ineffective in shielding one another. Ionic Radius Just as there are systematic differences in the sizes of atoms, there are also systematic differences in the sizes of ions. When a neutral atom is converted to an ion, we would expect a change in size. If the atom forms an anion, its size (or radius) increases, since the nuclear charge remains the same but the repulsion resulting from the additional electron(s) enlarges the domain of the electron cloud. The change in size is quite dramatic as the size of an anion formed from the halogens is nearly twice the size as the neutral atom from which it is formed. On the other hand, removing one or more electrons from an atom reduces the electron-electron repulsion but the nuclear charge remains the same, so the electron cloud shrinks, and the cation formed is smaller than the atom from which it is formed. There are parallel trends between atomic radii and atomic radii. However, for ions derived from elements in different groups, a size comparison is meaningful only if the ions are isoelectronic. When examining isoelectronic ions we find that cations are smaller than anions. Ionization Energy is the amount of energy necessary to remove an electron completely from an atom leaving behind an ion. This process is normally written as: energy + X (g) ---> X+ (g) + e- The magnitude of the ionization energy is a measure of how tightly an electron is held in the atom. The higher the ionization energy, the more difficult it is to remove the electron. As subsequent electrons are removed, the ionization energy increases because the nuclear charge remains constant but the repulsion between the remaining electrons decreases. From tables of data for multiple ionization energies for the elements it is possible to observe a very large stability when an atom or ion has an electronic configuration that is isoelectronic with a noble gas. From this data it is possible to determine the number of electrons in the valence shell, which correlates with the electronic configuration of the element. Additionally, ionization energies correlate with size. As you move down a group, the ionization energy decreases as the size of the atom increases. This makes sense when you realize that the electron that is being removed from an atom is in the valence shell that is farthest from the nucleus and thus more weakly held. Electron Affinity is defined as the energy change that occurs when an electron is accepted by an atom in the gaseous state to form an anion. The equation for this process is: X (g) + e- ---> X- (g) + energy NOTE: Electron affinities are generally negative because energy is usually released when a neutral atom gains an electron, which is an exothermic process (energy is released). The more negative the electron affinity, the greater the tendency of the atom to accept an electron, and the more stable the anion that results. As was true for ionization energies, electron affinities show a periodicity that is related to the electron configurations of the elements. The overall trend is an increase in the tendency to accept electrons (the electron affinity values become more negative) from left to right across a period. The electron affinities of metals are generally smaller than those of nonmetals. The values vary little within a group. CHEM 210 QM #’s and Periodic Trends _____ Post-lab, Show All Work for Credit Name _________________________ Section Electrons in an atom’s highest occupied shell (s and p electrons) and in partially filled subshells of lower shells d or f electrons). Example 1: K (potassium) has the electron configuration 1s22s22p63s23p64s1 Its highest occupied shell is 4 and it has one electron in the 4s orbital so it has one valence electron. Example 2: Co (cobalt) has the electron configuration 1s22s22p63s23p64s23d7. Its highest occupied shell is 4 and it has two electrons in the 4s orbital. Cobalt also has 7 electrons in a partially filled 3d orbital so it has a total of nine valence electrons. Example 3: Se (selenium) has the electron configuration 1s22s22p63s23p64s23d104p4. Its highest occupied shell is 4. Selenium has two electrons in the 4s orbital and four electrons in the 4p orbital. Selenium has a full 3d orbital so these ten electrons are NOT valence electrons. Selenium has six total valence electrons. 1. How many valence electrons do the following elements have? Elements F Mo Se Sb Rb Si Nd Ag+ Valance Electrons 7 1 6 5 1 4 2 0 Electron configurations can get cumbersome (think about the 107 electrons in Bohrium). They can be abbreviated using the noble gas (group 8) core notation (see examples below). Example1: Mg as the electron configuration: 1s22s22p63s2; Ne has the electron configuration 1s22s22p6 The electron configuration of Mg can be abbreviated: [Ne] 3s2 Example 2: Ru 1s22s22p63s23p64s23d104p65s24d6 which can be abbreviated: [Kr]5s24d6 2. Using the noble gas core notation, write the electron configurations for: Elements P Sr Co As Electronic Configuration [Ne] 3s² 3p³ [Kr] 5s2 [Ar] 3d7 4s2 [Ar] 3d¹⁰ 4s² 4p³ 3. Define Aufbau principle, Pauli Exclusion Principle, and Hund's Rule. Aufbau Principle According to the Aufbau principle, electrons first occupy those orbitals whose energy is the lowest. This implies that the electrons enter the orbitals having higher energies only when orbitals with lower energies have been completely filled. For Example Pauli Exclusion Principle According to it a single orbital contains a maximum of two electrons of opposite spin. For Example Hund’s Rule. According to Hund’s Rule, every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied. For Example 4. List and describe the four quantum numbers. The Principal Quantum Number(n) The first quantum number describes the electron shell, or energy level, of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom. For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6. The Azimuthal Quantum Number(l) The second quantum number, known as the angular or orbital quantum number, describes the subshell and gives the magnitude of the orbital angular momentum through the relation. In chemistry and spectroscopy, ℓ = 0 is called an s orbital, ℓ = 1 a p orbital, ℓ = 2 a d orbital, and ℓ = 3 an f orbital.. The Magnetic Quantum Number(ml) The magnetic quantum number describes the energy levels available within a subshell and yields the projection of the orbital angular momentum along a specified axis. The values of mℓ range from − to ℓ, with integer steps between them. The s subshell (ℓ = 0) contains one orbital, and therefore the mℓ of an electron in an s subshell will always be 0. The p subshell (ℓ = 1) contains three orbitals (in some systems depicted as three “dumbbell-shaped” clouds), so the mℓ of an electron in a p subshell will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with mℓ values of −2, −1, 0, 1, and 2. The value of the mℓ quantum number is associated with the orbital orientation. The Spin Projection Quantum Number(ms) The fourth quantum number describes the spin (intrinsic angular momentum) of the electron within that orbital and gives the projection of the spin angular momentum (s) along the specified axis. Analogously, the values of ms range from −s to s, where s is the spin quantum number, an intrinsic property of particles. An electron has spin s = ½, consequently ms will be ±, corresponding with spin and opposite spin. Each electron in any individual orbital must have different spins because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons 5. EVERY orbital can hold up to _____2________ electrons. 6. The s (lowercase) subshell can hold up to _____2______ electrons. 7. The p (lowercase) subshell can hold up to _____6_____ electrons. 8. The d (lowercase) subshell can hold up to _____10______ electrons. 9. The f (lowercase) subshell can hold up to ______14_______ electrons. 10. Give the electron configuration and the 4 quantum numbers for the last electron added for Co. Electronic Configuration n l ml ms 11. 1s22s22p63s23p63d74s2 3 2 2 -1/2 Electrons stay at the “Atomic Hotel” where the rooms are filled following Aufbau principle that states "ground" floors must be filled first and in order. The penthouse and higher floors costs more than the rooms on the lower floors. Money is equivalent to money, and excited electrons have more energy (money) and need to stay on the higher floors with better views. Recent accounting of the “Atomic Hotel” shows the following capacities shown by floors. Circle the excited electrons (ones with more money). May 23, 1997 June 11, 2005 October 10, 2013 A B C 12. The “Atomic Hotel” has a long standing tradition since 1925 to also follow the Pauli Exclusion Principle, where electrons can occupy the same room (energy level), only if the electrons have opposite spins as not to interfere with another electron guest. Circle the electrons that not follow the Pauli Exclusion Principle. April 1, 2005 July 4, 1976 May 11, 2012 A B C 13. The Pauli Exclusion Principle also identifies each electron within the hotel by giving each electron an address of four numbers (n, l, ml, and ms). The number n, is the principle quantum number which corresponds to the "floor", and l describes the shape of the room (s = 0, p = 1, d = 2, f = 3,...). However, there are set number of room shapes per floor. For each floor there is only one s room, three p rooms starting on the second floor, five d rooms starting on the third floor, and seven f type rooms starting on the fourth floor. The specific room number is identified using the ml number. Remember, electrons can pair up in a single room, only if the electrons have opposite spins as not to interfere with another electron guest, which is done using the ms number, the spin quantum number which is +½ (spin up) or -½ (spin down). 4f 6s -3 - 2 -1 0 +1 +2 +3 0 5p 4d 5s -1 -2 -1 0 0 +1 +2 0 4p +1 Floor n = 1, 2, 3…. Room type l s=0 p=1 d=2 f=3 ml specific room -1 0 +1 3d -2 -1 NOTE: d and f suites are more expensive than the s rooms on the next floor/s (3d > 4s). 0 +1 +2 4s 0 3p -1 0 +1 Example: an electron with the numbers 3, 0, 0, ½ is staying on the 3rd floor, in section s, in room #0, and is in + ½ spin state. 3s 0 2p -1 0 +1 2s 0 1s 0 Determine the address, four quantum numbers, of the circled electrons. April 22, 2012 2 1 1 -1/2 n l ml ms A August 27, 2009 4 2 1 +1/2 n l ml ms B February 29, 2012 4 0 0 +1/2 n l ml ms C 14. Hund's Rule, is the final policy at the “Atomic Hotel” that says, rooms on same floor that cost same, will be single occupancy until the floor is half-filled before sharing (pairing) of rooms occurs and the electron that arrives to the room first will be spin up and the electron that arrives last with be spin down. Please note Hund’s rule may not apply to those excited electrons that have purchased more expensive rooms. Circle the electrons and then explain how these electrons are violating Hund's Rule. December 12, 2007 January 16, 2011 B A March 25, 2008 C Because according to Hund’s Rule, every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied. In 4s electron is singly occupied but it has to be doubly occupied in the 3p energy level. Write out the ground state electron configurations and give the 4 quantum numbers for the last electron added for the following elements: 15. 16. 17. Na Electronic Configuration n l ml ms 1s22s22p63s1 3 0 0 -1/2 Electronic Configuration n l ml ms 1s22s22p63s23p3 3 1 1 -1/2 P Fe3+ Electronic Configuration n l ml ms 18. Ag Electronic Configuration n l ml ms 19. 1s22s22p5 2 1 1 -1/2 1s22s22p63s23p64s23d104p65s24d9 4 2 2 -1/2 Xe Electronic Configuration 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 5s2 5p6 There is no electron added because Xe is noble gas. For the following questions, identify the element and explain your answer. 20. What is the most likely charge of ions from elements in Group IIIA? 5 is the most likely charge of ions from elements in Group IIIA. Because, there is only 1 electron in p orbital of the Group IIIA. 21. Which element is larger: K or Rb? Rb is Larger Since potassium is located at the start of group 1A, and Rubidium at the end of the same group, Rubidium will have a larger atomic radius than Potassium. 22. Which element is larger: Si or P? P is Larger Since Silicon comes first when we move from left to right, therefore potassium will have a larger atomic radius than Silicon. 23. Which ion is larger: Na+ or K+? K+ is larger. Since Na is located at the start of group 1A, and k comes next to Na, therefore K will have a larger ion than Na. 24. Which ion is larger: O2- or F-? Both ions are of the same size because both have the same number of electrons which is 10. 25. Which element has a larger first ionization energy: Br or I? Br has larger First ionization energy. As we know that bromine comes first then iodine in Group 7A and First ionization energy decreases as we move down in a group. Therefore, Bromine has larger First ionization energy. 26. Which element has a larger (i.e., more negative) electron affinity: Cl or Na? Cl has larger electron affinity. As we know that Sodium comes first then chlorine in Period 3 and Electron Affinity increases as we move left to right in the Period. Therefore, chlorine has larger Electron Affinity. 27. What does it mean if the electron affinity is positive? When the electron affinity is positive, meaning energy is required to add an electron to the atom.