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The Principal Quantum Number ((n))The principal quantum number, (n), designates the principal electron shell. Because n describes the most probable distance of the electrons from the nucleus, the larger the number n is, the farther the electron is from the nucleus, the larger the size of the orbital, and the larger the atom is. N can be any positive integer starting at 1, as (n=1) designates the first principal shell (the innermost shell). The first principal shell is also called the ground state, or lowest energy state. This explains why (n) can not be 0 or any negative integer, because there exists no atoms with zero or a negative amount of energy levels/principal shells. When an electron is in an excited state or it gains energy, it may jump to the second principle shell, where (n=2).
This is called absorption because the electron is 'absorbing' photons, or energy. Known as emission, electrons can also 'emit' energy as they jump to lower principle shells, where n decreases by whole numbers. As the energy of the electron increases, so does the principal quantum number, e.g., n = 3 indicates the third principal shell, n = 4 indicates the fourth principal shell, and so on.n=1,2,3,4. The Orbital Angular Momentum Quantum Number ((l))The orbital angular momentum quantum number l determines the shape of an orbital, and therefore the angular distribution. The number of angular nodes is equal to the value of the angular momentum quantum number (l).
(For more information about angular nodes, see.) Each value of l indicates a specific s, p, d, f subshell (each unique in shape.) The value of l is dependent on the principal quantum number n. Unlike n, the value of l can be zero. It can also be a positive integer, but it cannot be larger than one less than the principal quantum number (n-1):l=0, 1, 2, 3, 4, (n-1). Principal ShellsThe value of the principal quantum number n is the level of the principal electronic shell (principal level). All orbitals that have the same n value are in the same principal level. For example, all orbitals on the second principal level have a principal quantum number of n=2. When the value of n is higher, the number of principal electronic shells is greater.
This causes a greater distance between the farthest electron and the nucleus. As a result, the size of the atom and its increases.Because the atomic radius increases, the electrons are farther from the nucleus. Thus it is easier for the atom to expel an electron because the nucleus does not have as strong a pull on it, and the decreases. OrbitalsThe number of in a subshell is equivalent to the number of values the magnetic quantum number ml takes on. A helpful equation to determine the number of orbitals in a subshell is 2l +1. This equation will not give you the value of ml, but the number of possible values that ml can take on in a particular orbital. For example, if l=1 and ml can have values -1, 0, or +1, the value of 2l+1 will be three and there will be three different orbitals.
The names of the orbitals are named after the subshells they are found in:s orbitalsp orbitalsd orbitalsf orbitalsl0123m l0-1, 0, +1-2, -1, 0, +1, +2-3, -2, -1, 0, +1, +2, +3Number of orbitals in designated subshell1357In the figure below, we see examples of two orbitals: the p orbital (blue) and the s orbital (red). The red s orbital is a 1s orbital. To picture a 2s orbital, imagine a layer similar to a cross section of a jawbreaker around the circle. The layers are depicting the atoms angular nodes. To picture a 3s orbital, imagine another layer around the circle, and so on and so on.
The p orbital is similar to the shape of a dumbbell, with its orientation within a subshell depending on m l. The shape and orientation of an orbital depends on l and m l.To visualize and organize the first three quantum numbers, we can think of them as constituents of a house. In the following image, the roof represents the principal quantum number n, each level represents a subshell l, and each room represents the different orbitals ml in each subshell. The s orbital, because the value of ml can only be 0, can only exist in one plane.
The p orbital, however, has three possible values of ml and so it has three possible orientations of the orbitals, shown by Px, Py, and Pz. The pattern continues, with the d orbital containing 5 possible orbital orientations, and f has 7:Another helpful visual in looking at the possible orbitals and subshells with a set of quantum numbers would be the electron orbital diagram. (For more electron orbital diagrams, see.) The characteristics of each quantum number are depicted in different areas of this diagram.
Restrictions.: In 1926, Wolfgang Pauli discovered that a set of quantum numbers is specific to a certain electron. That is, no two electrons can have the same values for n, l, ml, and ms. Although the first three quantum numbers identify a specific orbital and may have the same values, the fourth is significant and must have opposite spins.: Orbitals may have identical energy levels when they are of the same principal shell. These orbitals are called degenerate, or 'equal energy.' According to Hund's Rule, electrons fill orbitals one at a time.
This means that when drawing electron configurations using the model with the arrows, you must fill each shell with one electron each before starting to pair them up. Remember that the charge of an electron is negative and electrons repel each other. Electrons will try to create distance between it and other electrons by staying unpaired. This further explains why the spins of electrons in an orbital are opposite (i.e. +1/2 and -1/2).: According to the Heisenberg Uncertainty Principle, we cannot precisely measure the momentum and position of an electron at the same time.
As the momentum of the electron is more and more certain, the position of the electron becomes less certain, and vice versa. This helps explain integral quantum numbers and why n=2.5 cannot exist as a principal quantum number.
There must be an integral number of wavelengths (n) in order for an electron to maintain a standing wave. If there were to be partial waves, the whole and partial waves would cancel each other out and the particle would not move. If the particle was at rest, then its position and momentum would be certain. Because this is not so, n must have an integral value. It is not that the principal quantum number can only be measured in integral numbers, it is because the crest of one wave will overlap with the trough of another, and the wave will cancel out. Problems.
Suppose that all you know about a certain electron is that its principal quantum number is 3. What are the possible values for the other four quantum numbers?. Is it possible to have an electron with these quantum numbers: (n=2), (l=1), (ml=3), (ms=1/2)? Why or why not?. Is it possible to have two electrons with the same (n), (l), and (ml)?. How many subshells are in principal quantum level (n=3)?.
What type of orbital is designated by quantum numbers (n=4), (l=3), and (ml =0)? The LibreTexts libraries are and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739.
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