6.6: The Shapes of Atomic Orbitals (2024)

  1. Last updated
  2. Save as PDF
  • Page ID
    52808
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vectorC}[1]{\textbf{#1}}\)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}}\)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}\)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Introduction

    In the previous section we learned that electrons have wave/particle duality, exist in orbitals that are defined by the Schrödinger wave equation that involves the complex coordinate system and imaginary numbers, and that they can be defined by quantum numbers n, l and ml. We use the Greek symbol psi (\(\psi\) to represent a wave function. In this section we will look at the shapes of orbitals that have been transformed to the real coordinates of the x,y,z Cartesian coordinate system. The math of this transform is beyond the scope of this class, but for example, the \(\Psi _{P_{+1}}\) and the \(\Psi _{P_{-1}}\) orbitals can be combined in two ways that produce two new orbitals with real coordinates of the Cartesian coordinate system (the imaginary components of the wave functions cancel) the \(\Psi _{P_{x}}\) and \(\Psi _{P_{y}}\) orbitals. Since the original \(\Psi _{P_{+1}}\) and \(\Psi _{P_{+1}}\) were both solutions of the Schrödinger wave equation, their combinations are also solutions, and so we can visualize atomic orbitals as shapes along the x,y,z axes. All three quantum numbers influence the ultimate shape. The probability of finding an electron is \(\psi^2\) and in the following representations we are implictily defining the orbitals as the square of the wavefunction.

    \[1s \equiv \psi^2_{1s} \; and \; 2p_x \equiv \psi^2_{2p_x} \; and \; 3d_{xy} \equiv \psi^2_{3d_{xy}} \; .... \nonumber \]

    So when we say 1s or 3dxz we are orbital in terms of its location in space, and the images in Figure \(\PageIndex{1}\) represents the shapes of some common orbitals where there is roughly a 90% probability of finding the electron that resides in that orbital.

    6.6: The Shapes of Atomic Orbitals (2)

    Note in Figure \(\PageIndex{1}\) that there is one type of s orbital (l=0), three types of p (l=1), 5 types of d (l=2) and 7 types of d (l=3). These are not the orbitals described the the magnetic quantum numbers, but combinations of them that result in the x,y,z Cartesian coordinate system. It should also be indicated that these represent the geometry for the first principle quantum number where an azimuthal quantum number occurs, so for example, this is the s orbital of the first shell, the p orbital of the second shell, d orbital of the third shell and the f orbital of the fourth shell.

    We will look at each of these in turn.

    s-orbitals

    For l=0 the electron density function is spherically symmetric and the 1s orbital has no nodes. Figure \(\PageIndex{2}\) shows the square of the wavefunction

    6.6: The Shapes of Atomic Orbitals (3)

    As the principle quantum number increases both radial nodes occur and the average distance from the nucleus increases. The "onion skin" analogy has often been used to describe this where the flesh of the onion represents the probability of an electron exisiting (realize the greater r, the greater the surface area), and the nodes are the empty spaces between the layers of the onion. Of interest is that the higher principle quantum number has the peak of its inner peak closer to the nucleus (the first peak in the \(\psi^2_{3s}\) is at a shorter radius than for the \(\psi_{2s}\) or \(\psi_{1s}\).

    Exercise \(\PageIndex{1}\)

    How many radial nodes would a 5s orbit have?

    Answer

    Four

    p-orbitals

    For the n=2 shell and greater there are three p orbitals. In the Cartesian coordinate the pz correlates to the ml=0 and the px and py are mathematical combinations of the ml = +1 and ml = -1. (Note, the terms px, py and pz actually relate to the wavefunctions squared, as indicated above. For n=2 there is one node, in fact it is a nodal plane. If you look at the name, you can see that these are radially symmetric along one axis, which is the axis of the name, and the nodal plane is defined by the other two axes where they go through the origin.

    6.6: The Shapes of Atomic Orbitals (4) 6.6: The Shapes of Atomic Orbitals (5)

    Figure \(\PageIndex{3}\): 2 p orbitals along the x, y and z axes (left) and a 3 p orbital (right).

    Just as for s orbitals, as you move to higher principle quantum numbers the number of nodal surfaces increases, but they are no longer simple planar surfaces (see Figure \(\PageIndex{3}\)).

    d-orbitals

    Starting with the third principle quantum number d orbitals form, and there are 5 of these. Once again we are looking at ones that are defined by the Cartesian coordinate system. You will note that the 3 d orbits have two nodal surfaces.

    6.6: The Shapes of Atomic Orbitals (6)

    Three of the d-orbitals (dxy, dxz, and dyz) have electron density between two axes, with the nodal planes being the plane defined by the axes in the name with the remaining axes. So in the case of dxy, the nodal planes are the xz and yz planes. The \(d_{x^2-y^2}\) has electron density along the x and y axes, with the nodal planes being at 45 degrees to those axes. The \(d_{z^2}\) orbital has to conical surfaces with electron density forming in a lobe like a P orbital along the z axes, and a donut-like ring around the xy plane. Just as with the other orbitals, as number of nodal surfaces increases as you increase n.

    Exercise \(\PageIndex{2}\)

    What are the nodal planes for the dxy orbital?

    Answer

    The xz and yz planes

    f-orbitals

    There are seven f orbitals and these are the most complicated. As you have probably Figured by now, the first f orbitals appear in the n=4 "shell," and they have three nodal surfaces. ULAR students will not be required to know the names or shapes of the F orbitals, and the following embedded application from ChemTube3D can give you a feel for the f orbitals .

    Embedded Webpage from ChemTube 3D at the University of Liverpool. The direct link is https://www.chemtube3d.com/, if you click "Structure and Bonding/ atomic orbitals" you can look at the 3d structures of additional orbitals.

    Contributors and Attributions

    Robert E. Belford (University of Arkansas Little Rock; Department of Chemistry). The breadth, depth and veracity of this work is the responsibility of Robert E. Belford, rebelford@ualr.edu. You should contact him if you have any concerns. This material has bothoriginal contributions, and contentbuilt upon prior contributions of the LibreTexts Community and other resources,including but not limited to:

    Some material adopted, modified or adapted from

    • Steve Lower
    • Paul Flowers, et al. Open Stax
    • anonymous contributors to LibreText
      • some modified by Josh Halpern
    6.6: The Shapes of Atomic Orbitals (2024)

    FAQs

    What are the shapes of the atomic orbitals? ›

    An s-orbital is spherical with the nucleus at its centre, a p-orbitals is dumbbell-shaped and four of the five d orbitals are cloverleaf shaped. The fifth d orbital is shaped like an elongated dumbbell with a doughnut around its middle. The orbitals in an atom are organized into different layers or electron shells.

    What are the 5 orbital shapes? ›

    The shapes of the first five atomic orbitals are: 1s, 2s, 2px, 2py, and 2pz. The two colors show the phase or sign of the wave function in each region. Each picture is domain coloring of a ψ(x, y, z) function which depends on the coordinates of one electron.

    How many orbitals are in level 6? ›

    A principal shell with n = 6 contains six subshells. These subshells contain 1, 3, 5, 7, 9, and 11 orbitals, respectively, for a total of 36 orbitals.

    What are the 6f orbitals? ›

    In the general set of 6f orbitals, there are four distinct shapes, each of which possess a number of planar and conical nodes. The 6f orbitals possess two radial nodes. The 6fz 3 orbital (top row in the image above) has a planar node in the xy plane and two conical nodes with their exes along the z-axis.

    What are the 4 orbital shapes? ›

    There are four types of orbitals that you should be familiar with s, p, d and f (sharp, principle, diffuse and fundamental). Within each shell of an atom there are some combinations of orbitals.

    How to remember orbital shapes? ›

    There are four known shapes of atomic orbitals. First, what are the orbitals? They are s, p, d, and f - in that order You can remember them with the word "speedIfy", which may or may not be an actual word.

    What shapes are all orbits? ›

    All orbits are elliptical, which means they are an ellipse, similar to an oval. For the planets, the orbits are almost circular. The orbits of comets have a different shape.

    What is orbital shapes in chemistry? ›

    We already know that s-orbitals hold two electrons. The shape of this orbital is a sphere. The p-orbital (which holds a maximum of 6 electrons) is a peanut or dumbbell shape, and the d-orbital (holding a maximum of 10 electrons) is a cross peanut or cross dumbbell shape.

    How many 5 orbitals are there? ›

    n=5;l=(n–1)=4; hence the possible sub-shells for n=5 are: 5s, 5p, 5d, 5f and 5g. The number of orbitals in each would be 1, 3, 5, 7 and 9, respectively and summing them up gives the answer as 25.

    How many orbitals are in 6d? ›

    Answer and Explanation:

    6d has five d-orbitals.

    What are the 6 period orbitals? ›

    In the sixth period, the orbitals being filled are 6s,4f,5d,6p in that order. 32 elements (Z=55 (cesium) to 86 (radon)) are present. Along with 2sblock and 6p block elements, 10d block (transition) and 14f block (inner transition or lanthanoids) elements are present in sixth period.

    What are the orbitals of the 6th shell? ›

    In the sixth period (n = 6) of the periodic table, 6s, 4f, 5d and 6p orbitals are filled in the increasing order of the energy. Total 16 orbitals are available each of which contains a maximum 2 electrons. Thus, the sixth period can accommodate maximum 32 elements.

    Why doesn't 6f exist? ›

    Re: 6f orbital

    The reason why there isn't a sixth row on the periodic table is because there aren't discovered elements that can have ground states with electrons in the 6f subshell. Those would require a lot more electrons than the elements we have on the table.

    What is the shape of 6 hybrid orbitals? ›

    When both the 3dz2 and 3dx2-y2 orbitals are mixed with the 3s, 3px, 3py and 3pz orbitals, the result is a set of six sp3d2 hybrid orbitals that point toward the corners of an octahedron. The geometries of the five different sets of hybrid atomic orbitals (sp, sp2, sp3, sp3d and sp3d2) are shown in the figure below.

    What are the 7 types of orbitals? ›

    Different types of orbitals
    Orbital NameOrbital AbbreviationNumber of Degenerate Orbitals
    Sphericals orbital1 orbital
    Principalp orbital3 orbitals
    Diffused orbital5 orbitals
    Fundamentalf orbital7 orbitals
    Sep 30, 2021

    What are the shapes of all molecular orbitals? ›

    The main shapes of the molecules are linear, angular, trigonal planar, trigonal pyramidal, tetragonal, and bipyramidal.

    What are the shapes of the S and P orbitals? ›

    We already know that s-orbitals hold two electrons. The shape of this orbital is a sphere. The p-orbital (which holds a maximum of 6 electrons) is a peanut or dumbbell shape, and the d-orbital (holding a maximum of 10 electrons) is a cross peanut or cross dumbbell shape.

    What are the four types of atomic orbitals describe each? ›

    The s orbital, where the value of the azimuthal quantum number is equal to 0. The p orbital, where the value of the azimuthal quantum number is equal to 1. The d orbital, where the value of the azimuthal quantum number is equal to 2. The f orbital, where the value of the azimuthal quantum number is equal to 3.

    What is the shape of an f orbital? ›

    Tetrahedral is the shape of the f orbital generally.

    References

    Top Articles
    Latest Posts
    Recommended Articles
    Article information

    Author: Laurine Ryan

    Last Updated:

    Views: 5953

    Rating: 4.7 / 5 (77 voted)

    Reviews: 92% of readers found this page helpful

    Author information

    Name: Laurine Ryan

    Birthday: 1994-12-23

    Address: Suite 751 871 Lissette Throughway, West Kittie, NH 41603

    Phone: +2366831109631

    Job: Sales Producer

    Hobby: Creative writing, Motor sports, Do it yourself, Skateboarding, Coffee roasting, Calligraphy, Stand-up comedy

    Introduction: My name is Laurine Ryan, I am a adorable, fair, graceful, spotless, gorgeous, homely, cooperative person who loves writing and wants to share my knowledge and understanding with you.