Many of the concepts you'll encounter in quantum mechanics or quantum chemistry can be difficult to visualize. It's one thing to say, for example, that electrons exhibit both wave-like and particle-like properties, but what does that actually mean? Drawing atomic orbitals can be a great way to familiarize yourself with some basic concepts and get a more intuitive sense of how matter works down at the subatomic level. Here's how to draw the orbitals for a hydrogen atom.
- Skill level:
- Moderately Challenging
Instructions
Things you need
- Pencil
- Paper
- Calculator
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1
Recall that an electron in a hydrogen atom can only exist in certain discrete states, each described by a wave-like equation called a wave function. An atomic orbital is basically one of the states an electron can occupy. Squaring the wave function gives the probability that you will find the electron at any given point. If you were to draw a 3-D graph of the square of the wave function, and color areas of high probability a darker color and areas of low probability a lighter color, you would have a fair idea where the electron might be found.
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2
Note that rather than writing out the full wave function, chemists use a kind of shorthand, where each state an electron can occupy is described by four different quantum numbers. The first, the principal or n, denotes the size of the orbital and can be any integer value 1 or greater. All orbitals that share the same principal quantum number are called a shell. The second, the angular momentum, denotes the shape of the orbital.To avoid confusion, it is generally represented as a letter rather than a number; the letters are s, p, d where s = 0, p = 1, d = 2 and f = 3. (There are other possible values for angular momentum, but you will not encounter anything above d in an organic or general chemistry class.) All orbitals that share the same angular momentum quantum number are called a subshell. The third quantum number, the magnetic quantum number, denotes the orientation of the orbital. The fourth quantum number is the spin, or the orientation of the electron's spin axis. Electrons can have only two values for spin, +1/2 and -1/2. No electron can have the same set of quantum numbers. Consequently, a maximum of two electrons can occupy any given orbital, and if an orbital does contain two electrons, they must have opposite spins.
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3
Notice that because of the possible values for the magnetic quantum number and the angular momentum, higher shells contain more subshells, and higher subshells contain more possible orbitals. An s-subshell can only have one orbital, whereas a p subshell has three, a d subshell 5 and an f subshell 7. The allowed quantum numbers are as follows: The first shell (n = 1) contains only one s orbital. The second shell (n = 2) contains one s orbital and 3 p orbitals. The third shell (n=3) contains one s orbital, 3 p orbitals and 5 d orbitals. The fourth shell (n=4) contains one s orbital, 3 p orbitals, 5 d orbitals and 7 f orbitals.
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4
Open the link under the Resources section below and look at the animations for each type of orbital to get a feel for the different shapes.
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5
Draw an s-orbital. S-orbitals are spherical in shape and centered on the nucleus. The higher the principal quantum number, the larger the s-orbital will be. A non-ionized hydrogen atom in its lowest energy or ground state has only one electron in a 1s-orbital. (Notice that chemists usually write the principal quantum number followed by the subshell or angular momentum; therefore, 1s means first shell, s-type orbital). Remember that when you draw this shape, you are just drawing the approximate shape of the region of highest probability density -- i.e. where we expect the electron will most likely be. Since an electron exhibits wave-particle duality, we cannot precisely know both the position and the momentum of the electron, however, so there will always be some uncertainty, and if we wanted to be really accurate the edges of the orbital should be fuzzy like a cloud. In fact, a constantly shifting cloud of negative charge is perhaps the best and simplest way to imagine electrons in an orbital.
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6
Draw the three p-orbitals in the second shell. P-orbitals are roughly dumbbell shaped. Each one of the three lies along a different axis: along the x axis, along the y axis, and along the z axis. Notice that p-orbitals have a very different shape from s-orbitals.
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7
Draw the five d-orbitals in the third shell. D- orbitals are a little trickier to draw. Four of them form cloverleaf shapes. One cloverleaf is in the xy plane, with its "leaves" midway between the two axes on either side. Another cloverleaf is in the yz plane, and still another is in the xz plane. A fourth cloverleaf-shaped orbital is in the xy plane but with its "leaves" pointing directly along the axes. The fifth and final d-orbital is propeller-shaped and points straight along the z-axis with a ring around its middle.
Tips and warnings
- If electrons as waves and orbitals as mathematical functions seem counter-intuitive, don't worry about it -- it takes a little time to become used to thinking of electrons as both "waves" and "particles". No less a scientist than Albert Einstein was troubled by the bizarre and seemingly paradoxical implications of quantum mechanics; he famously remarked that "God does not play dice". Many experiments throughout the 20th century, however, verified that these theories predict the behavior of subatomic particles and systems with an uncanny level of accuracy. As counter-intuitive as these ideas may seem at first, they are the best description we have of how matter behaves at the atomic and subatomic level.
