The hydrogen atom is very important for understanding electrons in atoms in general as we use the knowledge of electrons in hydrogen as the basis for our model for electrons in all the elements.
There are several key ideas that we can take away from studying the hydrogen atom that we can apply to all atoms.
The first is that the electron in a hydrogen atom cannot have any energy but instead is only found in particular fixed energy levels. This is odd. This is unlike any of the macroscopic objects we deal with in our everyday lives. This is a strange and bizarre consequence of quantum mechanics (QM). This is a critical idea for understanding atoms, molecules, and chemistry.
Second, small mass particles like electrons obey a different set of rules than the macroscopic objects of our everyday lives. They are not the same. We should not try to imagine them as being the same. However, we constantly try to classify them in ways that fit with our own macroscopic world view. However, we must always remember that they are not macroscopic particles and therefore they will never fit in with our macroscopic pictures. Electrons are simply not small particles orbiting like planets around nuclei.
Third, electrons follow the rules of quantum mechanics (QM). QM provide us two useful ideas about electrons, their energy and their "wavefunction". Energies turn out to come in discrete or "quantized" units. Wavefunctions tell us something about the "spatial distribution" of the electron. For lack of a better description "where the electron is".
Fourth, we find that the energies and wavefunctions that describe an electron in a hydrogen atom are very systematic. They can be classified by a set of "quantum numbers". These define the actual mathematical function that is the wavefunction. They effectively describe the "size and shape" of the electron's wavefunction.
Finally, we use the ideas we get from the hydrogen atom to describe all the rest of the elements. So these solutions that we obtain from QM for hydrogen are used again and again in chemistry to describe atoms and bonding.
By looking at the specific wavelengths of light that are either absorbed or emitted from a sample of H atoms, we discover something about the energy of the electrons in the atom. First, we notice that only specific wavelengths are associated with transitions. This means that there are discrete energy levels that the electron is moving between. The energy of the light of the transition corresponds to the difference in energy between two of these levels. If the energy of the electron is increasing, this is from absorption of the light energy. If the light is being emitted, this is from the energy of the electron decreasing.
Here is a picture of the visible line spectrum for H atom emission. Notice the discrete wavelengths of emission above a zero background.
Here is a picture of the visible line spectrum for H atom absorption. Notice the lines appear in the same place, but they are now dark rather than light. That is because the energy has been removed from continuous white light background. Before encountering the H sample, the light source had intensity at all wavelengths. The only wavelengths that could excite the electrons were ones with energies that corresponded to the difference between energy levels in the H atom. These are the same differences as were observed in emission. Only now, the energy is exciting the electron up rather than being emitted after the electron is excited.
By examining all the lines in the spectrum of the hydrogen atoms, an empirical model was derived that explained the pattern of the emission. The specific wavelengths (or frequencies or energies) could be predicted based upon a constant and two integers. The interpretation was that one integer represented the initial state and one integer the final state. The wavelength (or frequency or energy) was related to the change that occurred moving between these two states.
The original formula related inverse wavelength (known as "wavenumber") to the integers that were related to the initial and final states.
\[{ 1 \over \lambda} = {\cal R} \left({1 \over n_f^2} - {1 \over n_i^2}\right)\]
Where \({ \cal R}\) is equal to \(1.097\times 10^7\) m-1 and \(n_f\) and \(n_i\) are integers (1,2,3,....) that describe the initial and final states of the electron.
While the original formula derived by Rydberg did not look directly at energies, we can rewrite the formula to have these units. Under these conditions, the change in energy of the electron is given by
\[\Delta E = {\cal R} \left({1 \over n_f^2} - {1 \over n_i^2}\right)\]
Now the constant \({\cal R}\) (the Rydberg constant) is equal to \(2.178\times 10^{-18}\) J. Note now the units are energy. \(n_f\) and \(n_i\) are integers (1,2,3,....) still describe the initial and final states of the electron. Many of you may have seen this in the past so you just take it for granted. However, this is quite remarkable. There are fixed energy states that can be related to integers (1,2,3....) for the energies of the electrons in a hydrogen atom. One energy level is exactly (1/22) different in energy than another. Another is different by (1/52). Two others have a ratio of (32/172). Everything is related to the square of an integer!
Note the formula here is given in terms of energy changes. The original formula from Rydberg simply related the inverse of the wavelength to two integers. However, we can convert wavelength to frequency to energy. So the above equation combines the Empirical ideas of Rydberg with the relation of energy to frequency derived from the photoelectric effect.
Also note that the above formula is for energy differences between two energy levels (\(\Delta E\)). If you want (or need) to calculate the potential energy value (\(E_n\)) of the individual quantum levels (\(n=1,2,3,...\)), then you use the formula:
\[ E_n = -{\cal R}\left({1 \over n}\right)^2\]
One more important thing... On the absolute energy scale, we assign a free electron with nothing near it (floating in space) a value of ZERO joules. This would correspond to an infinite level in the atom, or \(E_\infty\). This also means that all energy levels in the atom are below this value and are therefore negative.
A challenge to understanding electrons in atoms is that they are governed by the laws of quantum mechanics rather than the Newtonian laws that govern the physical world we interact with on a regular basis. One particular peculiarity is that we can no longer think about "particles". It would be nice if an atom were like the little picture that we all have of an electron orbiting a nucleus like the planets orbit the sun. However comforting this picture is (and we will often fall back on this picture as we try to "visualize" and draw atoms), it is not correct. There are many reasons for this but one of the most glaring is that we cannot think of electrons as particles that are localized in space. They are instead strange quantum mechanical beasts that at times appear to be particles and at other times seem to behave like waves. This wave particle duality is often discussed in terms of electrons being both waves and particles. It is better to remember that electrons are neither waves nor particles (as both of these are classical ideas) but instead are some new strange thing that only appears to us as being wave-like or particle-like depending on how we are interpreting different experiments.
This wave particle duality exists in other situations such as electro-magnetic radiation. We nearly always deal with EM radiation as a wave. However, the photoelectric experiments showed that the energy of light interacting with electrons requires us to take the new perspective that the energy appears to come from particle-like objects as it is proportional to the frequency of the light. This is why we invent the notion of a "photon" or packet of light. The "photon" is our idea of a light particle.
Because electrons (and other very low mass "particles") are quantum mechanical, they don't behave as we would expect from classical mechanics. One of the key manifestations of this is the fact that we can either know the precise location of a particle but not its momentum. Or we can know its momentum exactly but have no idea about its location! Just let that sink in for a moment. If it sounds strange (even impossible), it should. This is not the way the world around us appears. A baseball has a location and velocity. You know where it is. You know where it is going. This is not the case with quantum mechanical objects such as an electron. You might know something about its location, and something about where it is going. But you don't get to know both of these precisely. This idea is referred to as the Heisenberg uncertainty principle that states there is a minimum product of the uncertainties of position and momentum.
\[\Delta x \Delta p \geq { h \over 4 \pi } \]
Where \(\Delta x\) is the uncertainty in position (how well do you know the position). And \(\Delta p\) is the uncertainty in momentum (how well do you know the momentum or "where it is going"). If you knew the exact position of an electron then \(\Delta x\ = 0\) as you know it exactly. If you knew it to about 1 nm, then \(\Delta x = 1 nm\). The product of these two uncertainties must be not only finite but greater than \( {h / 4 \pi}\). Luckily, Planck's constant, \(h\), is a very, very small number. Thus we can know quite a bit about the location and momentum; we just can't know them exactly. Is this true for everything? Technically it would apply to everything not just systems we dubbed as quantum mechanical. However, since \(h\) is so small the only times these uncertainties are relevant is when we are interested in very small distances (like distances in atoms and molecules) and when we are interested in very small momentums (like those for particles with small masses like an electron).
How then do we deal with these small particles that are governed by different rules? We have to come up with a new model to explain their behavior since Newton's laws will fail us. The new model that we will use is Quantum Mechanics.
Electrons in atoms (and electrons in general) are governed by the laws of Quantum Mechanics (QM). The details of QM are beyond the time limits of first year chemistry. However, you should take a way a few key ideas regarding QM and chemistry. First, QM gives us "solutions" to problems. The problems we set up look a lot like physics problem. For example, a small negative particle (the electron) is interacting with a single massive positive particle (a proton). What is the solution? Where is the electron? Is it close to the proton? Far away? This is the problem of the hydrogen atom. There are two things you should take away from the idea. When we solve quantum mechanical problems we get two things: a wavefunction and the energy.
When we solve a QM problem we get an infinite number of solutions. How can we figure out which ones are of interest if there are an infinite number? We can order them in terms of their energies. In particular, we are most interested in the lowest energy solution as this is the "ground state" or most stable state of the electron in an H-atom. This solution has a wavefunction and an energy. If we look at the next highest energy solutions we discover that there are many solutions with the same energy. We call these degenerate solutions. As we move up in energy we find another group of degenerate solutions with a different higher energy. And we can keep moving up in energy with more and more solutions.
We can classify our wavefunctions with sets of quantum numbers. These help us to describe the mathematical form of the wavefunction with a simple set of integers. For a one electron system there are three quantum numbers.
The most important categorization of the solution to the hydrogen atom is the principle quantum number, n. We regard this as the most important because this number is related to the energy associated with that particular wavefunction. For hydrogen-like atoms (one electron and a nucleus) all solutions with the same n value are degenerate (they have the same energy). The energy of the level in joules is given by
\[E_n = -2.18\times 10^{-18} \left({Z^2 \over n^2}\right)\]
where Z is the nuclear charge. For hydrogen (Z=1), the energy levels start at n=1 with a value of -2.18 x 10-18J. Then energies are negative as we have defined E=0 as the energy of the electron and the nucleus at a distance of infinity. The electron is more stable (lower in energy) when it is closer to the nucleus. Since we have defined the separated nucleus and electron as zero, the more stable energies must all be negative. The more negative, the more stable it is. Then they move up in energy (closer to zero) with a spacing that decreases with each level until they all approach an energy of zero (n=infinity). The energy also depends on the square of the nuclear charge, Z. The ground state energy (n=1) for helium (1s2, Z=2) is 4 times lower than the ground state energy (n=1) for hydrogen (1s1, Z=1).
We can now use this formula to find the energy difference between any two states in the hydrogen atom. This will come out to be identical to the trends that were found for the Rhydberg equation.
One interesting application of this formula is that we can use it to find the energy required to remove the electron from a hydrogen-like atom. This is the energy difference between the ground state (lowest energy) \( n = 1\) and the final state in which the electron is separated from the nucleus. The electron separated from the nuclear corresponds to \( n = \infty \) . The energy for \( n = \infty \) is zero.
The next quantum number is the angular moment quantum number. This gets the symbol \(\ell\) (that's a cursive ell). This quantum number is related to the "shape" of the wavefunction.
\(\ell\) is an integer and can have any value starting at zero and going up \(n-1\).
Since the \(\ell\) values are also related to some historical experiments, they maintain other notations. \(\ell=0\) are called \(s\), \(\ell=1\) are \(p\), \(\ell=2\) are \(d\), \(\ell=3\) are \(f\), \(\ell=4\) are \(g\), \(\ell=5\) are \(h\)....
For a given \(n\) and \(\ell\), there are a number of degenerate solutions. The number of degenerate levels is equal to \(2\ell+1\). There are three \(p\) solutions (\(\ell=1\), 2(1)+1 = 3). There are seven \(f\) solutions (\(\ell=3\), 2(3)+1 = 7).
Each of these solutions gets a unique quantum number \(m_\ell\)
\(m_\ell\) is also an integer and can range from \(m_\ell = -\ell,...0...,+\ell\)
So if \(\ell=4\) the nine possible \(m_\ell\) values are -4,-3,-2,-1,0,+1,+2,+3,+4.
Orbital notation is simply a different means that chemists use to describe the wavefunction for a hydrogen atom. Rather than using the term wavefunction, instead we use the word "orbital".
The orbital notation uses only the n and \( \ell\) quantum numbers.
In this notation we simply state the principle quantum number \( n\) as a number. A letter is used to denote the \( \ell\) term as letters s,p,d...
So let's imagine a hydrogen atom in its lowest energy state. This is the ground state. The electron wavefunction is then described by n=1, \( \ell\) = 0, and \( m_{\ell}\)=0. We could also simply call this "orbital" a 1s orbital. (note: when doing this we ignore any mention of \( m_{\ell}\). We might use light to excite the hydrogen atom to a higher energy state. Let's say we went from 1s to 2p. The excited state could then have quantum numbers n=2, \( \ell\) =1. In the absence of any external perturbations such as a magnetic field all the \( m_{\ell}\) levels are the same. So any \( m_{\ell}\) solution could describe this state.
The hydrogen atoms orbitals are the "wavefunction" portion of the quantum mechanical solution to the hydrogen atom. The wavefunctions tell us about the probability of finding the electron at a certain point in space. Thus the orbitals offer us a picture of the electron in a hydrogen atom. However, this picture is not a simple one. First of all, the electron is spread out over space. We cannot pin-point its location. If this seems odd (even impossible), it should. Electrons are inherently quantum mechanical and behave differently than everything we encounter in our everyday life. As we build up from the hydrogen atom to multi-electron atom to bonding and molecules we will make extensive use of the wavefunctions that we first find for hydrogen. As such it is useful to become familiar with their shapes.
There are two key features for an orbital. The distribution of the electron away from the nucleus. This is known as the radial distribution. The other is the "shape" of the orbital and is the angular distribution. The radial distribution is mostly dependent on the principle quantum number n. The angular distribution depends on \( \ell\) and \(m_{\ell}\).
s orbitals are wavefunctions with \(\ell\) = 0. They have an angular distribution that is uniform at every angle. That means they are spheres.
This link shows some picture of this distribution. In fact, this link is to the orbitron page that contains lots of great plots of these orbitals.The radial probability distribution (the probability of finding the electron at a particular radius) starts at zero at the nucleus, increases, and then decays away to zero as the radius increases. A radial distribution plot for the 1s orbital (n=1, \(\ell\) =0) can be found here .
Higher energy s orbitals such as 3s (n=3, \(\ell\) =0) have "nodes" at particular distances. These are distances at which the electron has zero probability. This is a consequence of the quantum mechanical nature of the electron. It appears to have wave-like properties. As such there are regions in space where the amplitude (probability) is zero. You can see this clearly in a radial distribution function for the 3s . There is a node at two distances away from the nucleus (r = 0). Note: the probability is zero at the nucleus as well but this is the result of the finitely small volume at r=0 not a radial node.
p orbitals are wavefunctions with \(\ell\) = 1. They have an angular distribution that is not uniform at every angle. They have a shape that is best described as a "dumbbell". A picture of this distribution can be found here for a 2p
p orbitals have one angular node (one angle at which the probability of electron is always zero. The radial probability distribution (the probability of finding the electron at a particular radius) for a 2p, looks nearly identical to a 1s. What is different? Mostly the angular distribution. A plot for the 2p orbital (n=2, \(\ell\) =1) can be found here .
Higher energy p orbitals such as 3p (n=3, \(\ell\) =1) have the same angular distribution, but now the start to be "nodes" at particular distances in the radial distribution. You can see this clearly in a radial distribution function for the 3p . There is a node at one distance away from the nucleus (r = 0). Note: the probability is zero at the nucleus as well but this is the result of the infinitely small volume at r=0, not a radial node.
There are three different p orbitals that are nearly identical for the three different \( m_{\ell}\) values (-1,0,+1). These different orbitals essentially have different orientations.
d orbitals are wavefunctions with \(\ell\) = 2. They have an even more complex angular distribution than the p orbitals. For most of them it is a "clover leaf" distribution (something like 2 dumbbells in a plane). dorbitals have two angular nodes (two angles at which the probability of electron is always zero.
There are five different d orbitals that are nearly identical (n=2, \(\ell\) =1) for the five different \( m_{\ell}\) values (-2,-1,0,+1,+2). These different orbitals essentially have different orientations. There is one that is a little different than the others (this is the \(m_{\ell}\)=0). The shapes of the d orbitals can be seen here for a 3d
As n increases there are ever larger available \(\ell\) numbers. These give even more complex angular distributions with more angular nodes. After the d orbitals \(\ell\)=2, come the f \(\ell\)=3, then g \(\ell\)=4, then h\(\ell\)=5, ....