Detecting paramagnetism in a dioxygen and dinitrogen gases

A strong bar magnet is used to probe the magnetism of the two most common atmospheric gases.

Ingredients: oxygen gas, nitrogen gas, bar magnet, dish washing liquid

Procedure: A partial recipe follows.

1. Add water to a pyrex dish and add dish washing liquid to the water.

2. "Blow" a bubble in the soap solution using the oxygen gas.

3. Interrogate the bubble by bringing the bar magnet close to it; see if the bubble follows the magnet across the surface.

4. Repeat the procedure for nitrogen.

Understanding: We understand that a paramagnetic compound is attracted to a permanent magnet. The paramagnetism of the atom or molecule is due to the existence of unpaired electrons in half-filled orbitals, with unpaired spin magnetic moments of the electrons.

To experimentally determine the relative degree of paramagnetism, we interrogate the compounds using a strong permanent magnet. In our experiment, we find that the bubble filled with oxygen gas is attracted to the magnet while the bubble filled with nitrogen gas is not. We conclude that dioxygen gas is paramagnetic and dinitrogen gas is diamagnetic.

To theoretically determine the paramagnetism of the compound, we must determine the electron configurations of the molecules. The question is, how do we accurately describe the electron configuration of a molecule? We have an exact solution for the allowed wavefunction and corresponding energies for a one-electron atom or ion. We have an approximate theory for the allowed wavefunctions and energies for multi-electron atoms. Why not push our luck a bit and build a model for the molecular wave functions and corresponding energies, again using our results for the wavefunction and energies of the one-electron atom? Let's see how that works.

Consider dihydrogen. How can we understand the fact that dihydrogen is a stable molecule that is diamagnetic, with a "single bond" between the two hydrogen atoms? We think of two hydrogen atoms, each with one electron in an atomic 1s orbital. From our Lewis structures, we expect the two electrons to "pair" and form a covalent bond. The nature of the covalent bond is that the two electrons create significant negatively charged electron density between the two positively charged nuclei. How can we build a molecular orbital that embodies that basic idea?

Easy! We can build a molecular orbital orbital from the two 1s atomic orbitals by simply adding them together. That is, we add the wavefunctions representing the two 1s orbitals for atom A and atom B to form a wavefunction for the molecular orbital

ψσ1s = ψ1s(A) + ψ1s(B)

The orbital has the character of a bonding molecular orbital with substantial electron density between the two nuclei, encouraging bonding.

If we start with two atomic orbitals we need to end with two molecular orbitals. We can form an anti-bonding orbital by subtracting one wave function from the other

ψσ1s* = ψ1s(A) - ψ1s(B)

At a point halfway between the two nuclei, the magnitude and sign of the two atomic 1s wavefunctions are equal. We are subtracting the two atomic wavefunctions, so the magnitude of the molecular wave function will be zero everyone on that surface representing the points that are equidistant from the two hydrogen nuclei. That leads to a nodal surface halfway between the two atomic nuclei. That nodal surface characterizes the molecular orbital as being an anti-bonding orbital, which discourages the nuclei from being bound.

If we continue to add and subtract the atomic orbitals, moving to 2s, 2px, 2py, 2pz and beyond, we find a series of bonding and anti-bonding molecular orbitals: σ1s, σ1s*, σ2s, σ2s*, σ2pz, π2px, π2py, π2px*, π2py*, and σ2pz*. Note that the number of molecular orbitals we create is always equal to the number of atomic orbitals used to create them. The molecular orbital picture doesn't change the number of orbitals available to the electrons, it just changes the way we view them.

To build electron configurations for molecules, we simply follow the rules developed for forming electron configurations of multielectron atoms, defined by the Aufbau Principle, Pauli's Principle, and Hund's Rule. It's that simple. For the dihydrogen molecule we have 2 electrons, 1 contributed by each of the two hydrogen atoms. Filing the 2 electrons into the lowest lying σ1s molecular orbital, we predict an electron configuration

H2       σ1s2        

For the nitrogen molecule, each nitrogen atom contributes 7 electrons. We have 14 total electrons to file into our molecular orbitals, and we predict an electron configuration

N2       σ1s2 [σ1s*]2 σ2s2 [σ2s*]2 π2px2 π2py2 σ2pz2        

What are the properties of the nitrogen molecule that are predicted? Let's start with the strength of the nitrogen bond, measured by the bond order. The strength of the bond will be determined by the relative excess of electrons in bonding orbitals as opposed to anti-bonding orbitals

bond order = ½ [# electrons in bonding orbitals - # electrons in antibonding orbitals]

We ignore the core electrons, formed from the atomic 1s orbitals, and focus on the valence electrons, contributed by the atomic 2s and 2p orbitals. Dinitrogen has eight electrons in bonding molecular orbitals and two electrons in anti-bonding molecular orbitals. For dinitrogen we find

bond order(N2) = ½ [8-2] = 3

The surplus of electrons in bonding molecular orbitals defines the bond order which is roughly the number of bonds one would draw in a Lewis electron dot structure. For our best Lewis structure of dinitrogen, we find a triple bond, and that agrees with the bond order of three. But we get more than that. We can say that of the three pairs of electrons in bonding molecular orbitals, one pair forms a σ bond along the internuclear z-axis, a second forms a π bond above and below the internuclear axis in the x-direction, and the third forms a π bond above and below the internuclear axis in the y-direction.

Our electron configuration for the dinitrogen molecule also predicts that all electrons are found in doubly-occupied, filled molecular orbitals. Dinitrogen in its ground electronic state is predicted to be diamagnetic.

Moving on to dioxygen, we find an electron configuration

O2       σ1s2 [σ1s*]2 σ2s2 [σ2s*]2 σ2pz2 π2px2 π2py2 [π2py*]1 [π2py*]1        

The bond order for dioxygen is

bond order(O2) = ½ [8-4] = 2

which agrees with our result for the best Lewis structure for dioxygen. We now recognize one pair of bonding electrons to be in a σ bond and one to be in a π bond. More than that, we find that the dioxygen molecule has two half-filled π anti-bonding orbitals, making the molecule paramagnetic. That is exactly what we observed in our experiment!

Using a linear combination of atomic orbitals, we are able to transform our atomic orbitals into molecular orbitals. Following the standard rules for forming an electron configuration, we predict bond strength and magnetism for dinitrogen and dioxygen. The predictions of bond strength are in good agreement with previous predictions based on Lewis electron dot structures. Our prediction that dinitrogen is diamagnetic while dioxygen is paramagnetic is in perfect agreement with the result of our experimental interrogation of the two gases.

Trends in bond order in the homonuclear diatomic molecules

Question: Beginning with dinitrogen and ending with dineon, determine the bond order for the homonuclear diatomics. Plot the bond order as a function of the atomic number, Z.

You can check your answers here.

Trends in bond order in photoexcited homonuclear diatomic molecules

Question: Think of exciting one electron from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO). Beginning with dinitrogen and ending with dineon, plot the bond order as a function of the atomic number for the homonuclear diatomics. Compare your results with those for the molecules in the ground electronic state.

You can check your answers here.