Alkanes

The simples class of molecules is the simple straight-chain alkanes, which are the most prevalent in petroleum. The general formula for a linear alkane chain C$_n$H$_{2n+2}$. That is, linear alkanes contain $n$ carbons and $2n+2$ hydrogens and nothing else. Some examples are methane CH$_4$, ethane C$_2$H$_6$, propane C$_3$H$_8$, butane C$_4$H$_{10}$, etc. In a general alkane chain, each carbon is bonded to two other carbons and two hydrogens, execpt for the carbons at the ends of the chain, which are bonded to one other carbon and three hydrogens. Thus, butane is CH$_3$CH$_2$CH$_2$CH$_3$, for example.

In order to understand the bonding orbitals, we can apply the valence bond theory directly. Because each carbon is bonded to four other atoms, each carbon is $sp^3$ hybridized. In the CH bonds, an $sp^3$ hybrid orbital from carbon combines with the 1$s$ orbital of H to form $\sigma$ bonds. Recognizing that a hybrid orbital is just another type of atomic orbital and that each bond contains two electrons, the form of the two-particle valence-bond wave function is:

$$\Psi({\bf x}_1,{\bf x}_2) = C\left[\psi_{1s}^{\rm H}({\bf r}... ...rrow}(s_2) - \psi_{\uparrow}(s_2)\psi_{\downarrow}(s_1)\right]$$

for a hydrogen H bonding to a carbon C. Here, $\psi_{sp^3}$ is one of the $sp^3$ hybrids of the carbon, and $\psi_{1s}$ is a $1s$ orbital of H. For the CC bond, two $sp^3$ hybrids on the two different carbons combine to form another $\sigma$ bond whose wave function is

$$\Psi({\bf x}_1,{\bf x}_2) = C\left[\psi_{sp^3}^{{\rm C}_1}({... ...rrow}(s_2) - \psi_{\uparrow}(s_2)\psi_{\downarrow}(s_1)\right]$$

Here, we have to remember to choose among the $sp^3$ orbitals for each atom that will overlap substantially with the $sp^3$ orbitals of the bonding partners as the figure below illustrates for methane and ethane:

Figure: Overlapping 1s and sp$^3$ orbitals in methane or two sp$^3$ orbitals in ethane.

We could also combine the sp$^3$ orbitals using LCAO theory to form $\sigma$ and $\sigma^*$ molecular orbitals. The appropriate combinations would be

$$\sigma_{{\rm sp}^3} = {1 \over \sqrt{2(1-S)}} \left[\psi_{{\rm sp}^3}^{{\rm C}_1}({\bf r}) - \psi_{{\rm sp}^3}^{{\rm C}_2}({\bf r})\right]$$

$$\sigma^*_{{\rm sp}^3} = {1 \over \sqrt{2(1+S)}} \left[\psi_{{\rm sp}^3}^{{\rm C}_1}({\bf r}) + \psi_{{\rm sp}^3}^{{\rm C}_2}({\bf r})\right]$$

The $\sigma_{{\rm sp}^3}$ contains the bonding electron pair between the two carbons.

Because all of the CC bonds are single bonds, alkanes are quite flexible. In ethane, for example, the two CH$^3$ groups (called methyl groups can rotate about the C-C bond axis. This type of rotation is known as torsion (see figure below):

Figure: Rotation of the methyl group about the C-C bond axis.

Torsional motion depends on the so-called dihedral angle, which is the angle between the planes formed by four atoms. That is, we use the angle between the plane defined by the atoms 1, 2, and 3 and the plane defined by the atoms 2, 3, and 4, as the figure below illustrates:

Figure: Defining a dihedral angle.

Interestingly, if the bond axis between atoms 2 and 3 coincides with the $z$-axis, and the bond axis between atoms 1 and 2 coincides with the $x$-axis, then the spherical coordinates for the bond between atoms 3 and 4 can be used to define the dihedral angle; it will simply be the azimuthal angle. As a result of this flexibility, a long alkane chain can assume a compact structure as the figure below illustrates for C$_{400}$H$_{802}$:

Figure: Example conformation of C$_{400}$H$_{802}$.