The percent scharacter of carbon participating in a \(\mathrm{C}-\mathrm{H}\) bond can be established by measuring the \({ }^{13} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant and using the relationship $$ \text { Percent scharacter }=0.2 \mathrm{~J}\left({ }^{13} \mathrm{C}-{ }^{1} \mathrm{H}\right) $$ The \({ }^{15} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant observed for methane, for example, is \(125 \mathrm{~Hz}\), which gives \(25 \%\) scharacter, the value expected for an \(s p^{3}\) hybridized carbon atom. (a) Calculate the expected \({ }^{13} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant in ethylene and acetylene. (b) In cyclopropane, the \({ }^{19} \mathrm{C}-{ }^{1} \mathrm{H}\) coupling constant is \(160 \mathrm{~Hz}\). What is the hybridization of carbon in cyclopropane?

Understanding the percentage of s-character in a molecule provides insight into its bonding characteristics. 'Percent s-character' means the contribution of the s orbital to the bonding orbitals in a hybridized atom. This concept is critical because the amount of s-character influences molecular geometry, bond angles, and bond strengths.
In simple terms, when an atomic orbital undergoes hybridization, it blends its native s and p orbitals to create new hybrid orbitals, such as sp3, sp2, or sp for carbon atoms. With this mixture, the electron density around the atom will change, impacting chemical properties significantly. A higher s-character leads to a bond that is closer to the nucleus and thus stronger and shorter.
To calculate the percent s-character, we can use the relationship provided in the exercise, which links the 13C-1H coupling constant to the s-character. For example, in methane (CH_4), with a coupling constant of 125 Hz, the percent s-character comes out to be 25%, which is consistent with an sp3 hybridized carbon atom, having one part s character to three parts p character (1/4 s-character).

The 13C-1H coupling constant is a valuable tool used in nuclear magnetic resonance (NMR) spectroscopy to provide information about the molecular structure, specifically the relationship between carbon and hydrogen atoms. It measures the interaction between the magnetic fields of adjacent carbon-13 and hydrogen nuclei.

Relationship with Hybridization

By measuring the coupling constant, one can deduce the hybridization state of the carbon atom. This is because the coupling constant is affected by the electron's magnetic environment, which changes with the orbital hybridization. Within the same molecule, a carbon with more s-character will show a larger coupling constant.
For instance, in the step-by-step solution, to find ethylene and acetylene's 13C-1H coupling constants, the percent s-character was determined from their hybridizations (sp2 and sp, respectively), then applied the provided relationship. As the s-character increases, so does the measured coupling constant. This relationship allows us to calculate that ethylene (sp2 hybridized) has a coupling constant of 166.65 Hz and acetylene (sp hybridized) has a coupling constant of 250 Hz.

Let's delve into sp2 hybridization, a term that frequently surfaces in organic chemistry. When a carbon atom uses one s and two p orbitals for hybridization, it forms three equivalent sp2 hybrid orbitals. Each of these orbitals contains one part s and two parts p character, or roughly 33.33% s-character.
An atom with sp2 hybridization will have its orbitals arranged in a trigonal planar geometry, making an angle of 120 degrees with each other. This geometry provides stability through orbital overlap, often seen in alkenes like ethylene.

Effects on Physical Properties

This type of hybridization is also characterized by the presence of a pi bond resulting from the unhybridized p orbital. It affects properties such as bond lengths, strength, and even chemical reactivity. For example, the carbon-carbon double bond in ethylene is shorter and stronger than a single bond due to the increased s-character.
As demonstrated in the solution for cyclopropane's percent s-character and by evaluating the coupling constant, it was deductible that the carbon atoms in cyclopropane are also sp2 hybridized. This is a powerful instance of how hybridization affects and is deduced from molecular properties.