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Dialkyl urea group

Is it possible in very acidic solutions for the nitrogens ($\ce{N}$) to get another $\ce{H+}$, making it $\ce{R-NH2+-R}$? Since it has a free electron pair, it might be possible right? If so, at which pH?

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Reagents/solutions that are acidic enough will protonate the urea group, just not where you think. The most basic site is the oxygen. Perhaps this is a surprise.

The lone pairs on the nitrogen atoms are delocalized (see resonance depiction below). enter image description here

The oxygen atom has the most electron density. Below is an electron density map, where ${\mathrm{\color{red}{red}}}$ represents areas of high electron density and $\mathrm{\color{blue}{blue}}$ represents areas of low electron density. You can see the area around the oxygen atom is red and yellow, while the areas around the nitrogen atoms are green.

enter image description here

This image is from "The effect of intermolecular hydrogen bonding on the planarity of amides" by James A. Platts, et al. published in Phys. Chem. Chem. Phys., volume 2012, issue 14, page number 11944-11952, DOI:10.1039/C2CP41716B. The image is copyright the Royal Society of Chemistry and released under a Creative Commons Attribution 3.0 Unported license.

Protonating the oxygen atom leads to a cation where the positive charge is delocalized (better, more stable).

enter image description here

Protonating one of the nitrogen atoms leads to a cation where the positive charge is localized (worse, less stable).

enter image description here

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Well, yes but no. The nitrogen’s lone pair is in resonance with the $\ce{C=O}$ π bond. They form a π system not unlike the allyl anion. This is usually depicted for the amide case, but there are no significant differences in the urea case.

Amide resonance

Instead, the oxygen has an accesable lone pair perpendicular to the π system, so it is typically protonated first. This happens at low pH values; you typically need sulphuric acid or hydrochloric acid to get quantitive protonation.

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