The right answer IMHO for Overfull \hbox ... problems is microtype, but just for the record, specially if you are using xelatex (where the MWE of Heiko Oberdiek do not work), exactly the same ouput for this paragraph is obtained simply with \emergencystretch1em (or \setlength{\emergencystretch}{1em} for the purist):

\documentclass{article}
\emergencystretch1em
\begin{document}
The substance used in this experiment is 1,1-diphenyl-2-picryl-hydrazyl
(DPPH). This organic compound is a relatively stable free radical which has an
unpaired valence electron at one atom of the nitrogen bridge which is the source
of the paramagnetism of this compound. In a free nitrogen atom, 6 electrons
pair off and contribute no orbital or spin angular momentum. Nor do these
electrons as a whole have a net magnetic moment.
\end{document}
Edit
As also seem of interest in some comments to maintain the acronym in the first line, beside of the changes of the text or the layout as proposed Delia Ruby, a general package ninjutsu solution could be microtype with a reasonable settings of the shrink option.

\documentclass{article}
\usepackage[shrink=46]{microtype}
\begin{document}
The substance used in this experiment is 1,1-diphenyl-2-picryl-hydrazyl
(DPPH). This organic compound is a relatively stable free radical which has an
unpaired valence electron at one atom of the nitrogen bridge which is the source
of the paramagnetism of this compound. In a free nitrogen atom, 6 electrons
pair off and contribute no orbital or spin angular momentum. Nor do these
electrons as a whole have a net magnetic moment.
\end{document}
However, the minimum value in this case must be at least 46, that seem a little excessive, so whether this is a good idea just for this MWE ... I leave open.
Anyway, is worth to note that a strong shrinking could be limited just to the problematic paragraph:
\documentclass{article}
\usepackage{microtype}
\SetExpansion[context=bigformula,shrink=60]{encoding=OT1}{}
\begin{document}
Normal text that not need strong shrinking.
{\microtypecontext{expansion=bigformula} The substance used ... }
More text that not need strong shrinking.
\end{document}
chemformulato deal with the compound name. – Johannes_B Aug 08 '15 at 14:10\-at a break point, if you know the rules. – Bernard Aug 08 '15 at 14:10