How to determine the mass of ammonium nitrate needed and enthalpy of solution?
I am trying to figure out how to determine the mass of ammonium nitrate needed to reduce the temperature of solution from $\pu{23 ^\circ C}$ (measured room temperature at start) to $\pu{3 ^\circ C}$, once it dissolves in water.


The result will depend on the amount of water where you're dissolving the salt. For instance, the data you present refers to different solvent mass. And by the way, for your temperature difference ($\pu{20^\circ C}$) the problem is already solved: a $\pu{20^\circ C}$ temperature decrease, will require $\pu{10 g}$ of the salt.

It is important to distinguish the mass of the salt from the mass of the solvent.
\begin{align}
Q &amp;= m_\mathrm{salt} \times \Delta_\mathrm{solution} H^\circ_\mathrm{salt}\\
Q &amp;= m_\mathrm{water} \times C_{p(\mathrm{water})} \times \Delta T\\
\therefore m_\mathrm{salt} &amp;= \frac{m_\mathrm{water} \times C_{p(\mathrm{water})} \times \Delta T}{\Delta_\mathrm{solution} H^\circ_\mathrm{salt}}
\end{align}

Hope this helps.