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.
I have the formula:
$$Q=ms\Delta T$$
I looked up the enthalpy of solution for ammonium nitrate and found it to be $\pu{25.41 kJ/mol}$.
I also have data that if I add $\pu{15 g}$ of $\ce{NH4NO3}$ the temperature will decrease by $\pu{25 ^\circ C}$, if I add $\pu{10 g}$ the temperature will decrease by $\pu{20 ^\circ C}$, and if I add $\pu{5 g}$ the temperature will decrease $\pu{13 ^\circ C}$.
Clearly then, to get my solution to be $\pu{3 ^\circ C}$, I would need to add a mass somewhere in between $10$ and $\pu{15 g}$. In the lab I could just add random amounts between until the solution settled on $\pu{3 ^\circ C}$. Shouldn't there be a mathematical way to solve this rather than by random guessing?
I tried to use $Q=ms\Delta T$ by substituting the enthalpy of solution (in J) for $Q$, the specific heat of water for $s$, and $\pu{20 ^\circ C}$ for $\Delta T$ (since I am going from $23$ to $\pu{3 ^\circ C}$). However, solving $$\pu{25410 J} = m (\pu{4.186 J// ^\circ C})(\pu{20 ^\circ C})$$ gives me $\pu{303.66 g}$ which is clearly nowhere near the $\pu{10 - 1 g}$ that is sufficient to change the temperature.
How can I accurately determine the mass using a formula?
\pu{5 g}
for easier and correct typesetting. $\endgroup$like this
. See Jeff Atwood's answer. $\endgroup$