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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?

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    $\begingroup$ @Pritt There is always a space between the value and the unit. You can use \pu{5 g} for easier and correct typesetting. $\endgroup$
    – Martin - マーチン ♦
    Apr 27, 2017 at 12:55
  • $\begingroup$ @Martin-マーチンThanks for telling me this. I wasn't aware of that. $\endgroup$
    – Pritt says Reinstate Monica
    Apr 27, 2017 at 12:56
  • $\begingroup$ @Martin-マーチンAnd how did you write that syntax for \pu{5 g} with greying background? Moreover, how is that yellow highlight background made in the questions? $\endgroup$
    – Pritt says Reinstate Monica
    Apr 27, 2017 at 12:58
  • $\begingroup$ @Pritt you can include pieces of code in your comments by enclosing them within back-ticks. So the code gets shaded like this. See Jeff Atwood's answer. $\endgroup$
    – Berry Holmes
    Apr 27, 2017 at 13:39
  • $\begingroup$ The yellow highlighting is the quote option, you use that to add chunks of text copied from external references; it can be achieved by using > symbol before the text. $\endgroup$
    – Berry Holmes
    Apr 27, 2017 at 13:42

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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 &= m_\mathrm{salt} \times \Delta_\mathrm{solution} H^\circ_\mathrm{salt}\\ Q &= m_\mathrm{water} \times C_{p(\mathrm{water})} \times \Delta T\\ \therefore m_\mathrm{salt} &= \frac{m_\mathrm{water} \times C_{p(\mathrm{water})} \times \Delta T}{\Delta_\mathrm{solution} H^\circ_\mathrm{salt}} \end{align}

Hope this helps.

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