How to Find Volume from Moles?


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> What volume of $\pu{12.0 M }\ce{HCl}$ is required to make $\pu{75.0 mL}$ of $\pu{3.50 M }\ce{HCl}$?
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I didn't know how to solve it at first until I multiplied $\pu{75mL}$ by $\pu{3.50 M}$ and got $\pu{262.5 mM}$ and then divided that by $\pu{12.0M}$ to get the answer. I was just wondering if there was an equation that I can use for these kinds of questions? Or is it just simple math? Ex. $\pu{75.0mL} \times \pu{3.50 M }\ce{HCl}$. 




The equation you are looking for is the definition of concentration


$$c=\frac nV$$


where  

$c$ is concentration,  

$n$ is amount of substance, and  

$V$ is volume.


You can rearrange this equation to solve for concentration, amount of substance, or volume as required.


Furthermore, you know that the amount of solute does not change when you dilute a solution, i.e.


$$n\_1=n\_2$$


and thus using the above-mentioned equation


$$c\_1\cdot V\_1=c\_2\cdot V\_2$$


You can rearrange this equation to solve the problem that is given in the question:


$$\begin{align}
c\_1\cdot V\_1&=c\_2\cdot V\_2\\[6pt]
V\_1&=\frac{c\_2\cdot V\_2}{c\_1}\\[6pt]
&=\frac{3.50\ \mathrm{mol\ l^{-1}}\times75.0\ \mathrm{ml}}{12.0\ \mathrm{mol\ l^{-1}}}\\[6pt]
&=21.875\ \mathrm{ml}\\[6pt]
&\approx21.9\ \mathrm{ml}
\end{align}$$


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"$\pu{12 M }\ce{ HCl}$" means that there are $\pu{12 mol}$ of $\ce{HCl}$ per liter of solution (or 12 mmol per milliliter).


You are looking for a volume of $\pu{12 M }\ce{ HCl}$ that contains ($\pu{3.5\times 75 = 262.5 mmol}$ ) of $\ce{HCl}$.


Since you need $\pu{262.5 mmol}$, and the $\pu{12 M }$ solution has $\pu{12 mmol}$ per $\pu{mL}$, then you need $\pu{262.5mmol}\div \pu{12M} = \pu{21.875 ml}$ of $\pu{12 M }\ce{ HCl}$.


This is to explain why it worked, but I suggest you to learn what the equations mean, rather than just learning "how to do it": once you understand what you are doing, it's not hard.


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