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Why is $M_1V_1 = M_2V_2$ only used for diluting solutions? Why can it not be used to figure out how many moles to add to make a solution more dilute?

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    $\begingroup$ Because it is based on conservation of moles. $M_1V_1=\text{moles before dilution}=M_2V_2=\text{moles after dilution}$ $\endgroup$ – Satwik Pasani Jan 13 '14 at 16:55
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As Satwik has stated in the comments, the equation relies on the conservation of moles principle.

$$\dfrac{mol_1}{V_1} \times {V_1} = \dfrac{mol_2}{V_2} \times {V_2}$$

Based on the above, given a constant volume of a solution, adding more moles of solute will only make the concentration increase, not decrease. Adding in more solvent is the only answer in that case.

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