Question

Derive a formula for the volume of water ${V_2}$, which must be added to ${V_1}{\text{ }}mL$ of concentrated solution of molarity ${M_1}$ to give a solution of molarity ${M_2}$.

Answer 1

Hint: This question can be solved by the concept that the number of moles of solute before dilution is equal to the number of moles of solute after dilution. This is known as the principle of dilution.

Complete answer:
As we know that molar concentration (which is also called molarity) is defined as the number of moles of target substance (solute) dissolved in one liter of solution. Molarity is given as concentration of solute divided by volume of the solution. The formula of molarity can be represented as:
$M = \dfrac{n}{V}$ $ - (1)$
$M = $ Molarity of the solution
$n = $ number of moles of solute
$V = $ Volume of the solution
As in this question, we are adding water that is a solvent to the solution which means we are diluting the solution. As we know that dilution is the process of decreasing the concentration of a solute in a solution which can be done by adding more solvent to the solution. Here, dilution means addition of solvent to the solution without the addition of more solvent. Therefore, we can say that while dilution the number of moles of solute remains constant. This is also known as the principle of dilution. The principle of dilution can be represented as:
Initial number of moles $ = $ final number of moles
Now, from equation $ - (1)$ the number of moles of solute can be given as:
That is, Initial molarity $ \times $ initial volume $ = $ final molarity $ \times $ final volume $ - (2)$
As we know that initially the volume of solution was ${V_1}{\text{ }}mL$ and initially the molarity is ${M_1}$. Also, when we add water then the total volume becomes ${V_1} + {V_2}$ and the molarity of the final solution is given as ${M_2}$. Now from equation $ - (2)$, by substituting values we get:
$
  {M_1}{V_1} = {M_2}({V_1} + {V_2}) \\
  {V_2} = \dfrac{{{V_1}({M_1} - {M_2})}}{{{M_2}}} \\
 $
Therefore, the final volume (${V_2}$) after adding water in the solution is $\dfrac{{{V_1}({M_1} - {M_2})}}{{{M_2}}}$.

Note:
Always remember that whenever we add a solvent like water in the solution the process of dilution occurs in which the final volume will be the volume of solution without adding solvent plus the volume of solvent.