Dilution Equation M1V1 = M2V2

M1V1 = M2V2: diluting adds only solvent, so the moles of solute — concentration times volume — stay the same.

M_1 V_1 = M_2 V_2

M1 x V1 = M2 x V2

What it means

Before and after a dilution you hold the same moles of solute, and moles equal concentration times volume. So M₁V₁ = M₂V₂: know any three of the four and you can solve for the fourth. The equation only works when dilution is the only thing happening — if solute is added or removed, or a reaction consumes it, the mole balance behind it no longer holds.

Variables

Symbol	Name	Unit	Meaning
$M1$	Initial molarity	mol/L (M)	The concentration of the concentrated starting solution.
$V1$	Initial volume	mL or L	The volume of the concentrated solution taken, in the same units as V2.
$M2$	Final molarity	mol/L (M)	The concentration of the diluted final solution.
$V2$	Final volume	mL or L	The total volume of the diluted final solution.

Assumptions

Only solvent is added — no solute enters or leaves during the dilution.
The solute is the same on both sides of the equation.
Volumes on both sides use the same units (mL with mL, L with L).
The solution mixes completely, with no volume contraction effects beyond ordinary mixing.

Worked example

Problem

How many milliliters of 12.0 M HCl stock are needed to prepare 500.0 mL of 0.500 M HCl?

Given:

Stock concentration M1 = 12.0 M
Desired concentration M2 = 0.500 M
Desired volume V2 = 500.0 mL

Rearrange for the unknown: V1 = M2V2 / M1.
Substitute: V1 = (0.500 M)(500.0 mL) / 12.0 M.
Multiply the numerator: 0.500 × 500.0 = 250.
Divide: 250 / 12.0 = 20.8 mL.

20.8 mL of 12.0 M stock, diluted to 500.0 mL

Only 20.8 mL of the concentrated stock is needed — always add the acid to water, then dilute to the final volume mark.