Can I use milliliters in the dilution equation?

Yes, as long as both $V_1$ and $V_2$ are in the same units. The volume units cancel out when evaluating the ratio.

What if two different solutions of the same solute are mixed?

The $M_1 V_1 = M_2 V_2$ equation does not apply directly. Calculate the total moles of solute from both solutions, sum the volumes, and divide total moles by total volume.

Does molarity change with temperature?

Yes. Because the volume of a liquid expands or contracts with temperature changes, the molarity of a solution is temperature-dependent.

How to calculate molarity and dilutions

Molarity ( $M$ ) is the concentration of a solution expressed as moles of solute per liter of solution. The dilution equation, $M_1 V_1 = M_2 V_2$ , applies when adding solvent to a solution to decrease its concentration without changing the total moles of solute.

The setup

Identify the known and unknown variables. For molarity, you need moles of solute ( $n$ ) and volume of solution in liters ( $V$ ). The formula is $M = \frac{n}{V}$ . For dilutions, identify the initial molarity ( $M_1$ ), initial volume ( $V_1$ ), final molarity ( $M_2$ ), and final volume ( $V_2$ ). The governing equation is $M_1 V_1 = M_2 V_2$ .

The steps

For molarity: Convert the given mass of solute to moles using the molar mass. Convert the solution volume to liters. Divide moles by liters. 2. For dilutions: List $M_1$ , $V_1$ , $M_2$ , and $V_2$ . Convert volumes to the same unit (liters or milliliters). Rearrange $M_1 V_1 = M_2 V_2$ to isolate the unknown variable. Substitute the known values and compute the result.

Checking the result

Verify that units cancel correctly. In a dilution, adding solvent increases the volume ( $V_2 > V_1$ ), so the final concentration must be lower than the initial concentration ( $M_2 < M_1$ ).

Common errors

Failing to convert milliliters to liters when calculating molarity from moles. For dilutions, using the volume of solvent added instead of the total final volume ( $V_2$ ). $V_2$ is the sum of the initial volume and the added solvent volume.

Worked example

Calculate the molarity of a solution prepared by dissolving 11.69 g of NaCl (molar mass = 58.44 g/mol) in water to make 250.0 mL of solution. Then, find the final concentration if 50.0 mL of this solution is diluted to a total volume of 500.0 mL.

$n = \frac{11.69 ext{ g}}{58.44 ext{ g/mol}} = 0.200 ext{ mol}$ . $V = 250.0 ext{ mL} = 0.2500 ext{ L}$ . $M_1 = \frac{0.200 ext{ mol}}{0.2500 ext{ L}} = 0.800 ext{ M}$ . For the dilution: $M_1 = 0.800 ext{ M}$ , $V_1 = 50.0 ext{ mL}$ , $V_2 = 500.0 ext{ mL}$ . $M_2 = \frac{M_1 V_1}{V_2} = \frac{(0.800 ext{ M})(50.0 ext{ mL})}{500.0 ext{ mL}} = 0.0800 ext{ M}$ .

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References: OpenStax Chemistry 2e, Chapter 3.3 Molarity · Zumdahl Chemistry 10th Edition, Chapter 4.3 The Composition of Solutions