Calculating Concentrations with Units and Dilutions

Calculating the concentration of a chemical solution is a basic skill all students of chemistry must develop early in their studies. What is concentration? Concentration refers to the amount of solute that is dissolved in a solvent. We normally think of a solute as a solid that is added to a solvent (e.g., adding table salt to water), but the solute could easily exist in another phase. For example, if we add a small amount of ethanol to water, then the ethanol is the solute, and the water is the solvent. If we add a smaller amount of water to a larger amount of ethanol, then the water could be the solute.

How to Calculate Units of Concentration

Once you have identified the solute and solvent in a solution, you are ready to determine its concentration. Concentration may be expressed several different ways, using percent composition by mass, volume percent, mole fraction, molarity, molality, or normality.

Percent Composition by Mass (%)

This is the mass of the solute divided by the mass of the solution (mass of solute plus mass of solvent), multiplied by 100.
Example:
Determine the percent composition by mass of a 100 g salt solution which contains 20 g salt.
Solution:
20 g NaCl / 100 g solution x 100 = 20% NaCl solution

Volume Percent (% v/v)

Volume percent or volume/volume percent most often is used when preparing solutions of liquids. Volume percent is defined as:
v/v % = [(volume of solute)/(volume of solution)] x 100%
Note that volume percent is relative to the volume of the solution, not the volume of solvent. For example, wine is about 12% v/v ethanol. This means there is 12 ml ethanol for every 100 ml of wine. It is important to realize liquid and gas volumes are not necessarily additive. If you mix 12 ml of ethanol and 100 ml of wine, you will get less than 112 ml of solution.
As another example, 70% v/v rubbing alcohol may be prepared by taking 700 ml of isopropyl alcohol and adding sufficient water to obtain 1000 ml of solution (which will not be 300 ml).

Mole Fraction (X)

This is the number of moles of a compound divided by the total number of moles of all chemical species in the solution. Keep in mind, the sum of all mole fractions in a solution always equals 1.
Example:What are the mole fractions of the components of the solution formed when 92 g glycerol is mixed with 90 g water? (molecular weight water = 18; molecular weight of glycerol = 92)
Solution:
90 g water = 90 g x 1 mol / 18 g = 5 mol water
92 g glycerol = 92 g x 1 mol / 92 g = 1 mol glycerol
total mol = 5 + 1 = 6 mol
x_water = 5 mol / 6 mol = 0.833
x _glycerol = 1 mol / 6 mol = 0.167
It's a good idea to check your math by making sure the mole fractions add up to 1:
x_water + x_glycerol = .833 + 0.167 = 1.000

Molarity (M)

Molarity is probably the most commonly used unit of concentration. It is the number of moles of solute per liter of solution (not necessarily the same as the volume of solvent!).
Example:
What is the molarity of a solution made when water is added to 11 g CaCl₂ to make 100 mL of solution? (The molecular weight of CaCl₂ = 110)
Solution:
11 g CaCl₂ / (110 g CaCl₂ / mol CaCl₂) = 0.10 mol CaCl₂
100 mL x 1 L / 1000 mL = 0.10 L
molarity = 0.10 mol / 0.10 L
molarity = 1.0 M

Molality (m)

Molality is the number of moles of solute per kilogram of solvent. Because the density of water at 25°C is about 1 kilogram per liter, molality is approximately equal to molarity for dilute aqueous solutions at this temperature. This is a useful approximation, but remember that it is only an approximation and doesn't apply when the solution is at a different temperature, isn't dilute, or uses a solvent other than water.
Example:What is the molality of a solution of 10 g NaOH in 500 g water? (Molecular weight of NaOH is 40)
Solution:
10 g NaOH / (40 g NaOH / 1 mol NaOH) = 0.25 mol NaOH
500 g water x 1 kg / 1000 g = 0.50 kg water
molality = 0.25 mol / 0.50 kg
molality = 0.05 M / kg
molality = 0.50 m

Normality (N)

Normality is equal to the gram equivalent weight of a solute per liter of solution. A gram equivalent weight or equivalent is a measure of the reactive capacity of a given molecule. Normality is the only concentration unit that is reaction dependent.
Example:
1 M sulfuric acid (H₂SO₄) is 2 N for acid-base reactions because each mole of sulfuric acid provides 2 moles of H⁺ ions. On the other hand, 1 M sulfuric acid is 1 N for sulfate precipitation, since 1 mole of sulfuric acid provides 1 mole of sulfate ions.

1. Grams per Liter (g/L)
   This is a simple method of preparing a solution based on grams of solute per liter of solution.
2. Formality (F)
   A formal solution is expressed regarding formula weight units per liter of solution.
3. Parts per Million (ppm) and Parts per Billion (ppb)Used for extremely dilute solutions, these units express the ratio of parts of solute per either 1 million parts of the solution or 1 billion parts of a solution.
   Example:
   A sample of water is found to contain 2 ppm lead. This means that for every million parts, two of them are lead. So, in a one gram sample of water, two-millionths of a gram would be lead. For aqueous solutions, the density of water is assumed to be 1.00 g/ml for these units of concentration.

How to Calculate Dilutions

You dilute a solution whenever you add solvent to a solution. Adding solvent results in a solution of lower concentration. You can calculate the concentration of a solution following a dilution by applying this equation:

M_iV_i = M_fV_f

where M is molarity, V is volume, and the subscripts i and f refer to the initial and final values.

Example:
How many milliliters of 5.5 M NaOH are needed to prepare 300 mL of 1.2 M NaOH?

Solution:
5.5 M x V₁ = 1.2 M x 0.3 L
V₁ = 1.2 M x 0.3 L / 5.5 M
V₁ = 0.065 L
V₁ = 65 mL

So, to prepare the 1.2 M NaOH solution, you pour 65 mL of 5.5 M NaOH into your container and add water to get 300 mL final volume