# How to prepare solutions and dilutions

## Calculating a dilution

The quantity of substance introduced via a stock solution is the same as that of the final solution

*n*_{i}= n_{f}

### Starting from a solution with a known concentration

**Example**: prepare 100 mL of a 10^{-1} mol/L solution of Na^{+} from a 1 mol/L solution

Starting from the above equation, we obtain:

Where

**C**= stock solution concentration (in mol/L)_{i}**V**= stock solution volume (in L)_{i}**C**= final solution concentration (in mol/L)_{f}**V**= final solution volume (in L)_{f}Thus, the volume to be withdrawn from the the stock solution of known concentration is:

In order to prepare a 100 mL solution of Na^{+} at 10^{-1} mol/L from a 1 mol/L solution, we will need to withdraw 10 mL then make it up to 100 mL with the solvent (water or other solution).

### Starting from a commercial solution

**Example**: prepare a 100 mL solution at 10^{-1} mol/L of HCl from a commercialised solution.

Most labels of commercial solutions do not give the concentration level but instead the molar mass: M (36.46g/mol), the density: d (1.18) and the percentage by mass: %m (0.35)

#### Calculating a commercial solution's concentration

µ = volumic mass of the stock solution in kg/m

µ0 = volumic mass of water = 1 000 kg/m

^{3}(g/L)µ0 = volumic mass of water = 1 000 kg/m

^{3}(g/L)
Where

**C**= commercial solution concentration_{sol}**µ**= water density = 1000 kg/m_{0}^{3}(g/L)**d**= commercial solution density**%m**= percentage by mass of commercial solution**M**= molar mass of the commercial solution_{sol}#### Calculating the volume to be withdrawn for a dilution

Where

**C**= commercial solution concentration_{sol}**µ**= water density = 1000 kg/m_{0}^{3}(g/L)**d**= commercial solution density**%m**= percentage by mass of commercial solution**M**= molar mass of the solution_{sol}Numerical application:

In order to prepare a 100 mL solution of HCl at 10^{-1} mol/L from a (pure) commercial solution, we need to withdraw a sample of 883 µl then make up to 100 mL with the solvent (water or other solution).