Q. how to calculate concentration of diluted solution

Goal. Calculate the concentration of a diluted solution, using a step by step method and clear algebraic formulas.

Principle. Dilution conserves the amount (moles or mass) of solute. If the concentration and volume of the stock (before dilution) are \(C_{1}\) and \(V_{1}\), and the concentration and volume after dilution are \(C_{2}\) and \(V_{2}\), then the amount of solute initially equals the amount finally. In terms of concentration and volume this gives the fundamental dilution equation:

\[ C_{1} V_{1} = C_{2} V_{2} \]

What each symbol means. \(C_{1}\) is the initial (stock) concentration. \(V_{1}\) is the volume of stock taken for the dilution (the aliquot). \(C_{2}\) is the desired final concentration. \(V_{2}\) is the total final volume after adding solvent.

Algebraic rearrangements. Depending on which quantity you need, solve the fundamental equation for that variable.

To find the final concentration \(C_{2}\):

\[ C_{2} = \frac{C_{1} V_{1}}{V_{2}} \]

To find the required aliquot volume \(V_{1}\) of stock to make a solution of concentration \(C_{2}\) and volume \(V_{2}\):

\[ V_{1} = \frac{C_{2} V_{2}}{C_{1}} \]

Units and consistency. Always use consistent concentration units for \(C_{1}\) and \(C_{2}\) (for example both in mol L^{-1} or both in \% w/v). Always use the same volume units for \(V_{1}\) and \(V_{2}\) (for example both in L or both in mL). If necessary convert mL to L by dividing by 1000 before using molarity formulas, or keep everything in mL if you are using percent by volume or w/v, since the algebra only requires consistent units.

Worked example 1. You have a 5.00 M stock solution and you need 250.0 mL of a 0.500 M solution. Find the volume of stock to pipette.

Step 1. Convert volumes to liters to match molarity units, if you prefer: \(V_{2}=250.0\ \mathrm{mL}=0.250\ \mathrm{L}\).

Step 2. Use \(V_{1} = \dfrac{C_{2} V_{2}}{C_{1}}\).

\[ V_{1} = \frac{0.500\ \mathrm{M} \times 0.250\ \mathrm{L}}{5.00\ \mathrm{M}} \]

\[ V_{1} = \frac{0.125\ \mathrm{mol}}{5.00\ \mathrm{M}} = 0.0250\ \mathrm{L} \]

Step 3. Convert to mL if desired: \(0.0250\ \mathrm{L} = 25.0\ \mathrm{mL}\). So pipette 25.0 mL of the 5.00 M stock and dilute to 250.0 mL total volume.

Worked example 2 (percent solution). You have a 10\% w/v stock and need 500 mL of a 2\% w/v solution. Find the volume of stock to use.

Use the same formula with percent units (consistent units cancel):

\[ V_{1} = \frac{C_{2} V_{2}}{C_{1}} = \frac{2\% \times 500\ \mathrm{mL}}{10\%} \]

\[ V_{1} = \frac{1000\% \cdot \mathrm{mL}}{10\%} = 100\ \mathrm{mL} \]

So take 100 mL of the 10\% stock and add solvent to reach 500 mL total.

Serial dilutions. For large dilution factors it is often convenient to perform serial dilutions. The dilution factor \(DF\) is the ratio of final volume to aliquot volume: \(DF = \dfrac{V_{2}}{V_{1}}\). The final concentration after a single dilution is \(C_{2} = \dfrac{C_{1}}{DF}\). For multiple sequential dilutions multiply the dilution factors: total dilution factor is the product of the individual \(DF\) values, and the final concentration is \(C_{\text{final}} = \dfrac{C_{\text{stock}}}{DF_{\text{total}}}\).

Checklist and common mistakes.

1. Ensure units for volumes are the same. 2. Ensure concentration units match. 3. Remember that \(V_{1}\) is the aliquot of stock, and \(V_{2}\) is the final total volume (not the volume of solvent added). 4. If using percent, include the \% when calculating, or treat percent numerically provided both concentrations use the same percent definition. 5. When necessary convert mL to L for molarity calculations.

Summary. Use the conservation formula \[ C_{1} V_{1} = C_{2} V_{2} \] and solve for the unknown: \[ C_{2} = \frac{C_{1} V_{1}}{V_{2}} \] or \[ V_{1} = \frac{C_{2} V_{2}}{C_{1}} \]. Apply consistent units and follow the step by step examples above to calculate diluted concentrations or required aliquot volumes.