1. Dilution Factor Calculation
2. Single Dilution Using C₁V₁ = C₂V₂
3. Serial Dilution Calculation
4. Practical Examples
5. Key Formulas Summary

Dilution Factor Calculation

Basic Formula

The dilution factor (DF) represents how many times a solution has been diluted:

$$\text{Dilution Factor} = \frac{\text{Final Volume}}{\text{Sample Volume}}$$

Notation Types

S:D notation (Stock to Diluent)

• Format: Stock Volume : Diluent Volume

S:T notation (Stock to Total)

• Format: Stock Volume : Total Volume

Example

If you add 10 mL of stock solution to 90 mL of solvent:

• Total volume = 100 mL
• DF = 100 mL ÷ 10 mL = 10 (represents a 1:10 dilution)

Single Dilution Using C₁V₁ = C₂V₂

The Dilution Equation

$$C_1V_1 = C_2V_2$$

Variables

• C₁ = initial (starting) concentration
• V₁ = initial (starting) volume
• C₂ = final concentration
• V₂ = final volume

Solving for Unknown Variables

To find initial volume (V₁): $$V_1 = \frac{C_2 \times V_2}{C_1}$$

To find final concentration (C₂): $$C_2 = \frac{C_1 \times V_1}{V_2}$$

To find final volume (V₂): $$V_2 = \frac{C_1 \times V_1}{C_2}$$

Step-by-Step Process

1. Identify all known values (C₁, V₁, C₂, or V₂)
2. Determine which variable you need to solve for
3. Rearrange the equation accordingly
4. Substitute values and calculate
5. Verify units match before calculation

Serial Dilution Calculation

Definition

A serial dilution is a step-wise series of dilutions where the dilution factor remains constant for each step. Common dilution factors are 2-fold (1:2) and 10-fold (1:10).

Common Serial Dilution Setup

For a 10-fold serial dilution:

• Combine 1 mL of sample (or previous dilution) with 9 mL of fresh diluent
• Dilution factor per step = 10⁻¹ (or 1/10)

Total Dilution Factor Formula

When performing multiple sequential dilutions:

$$\text{Total Dilution Factor} = \text{DF}1 \times \text{DF}2 \times \text{DF}_3 \times \ldots$$

Example: 7-Step 10-Fold Serial Dilution

$$\text{Total DF} = 10^{-1} \times 10^{-1} \times 10^{-1} \times 10^{-1} \times 10^{-1} \times 10^{-1} \times 10^{-1} = 10^{-7}$$

The final concentration = 1/10,000,000 of the original

Required Volume Calculations

Transfer Volume (amount to move to next tube): $$\text{Transfer Volume} = \frac{\text{Final Volume}}{\text{Dilution Factor}}$$

Diluent Volume (amount of fresh diluent): $$\text{Diluent Volume} = \text{Final Volume} - \text{Transfer Volume}$$

Practical Examples

Example 1: Basic Dilution Calculation

Question: What volume of a 10 mM stock solution is required to make 20 mL of a 50 μM solution?

Given:

• C₁ = 10 mM (starting concentration)
• C₂ = 50 μM = 0.05 mM (final concentration)
• V₂ = 20 mL (final volume)

Solution: $$V_1 = \frac{C_2 \times V_2}{C_1} = \frac{0.05 \text{ mM} \times 20 \text{ mL}}{10 \text{ mM}} = \frac{1}{10} = 0.1 \text{ mL}$$

Answer: Use 0.1 mL (100 μL) of stock solution + 19.9 mL of diluent = 20 mL total

Example 2: Serial Dilution Planning

Question: Plan a 10-fold serial dilution with 10 mL per tube for 5 steps.

Calculations for each tube:

• Transfer volume = 10 mL ÷ 10 = 1 mL
• Diluent volume = 10 mL - 1 mL = 9 mL

Procedure:

1. Tube 1 (10⁻¹): Transfer 1 mL of original sample to 9 mL diluent
2. Tube 2 (10⁻²): Transfer 1 mL from Tube 1 to 9 mL fresh diluent
3. Tube 3 (10⁻³): Transfer 1 mL from Tube 2 to 9 mL fresh diluent
4. Tube 4 (10⁻⁴): Transfer 1 mL from Tube 3 to 9 mL fresh diluent
5. Tube 5 (10⁻⁵): Transfer 1 mL from Tube 4 to 9 mL fresh diluent

Example 3: Finding Initial Concentration After Serial Dilution

Question: You measure a bacterial count of 100 cells/mL in a sample that was diluted 10⁻⁶. What is the original concentration?

Given:

• Measured concentration = 100 cells/mL
• Final dilution factor = 10⁻⁶

Solution: $$\text{Initial Concentration} = 100 \text{ cells/mL} \times 10^6 = 100,000,000 \text{ cells/mL} = 10^8 \text{ cells/mL}$$

Answer: The original concentration was 10⁸ cells/mL or 100 million cells/mL

Key Formulas Summary

Dilution Factor

$$\text{DF} = \frac{\text{Final Volume}}{\text{Sample Volume}}$$

Single Dilution (C₁V₁ = C₂V₂)

• Stock Volume: $V_1 = \frac{C_2 \times V_2}{C_1}$
• Final Concentration: $C_2 = \frac{C_1 \times V_1}{V_2}$

Serial Dilution

• Transfer Volume: $\frac{\text{Final Volume}}{\text{Dilution Factor}}$
• Diluent Volume: $\text{Final Volume} - \text{Transfer Volume}$
• Total DF: $\text{DF}1 \times \text{DF}2 \times \text{DF}_3 \times \ldots$

Back-Calculating Original Concentration

$$\text{Original Concentration} = \text{Measured Concentration} \times \text{Total Dilution Factor}$$

Quick Reference Tips

• Always ensure units are consistent before performing calculations
• Mix thoroughly at each dilution step, especially in serial dilutions
• Use sterile techniques when preparing dilutions
• Label all tubes clearly with the dilution factor (e.g., 10⁻³, 10⁻⁶)
• For 10-fold dilutions: 1 mL sample + 9 mL diluent = 10 mL total
• For 2-fold dilutions: 1 mL sample + 1 mL diluent = 2 mL total
• Verify calculations by working backwards from the final solution