Dilution is a fundamental process in laboratory science, healthcare, and chemistry, involving the reduction of a solute’s concentration by mixing it with a greater volume of solvent. This action allows researchers and technicians to create solutions with precise, lower concentrations from a more concentrated starting material, known as the stock. Mastering the calculations behind this process is a foundational skill necessary for preparing accurate medical doses, calibrating laboratory instruments, and ensuring the success of biological experiments.
The Core Mathematical Principle
The mathematics of dilution are governed by the principle of conservation of mass: the total amount of the dissolved substance, or solute, remains unchanged during the process. When a concentrated solution is mixed with a solvent, the mass of the solute does not change; only the volume of the mixture increases, causing the concentration to decrease. This relationship is mathematically expressed by the formula \(C_1V_1 = C_2V_2\).
In this equation, \(C_1\) and \(V_1\) represent the initial concentration and volume of the stock solution, while \(C_2\) and \(V_2\) represent the final concentration and total volume of the diluted solution. The product \(C_1V_1\) calculates the total amount of solute in the initial sample, which must equal the total amount of solute calculated by \(C_2V_2\). To ensure accurate calculations, both concentration units (e.g., Molarity) and both volume units (e.g., milliliters) must be consistent on both sides of the equation.
Step-by-Step Single Dilution Calculations
Applying the conservation equation begins by clearly identifying the three known values and the single unknown quantity. A common problem involves determining the necessary volume of a concentrated stock solution (\(V_1\)) needed to prepare a specific volume (\(V_2\)) at a target lower concentration (\(C_2\)). For instance, if a scientist needs to prepare 500 milliliters of a 0.2 M solution from a 2.0 M stock, the known values are \(C_1 = 2.0 \text{ M}\), \(C_2 = 0.2 \text{ M}\), and \(V_2 = 500 \text{ mL}\).
To solve for the required volume of stock, \(V_1\), the equation is algebraically rearranged: \(V_1 = (C_2V_2) / C_1\). Plugging in the example values, \(V_1\) equals \((0.2 \text{ M} \times 500 \text{ mL}) / 2.0 \text{ M}\). This simplifies to \(100 \text{ M}\cdot\text{mL} / 2.0 \text{ M}\). The molarity units cancel out during the division, leaving the final result in milliliters.
The calculation yields \(V_1 = 50\) milliliters. This means 50 milliliters of the 2.0 M stock solution must be measured out precisely. This volume contains the exact amount of solute required for the final concentration.
The remaining volume must be composed of the pure solvent, often distilled water, calculated by subtracting the stock volume from the final volume (\(V_{\text{solvent}} = V_2 – V_1\)). In this example, 50 milliliters of stock is added to 450 milliliters of solvent to reach the final 500-milliliter volume of the 0.2 M solution. This two-part calculation ensures both the correct amount of solute and the correct final concentration are achieved.
Calculating the Dilution Factor
Beyond calculating volumes, scientists often refer to the Dilution Factor (DF), which is a dimensionless number expressing the extent to which the concentration has been reduced. The DF indicates how many times more concentrated the stock solution is compared to the final dilute solution. The Dilution Factor can be calculated directly by taking the ratio of the stock concentration to the final concentration (\(C_{\text{stock}} / C_{\text{final}}\)).
Alternatively, the factor can be calculated using volumes by dividing the total final volume by the volume of the stock solution used (\(V_{\text{final}} / V_{\text{stock}}\)). Using the previous example where 50 milliliters of stock was diluted into a total volume of 500 milliliters, the DF is \(500 \text{ mL} / 50 \text{ mL}\), which equals 10. This result is often expressed as a factor of “10x” (common in chemistry) or as a ratio of “1:10” (common in biology).
The ratio notation, such as 1:10, means that one part of the stock solution is combined with nine parts of solvent to create ten total parts of the final solution. The Dilution Factor provides a quick, standardized way to compare relative concentrations, which is particularly useful when communicating protocols between laboratories.
Solving Multi-Step Serial Dilutions
When extremely high dilutions are required, such as in microbiology to count bacterial colonies or in immunology to titer antibodies, a technique called serial dilution is employed. This method involves performing a sequence of smaller, identical dilution steps, where the diluted solution from one step becomes the stock solution for the next. Serial dilutions are far more accurate than attempting to measure the minuscule volume of stock needed for a single, massive dilution, since measurement errors are minimized across smaller transfers.
Consider a laboratory procedure that requires a total dilution factor of 10,000x, which can be achieved through four consecutive 1:10 dilutions. In the first step, one milliliter of the original stock solution is added to nine milliliters of solvent, creating a 1:10 dilution with a factor of 10. This newly created solution then serves as the stock for the second step.
For the second step, one milliliter of the first dilution is transferred to a new tube containing nine milliliters of solvent, resulting in a second 1:10 dilution. To find the total Dilution Factor after these two steps, the individual factors are multiplied together: \(10 \times 10\) equals a total factor of 100x. The final concentration is found by dividing the original stock concentration by the total cumulative Dilution Factor.
If the initial stock concentration was 10 M, after the first 1:10 dilution, the concentration becomes 1 M. After the second 1:10 dilution, the concentration is reduced to 0.1 M. By repeating the 1:10 step four times, the total dilution factor becomes \(10 \times 10 \times 10 \times 10\), or \(10^4\), which is 10,000x. This accurately reduces the final concentration of the original 10 M stock down to a final concentration of 0.001 M.