The dilution equation, \(C_1V_1 = C_2V_2\), is a fundamental relationship in chemistry and biology used to calculate the concentration and volume of solutions before and after dilution. This algebraic formula allows scientists to precisely prepare solutions of a desired concentration from a more concentrated starting material, often called a stock solution. The equation is applied in laboratory settings for preparing reagents, calibrating instruments, and conducting serial dilutions. Understanding this relationship is a foundational skill for anyone working with solutions.
What Each Variable Represents
The equation contains four variables representing the solution’s properties at two points in the dilution process. The subscript “1” denotes the initial state (the concentrated stock solution), and “2” refers to the final, diluted solution. \(C_1\) is the initial concentration, and \(V_1\) is the initial volume, representing the amount of stock solution measured out for dilution.
The variables on the right side describe the target solution. \(C_2\) is the final concentration, which is the desired strength of the diluted solution. \(V_2\) is the final volume, representing the total volume after dilution is complete. For the equation to be solved correctly, the units for both concentrations (\(C_1\) and \(C_2\)) must be identical (e.g., Molarity or ppm). Similarly, the units for both volumes (\(V_1\) and \(V_2\)) must also match (e.g., liters or milliliters).
The Underlying Principle of Conservation
The \(C_1V_1 = C_2V_2\) equation is based on the principle of the conservation of mass, specifically the conservation of solute. A solution consists of a solute (the dissolved substance) and a solvent (the dissolving medium). When a solution is diluted, only the solvent is added.
This means the quantity of solute particles—the number of moles—remains the same before and after dilution. The product of concentration (\(C\)) and volume (\(V\)) mathematically represents the total amount of solute (moles = concentration \(\times\) volume). Because the amount of solute must be conserved, the initial amount (\(C_1V_1\)) must equal the final amount (\(C_2V_2\)). The concentration decreases only because the fixed amount of solute is spread over a larger total volume.
How to Perform Dilution Calculations
The most common application of this equation is determining the volume of stock solution (\(V_1\)) required to prepare a specific volume of a less concentrated solution. To solve for \(V_1\), three variables must be known: the stock concentration (\(C_1\)), the target concentration (\(C_2\)), and the final target volume (\(V_2\)). The equation is algebraically rearranged to isolate the unknown variable: \(V_1 = (C_2V_2) / C_1\).
For example, consider preparing 250 milliliters of a 0.25 M solution from a 5.0 M stock. The known values are \(C_1 = 5.0\) M, \(C_2 = 0.25\) M, and \(V_2 = 250\) mL. Plugging these values into the formula gives \(V_1 = (0.25 \text{ M} \times 250 \text{ mL}) / 5.0 \text{ M}\).
The calculation yields \(V_1 = 12.5\) mL. The Molarity units cancel out, leaving the answer in milliliters. This means 12.5 mL of the 5.0 M stock solution must be measured and then diluted with solvent to a total volume of 250 mL. The equation can be similarly rearranged to solve for any of the other variables (\(C_2\), \(C_1\), or \(V_2\)).
Important Considerations for Use
This equation applies only to simple dilution, which involves adding a pure solvent to a single stock solution. It is not applicable when two different solutions are mixed or when a chemical reaction occurs between the solute and the solvent. Remember that \(V_2\) is the total volume of the diluted solution, not just the volume of solvent added.
In laboratory practice, accuracy relies on using precise measuring tools, such as volumetric flasks, calibrated to contain the exact final volume \(V_2\). When working with highly concentrated reagents, especially strong acids, strict safety protocols must be followed. Dilution often releases heat in an exothermic reaction, which can cause splattering if not handled correctly. Always add the concentrated component slowly to the larger volume of water, allowing the water to safely absorb the generated heat.