Solutions are mixtures where the solute is evenly dispersed throughout the solvent. The amount of solute present in a given volume determines the solution’s concentration. It is often necessary to reduce a solution’s concentration in scientific research, industrial processes, and household applications. This process, called dilution, involves adding more solvent to make the mixture less concentrated. A straightforward mathematical tool provides a precise method for managing these concentration changes.
The Dilution Equation
The relationship \(C_1V_1 = C_2V_2\) is formally known as the Dilution Equation or the Dilution Formula. This mathematical expression is used to accurately prepare a new solution of a desired concentration from a more concentrated starting material, often called the stock solution.
The formula relies on the conservation of moles, a fundamental scientific principle. Since no solute is added or removed during dilution, the total amount of solute remains constant. The quantity of solute in the initial solution must equal the quantity of solute in the final, diluted solution. The equation states that the initial amount of solute (\(C_1V_1\)) equals the final amount of solute (\(C_2V_2\)).
Decoding Concentration and Volume
The letters in the Dilution Equation represent specific properties of the solution. \(C\) stands for concentration, and \(V\) represents volume. The subscript ‘1’ refers to the initial, more concentrated stock solution, while ‘2’ refers to the final, diluted solution.
The concentration variable (\(C\)) is most commonly expressed in molarity (M), defined as moles of solute per liter of solution. \(C\) can also use other units, such as mass per volume or a percentage, provided the concentration units for \(C_1\) and \(C_2\) are identical. The volume variable (\(V\)) is typically measured in liters (L) or milliliters (mL).
Accurate use of the formula requires that the volume units also be consistent on both sides of the equation. When concentration (\(C\)) is multiplied by volume (\(V\)), the volume units cancel out. This calculation confirms the principle of conserved quantity by leaving the amount of solute in moles or a proportional mass.
Calculating Dilution Problems
The Dilution Equation is a practical tool because it allows chemists to solve for any one of the four variables, provided the other three are known. The first step is to identify the three known values, such as the initial concentration (\(C_1\)), the target concentration (\(C_2\)), and the final volume (\(V_2\)). The next step involves algebraically rearranging the formula to isolate the unknown variable.
A common scenario is determining the volume of the stock solution needed, which requires solving for \(V_1\). The formula is rearranged to \(V_1 = (C_2 \times V_2) / C_1\). For instance, if a laboratory needs to prepare 5 liters (\(V_2\)) of a 1.5 M solution (\(C_2\)) from a 6 M stock solution (\(C_1\)), the formula reveals the precise volume of stock solution needed. This calculated volume (\(V_1\)) is then combined with enough solvent to reach the desired final volume (\(V_2\)).