Question

The recommended procedure for preparing a very dilute solution is not to weigh out a very small mass or measure a very small volume of a stock solution. Instead, it is done by a series of dilutions. A sample of $0.8214\mathrm{g}$ of ${\mathrm{KMnO}}_4$ was dissolved in water and made up to the volume in a $500-\mathrm{mL}$ volumetric flask. A $2.000-\mathrm{mL}$ sample of this solution was transferred to a $1000-\mathrm{mL}$ volumetric flask and diluted to the mark with water. Next, $10.00\mathrm{mL}$ of the diluted solution was transferred to a $250-\mathrm{mL}$ flask and diluted to the mark with water. (a) Calculate the concentration (in molarity) of the final solution. (b) Calculate the mass of ${\mathrm{KMnO}}_4$ needed to directly prepare the final solution.

Short answer

The final concentration is 0.00000083152 M and mass required is 0.0000328319 g.

Step-by-step solution

Determine the initial concentration

First, calculate the number of moles of KMnO₄ in the initial solution using its molar mass (158.04 g/mol). The given mass is 0.8214 g, so the number of moles $n$ is:$$n=\frac{0.8214\text{g}}{158.04\text{g/mol}}=0.005197\text{mol}$$The initial concentration $C_1$ in the 500-mL flask is:$$C_1=\frac{0.005197\text{mol}}{0.500\text{L}}=0.010394\text{M}$$

Calculate the concentration after the first dilution

A 2.000-mL sample is taken from the initial solution and diluted in a 1000-mL flask. Use the dilution formula $C_1{V}_1=C_2{V}_2$:- $C_1=0.010394\text{M}$- $V_1=0.002\text{L}$- $V_2=1.000\text{L}$Solve for $C_2$:$$C_2=\frac{C_1{V}_1}{V_2}=\frac{0.010394\text{M}\times 0.002\text{L}}{1.000\text{L}}=0.000020788\text{M}$$

Calculate the concentration after the second dilution

Next, a 10.00-mL sample of the solution (from Step 2) is diluted in a 250-mL flask. Again using the dilution formula:- $C_2=0.000020788\text{M}$- $V_1=0.010\text{L}$- $V_2=0.250\text{L}$Solve for $C_3$:$$C_3=\frac{C_2{V}_1}{V_2}=\frac{0.000020788\text{M}\times 0.010\text{L}}{0.250\text{L}}=0.00000083152\text{M}$$

Calculate the mass needed to directly prepare the final solution

For direct preparation of the final solution in the 250-mL flask at the same concentration (from Step 3) without dilution steps:- Find moles needed: $n=C_{3}V=0.00000083152\text{M}\times 0.250\text{L}=2.0788\times {10}^{-7}\text{mol}$- Calculate mass: $m=n\times \text{molar mass}=2.0788\times {10}^{-7}\text{mol}\times 158.04\text{g/mol}=0.0000328319\text{g}$

Key concepts

Dilution Process

The dilution process is an important technique used in chemistry to decrease the concentration of a solution by increasing its volume. In this method, more solvent, usually water, is added to a given amount of a more concentrated solution. This does not change the amount of solute present but significantly increases the total volume of the solution.

This process is especially useful when dealing with solutions that need to be measured accurately, like in the preparation of dilute solutions from a concentrated stock. Here's how it typically works:

• Start with a known concentration and volume of the stock solution.
• Add a specific volume of solvent to reach the desired lower concentration.
• Calculate the new concentration using the formula: $C_1{V}_1=C_2{V}_2$ where $C_1$ and $V_1$ are the initial concentration and volume, and $C_2$ and $V_2$ are the final concentration and volume.

Using a series of dilutions, just as described in the original exercise, is common practice when the final desired concentration is extremely low. This step-by-step dilution avoids the need to measure very small amounts directly, which can result in significant errors.

Molarity Calculation

Molarity, often denoted as $M$, is a measure of the concentration of a solute in a solution, expressed as moles of solute per liter of solution. Calculating molarity involves understanding the relationship between the mass of the solute, its molar mass, and the volume of the solution.

To calculate molarity:

• First, determine the number of moles of solute using its molar mass. For example, using potassium permanganate (KMnO₄), we find moles by dividing its mass by its molar mass.
• Then, measure the volume of the solution in liters.
• Finally, use the formula: $M=\frac{n}{V}$, where $n$ is the number of moles and $V$ is the volume in liters.

These calculations are pivotal in preparing and analyzing solutions to ensure that the correct concentrations are applied, which can affect the outcomes of chemical reactions.

KMnO₄ Solution

Potassium permanganate (KMnO₄) is a versatile compound frequently used in laboratories and industries due to its oxidizing properties. It appears as a deep purple solid and is easily soluble in water, forming a bright purple solution.

Important aspects of working with KMnO₄ solutions include:

• Safety: KMnO₄ is a strong oxidizer and can potentially cause burns. Handle with care using appropriate personal protective equipment (PPE).
• Stability: A freshly prepared KMnO₄ solution should be kept in a dark container to prevent decomposition by light.
• Applications: It's commonly used in titrations and as a disinfectant or antiseptic due to its ability to deactivate many pathogens.

When preparing a KMnO₄ solution, detailed precision is crucial. This involves careful weighing of the solid, proper dilution steps, and ensuring the solution is homogenous before use.