Question

An aqueous solution that contains 285 ppm of potassium nitrate \(\left(\mathrm{KNO}_{3}\right)\) is being used to feed plants in a garden. What volume of this solution is needed to prepare \(2.0 \mathrm{~L}\) of a solution that is \(75 \mathrm{ppm}\) in \(\mathrm{KNO}_{3}\) ?

Short answer

Approximately 0.53 L of 285 ppm solution is needed.

Step-by-step solution

Understand the Concept of PPM

PPM stands for 'parts per million.' It's a way to express very dilute concentrations of substances. In this case, 285 ppm means 285 grams of potassium nitrate in 1,000,000 grams (or 1,000 liters) of solution.

Apply the Dilution Formula

Using the dilution formula, \(C_1V_1 = C_2V_2\), where \(C_1\) and \(C_2\) are the initial and final concentrations, and \(V_1\) and \(V_2\) are the initial and final volumes of the solution. Here, \(C_1 = 285\) ppm, \(C_2 = 75\) ppm, and \(V_2 = 2.0\) L.

Solve for Initial Volume

Rearrange the dilution formula to solve for \(V_1\): \(V_1 = \frac{C_2V_2}{C_1}\). Substitute the values in to get \(V_1 = \frac{75 \times 2.0}{285}\).

Perform the Calculation

Calculate \(V_1\): \(V_1 = \frac{150}{285} = 0.5263\) L, or approximately \(0.53\) L when rounded to two decimal places.

Key concepts

PPM (Parts Per Million)

PPM, or 'Parts Per Million,' is a unit of measurement used to express very small concentrations of a substance within a solution. It’s akin to measuring a few drops of ink in a large swimming pool. Essentially, one ppm means one part of a solute per one million parts of solution. In this exercise, 285 ppm of potassium nitrate (\(\mathrm{KNO}_{3}\)) indicates that there are 285 grams of potassium nitrate in one million grams (or 1000 liters) of the solution. This unit is particularly useful in various fields like chemistry, environmental science, and medicine where tiny concentrations have significant effects. Understanding ppm is crucial for assessing how much of a chemical is present in given conditions without the complexity of other units like molarity or molality.

Concentration Units

Concentration units are essential for quantifying the amount of solute in a given amount of solvent or solution. Common concentration units include:

• PPM (parts per million): Useful for indicating small concentrations as discussed earlier.
• Molarity (M): Expressed as moles of solute per liter of solution.
• Molality (m): Expressed as moles of solute per kilogram of solvent.
• Percentage (%): Represents the ratio of solute to solution mass or volume.

These units allow for the precise communication of how much substance is dissolved, which is vital for experiments and solution preparations. In the context of the problem, the use of ppm provides a straightforward way to express the concentration of potassium nitrate suitable for plant feeding.

Dilution Formula

The dilution formula is a simple mathematical equation used when diluting a solution. It's expressed as\[C_1V_1 = C_2V_2\]where:

• \[C_1\] is the initial concentration of the solution,
• \[V_1\] is the initial volume,
• \[C_2\] is the final concentration, and
• \[V_2\] is the final volume.

This formula helps ensure that the amount of substance remains constant before and after dilution. It's like spreading the same amount of butter over a different-sized slice of bread. By rearranging the formula to solve for the unknown variable, you can adjust the solution's concentration appropriately. In the original problem, using this formula allows you to calculate how much of the 285 ppm solution is needed to achieve a 75 ppm concentration in a 2.0 L solution.

Solution Preparation

Preparing a solution involves a systematic approach to ensure the correct concentration and volume. Follow these steps:

• Identify the Final Volume and Concentration: Know how much solution you want to prepare and its concentration. For example, 2.0 L of a 75 ppm solution.
• Use the Dilution Formula: Determine how much of the more concentrated stock solution is needed, as seen with \(V_1 = \frac{C_2V_2}{C_1}\).
• Measure Accurately: Use precise measuring instruments like pipettes or graduated cylinders to ensure the right amount of stock solution is added.
• Mix Thoroughly: Combine the measured volume with additional solvent up to the desired volume, ensuring the solute is evenly distributed.

When done carefully, these steps will result in a solution with the intended concentration, ready for use, such as in fertilizing plants.