Question

What is the final volume of an orange juice prepared from \(100.0 \mathrm{~mL}\) of orange juice concentrate if the final juice is to be \(20.0 \%\) of the strength of the original?

Short answer

The final volume of the orange juice is 500 mL.

Step-by-step solution

Understand the problem

We are asked to dilute an orange juice concentrate so that its final concentration is 20% of its original strength. We start with 100.0 mL of concentrate.

Define key variables

Let the initial volume of concentrate be \( V_c = 100.0 \ mL \) and the initial concentration be \( C_c = 100\% \). The final concentration \( C_f \) should be 20% of the initial concentration, so \( C_f = 20\% \). Let \( V_f \) be the final volume of the diluted juice.

Set up the dilution equation

We use the equation for dilutions: \( C_c \times V_c = C_f \times V_f \). Substitute the values \( 100 \times 100.0 = 20 \times V_f \).

Solve for the final volume

Rearrange the dilution equation to solve for \( V_f \): \( V_f = \frac{100 \times 100.0}{20} \). Calculate \( V_f \): \( V_f = \frac{10000}{20} = 500 \ mL \).

Interpret the result

The final volume of the orange juice after dilution is 500 mL. This means we need to add enough water to the 100 mL of concentrate to make a total of 500 mL of juice.

Key concepts

Concentration

Concentration refers to how much of a substance is present in a mixture. In the context of the given exercise, concentration describes how strong or pure the orange juice is before and after dilution.
When we talk about a 100% concentration, we mean that the substance is entirely pure — in this case, pure orange juice concentrate. By diluting it to 20%, we are essentially making the juice weaker or less concentrated by adding more liquid, usually water.

• Original Concentration: The initial concentration of our orange juice, which is 100% pure concentrate.
• Final Concentration: After dilution, the final concentration is reduced to 20% of the original.

This concept is crucial because understanding it allows us to calculate how much more liquid we need to add to reach the desired concentration. This is different from the amount or volume of juice and helps us to change the potency without altering the quality too much. Concentrating how much concentrate is present in a mixture directly impacts the flavor and consistency of the final product.

Volume

Volume represents the amount of space that a substance occupies. In this exercise, we begin with a starting volume of orange juice concentrate and end with a final volume of diluted juice.

• Initial Volume: The initial volume of the concentrate is 100.0 mL.
• Final Volume: After dilution, the total volume becomes 500 mL.

Understanding volume is essential because it informs us how much additional liquid (water) is needed to reach a specific concentration. To dilute the juice from a concentrate, one must add a sufficient amount of water to the initial volume to achieve the desired final volume. By calculating the final volume, one knows how much physical space the diluted mixture will occupy and how much product (in terms of quantity) they will end up with. This makes sure the blend is both sufficient in quantity for consumption and consistent in flavor.

Dilution Equation

The dilution equation is a mathematical formula used to calculate how to dilute a concentrated solution. In this scenario, it helps us find out how much final juice we have after diluting it to 20% strength of the original. The equation is expressed as:\[ C_c \times V_c = C_f \times V_f \]where:

• \( C_c \) is the initial concentration (100%).
• \( V_c \) is the initial volume (100.0 mL).
• \( C_f \) is the final concentration (20%).
• \( V_f \) is the final volume, which we need to calculate.

Using this formula, we can substitute the known values and rearrange to find:\[ V_f = \frac{C_c \times V_c}{C_f} = \frac{100 \times 100.0}{20} = 500 \, \text{mL} \]This calculation is handy because it provides a clear process for diluting a concentrate to a certain strength, making it applicable in many real-world situations, such as food preparation or chemical solutions. Knowing how to use the dilution equation ensures the right proportion is achieved for any dilution task.