Question

Nalorphine, a relative of morphine, is used to combat withdrawal symptoms in heroin users. How many milliliters of a \(0.40 \%(\mathrm{~m} / \mathrm{v})\) solution of nalorphine must be injected to obtain a dose of \(1.5 \mathrm{mg}\) ?

Short answer

0.375 mL of nalorphine solution is needed.

Step-by-step solution

Understanding the Problem

We need to determine how many milliliters of a 0.40% (mass/volume) nalorphine solution contains a dose of 1.5 mg of nalorphine. A 0.40% (m/v) solution means that there are 0.40 grams of nalorphine per 100 mL of solution.

Convert Grams to Milligrams

Since the dose required is in milligrams, convert the grams from the percentage to milligrams. 1 gram = 1000 milligrams, thus: 0.40 grams = 0.40 x 1000 milligrams = 400 milligrams of nalorphine per 100 mL of solution.

Set Up a Proportion

Set up a proportion to find the volume in milliliters that corresponds to 1.5 mg of nalorphine. The proportion is set as follows: \[ \frac{400 ext{ mg}}{100 ext{ mL}} = \frac{1.5 ext{ mg}}{x ext{ mL}} \]

Solve the Proportion

Cross-multiply and solve for \(x\):\[ 400 ext{ mg} imes x ext{ mL} = 1.5 ext{ mg} imes 100 ext{ mL} \]\[ 400x = 150 \]\[ x = \frac{150}{400} \]\[ x = 0.375 ext{ mL} \]

Conclusion

The required volume of nalorphine solution is therefore 0.375 milliliters.

Key concepts

Mass-Volume Percentage

Mass-volume percentage ( ext{m/v}) is an essential concept when working with solutions. It allows you to express the concentration of a substance in a solution in a straightforward way. Specifically, it provides a ratio of the solute's mass to the volume of the solution. For instance, in a 0.40% ( ext{m/v}) solution, there are 0.40 grams of solute per 100 milliliters of the solution. This percentage helps you understand how concentrated the solution is, which is crucial for calculating the needed volume to achieve a certain dosage.

To find out how much the solution you need for a specific amount of solute, start by converting the mass you've calculated from the percentage to the actual amount needed. This involves understanding the percentage as a ratio, which in this case is 0.40 grams in 100 mL, and then using it to guide the calculation of other needed amounts through conversion processes or setting up proportions.

Unit Conversions

Unit conversions are key to solving problems in chemistry. In this exercise, converting units from grams to milligrams is necessary because the desired dose is given in milligrams. Such conversions help to bridge different measurement units to arrive at a useful result.

Here’s a fundamental conversion step: 1 gram equals 1000 milligrams. Converting 0.40 grams to milligrams involves this multiplication:

• 0.40 grams × 1000 milligrams/gram = 400 milligrams

Understanding these conversions allows you to adjust measurements as needed to align with the units given in the problem, ensuring accuracy and consistency in your solution finding process.

Proportion Method

The proportion method is a powerful and often used strategy to solve real-world problems, like finding how much of a solution is needed for a specific dose. The key steps involve setting up relationships between known quantities and the unknown quantity you are solving for.

In the exercise, you apply the proportion method by equating the concentration in milligrams to the volume in milliliters of the known solution with the desired condition:

• Set up the equation: \[ \frac{400 \text{ mg}}{100 \text{ mL}} = \frac{1.5 \text{ mg}}{x \text{ mL}} \]
• Cross-multiply to solve the equation for \(x\): \[ 400 \cdot x = 1.5 \times 100 \]
• Simplify to find \(x\): \[ x = \frac{150}{400} = 0.375 \text{ mL} \]

Using this method ensures that you maintain a balanced relationship between varying parts of your solution, leading directly to the correct answer.