Problem 43 - Chemistry and Chemical Reactivity

Question

The anesthetic procaine hydrochloride is often used to deaden pain during dental surgery. The compound is packaged as a \(10 . \% \text { solution (by mass; } d=1.0 \mathrm{g} / \mathrm{mL})\) in water. If your dentist injects 0.50 mL of the solution, what mass of procaine hydrochloride (in milligrams) is injected?

Step-by-Step Solution

Answer

50 mg of procaine hydrochloride is injected.

Step 1: Determine Mass of Solution Injected

First, we need to find the mass of the solution injected. Since the density \(d\) of the solution is given as \(1.0\, \mathrm{g/mL}\), the mass of \(0.50\, \mathrm{mL}\) of the solution can be calculated using the formula \( \text{mass} = \text{density} \times \text{volume} \). Thus, the mass of the solution is \(0.50\, \mathrm{g}.\)

Step 2: Calculate Mass of Procaine Hydrochloride

The solution is a 10% solution by mass, which means that there are 10 grams of procaine hydrochloride in every 100 grams of solution. To find the mass of procaine hydrochloride in \(0.50\, \mathrm{g}\) of solution, we set up the proportion \(\frac{10\, \mathrm{g}}{100\, \mathrm{g}} = \frac{x\, \mathrm{g}}{0.50\, \mathrm{g}}\), where \(x\) is the mass of procaine hydrochloride. Solving for \(x\), we find \(x = 0.05\, \mathrm{g}.\)

Step 3: Convert Grams to Milligrams

Finally, convert the mass of procaine hydrochloride from grams to milligrams. Since there are 1000 milligrams in a gram, we multiply the mass in grams by 1000. Therefore, \(0.05\, \mathrm{g} = 0.05 \times 1000 = 50\, \mathrm{mg}.\)

Key Concepts

Density and Volume relationship

Understanding the relationship between density and volume is key to solving concentration problems like injecting procaine hydrochloride in dental procedures. Density (\(d\)) is defined as the mass (\(m\)) per unit of volume (\(V\)). This is expressed through the formula:
\(d = \frac{m}{V}\).
To find the mass of a substance when you know its density and volume, you can rearrange this formula to:
\(m = d \times V\).
In this specific example, since the density of the solution is \(1.0\, \text{g/mL}\), and the volume injected is \(0.50\, \text{mL}\), the mass of the solution is computed as follows:
\(0.50\, \text{mL} \times 1.0\, \text{g/mL} = 0.50\, \text{g}\).
This simple calculation shows how directly mass is influenced by volume when the density is constant. This forms the foundation for understanding more complex solution chemistry.

Mass Percentage Solution

Mass percentage solution is an important concept that tells you how much of a solute is present in a solution, expressed as a percentage of the total mass of the solution. It is given by:
\(\text{Mass \, Percentage} = \frac{\text{mass of solute}}{\text{mass of solution}} \times 100\%\).
In the problem of procaine hydrochloride, we are dealing with a 10% mass percentage. This means there are 10 grams of procaine hydrochloride in every 100 grams of the entire solution.
To find out how much procaine hydrochloride is in the \(0.50\, \text{g}\) of total solution, we set up a proportion:

• \(\frac{10\, \text{g}}{100\, \text{g}} = \frac{x\, \text{g}}{0.50\, \text{g}}\)

Solving for \(x\) gives us \(x = 0.05\, \text{g}\).
The proportion demonstrates the simplicity of using percentage to scale quantities accurately. This method is particularly useful for ensuring the correct dosage in medical settings.

Conversion of Units (grams to milligrams)

Converting between units, such as grams to milligrams, is a essential skill in science and daily life applications, especially in medical dosing. Remember:
\(1\, \text{gram} = 1000\, \text{milligrams}\).
To convert a given mass in grams to milligrams, you simply multiply the number of grams by 1000.
In the procaine hydrochloride problem, the mass of procaine hydrochloride was calculated as \(0.05\, \text{g}\). Converting this to milligrams, you compute:

• \(0.05\, \text{g} \times 1000 = 50\, \text{mg}\)

This conversion ensures that measurements are precise and appropriate for the context in which they are needed, like providing an accurate dose of anesthetic during a dental procedure.