Problem 84 - Chemistry: The Central Science in SI Units, Global Edition

Question

Saline solution used in hospital contains \(0.9 \%\) sodium chloride by mass. Calculate the number of grams of sodium chloride in 0.5 gal of saline solution if the solution has a density of \(1.01 \mathrm{~g} / \mathrm{mL}\)

Step-by-Step Solution

Answer

There are approximately 17.19 grams of sodium chloride in 0.5 gallons of saline solution.

Step 1: Convert gallons to milliliters

First, we need to convert 0.5 gallons to milliliters using the conversion factor: \(1 \mathrm{~gal} = 3785.41 \mathrm{~mL}\). To do so, multiply 0.5 gallons by the conversion factor: \[0.5 \mathrm{~gal} \times 3785.41 \mathrm{~mL/gal} = 1892.705 \mathrm{~mL}\]

Step 2: Find the mass of the solution

Now, we can use the given density to find the mass of the saline solution. The formula to find mass is: \[mass = volume \times density\] So, the mass of the solution is: \[1892.705 \mathrm{~mL} \times 1.01 \mathrm{~g/mL} = 1910.53205 \mathrm{~g}\]

Step 3: Calculate the mass of sodium chloride

The saline solution is 0.9% sodium chloride by mass. To find the mass of sodium chloride, simply multiply the mass of the solution by 0.9%. \[mass_{NaCl} = mass_{solution} \times 0.9 \% \] \[mass_{NaCl} = 1910.53205 \mathrm{~g} \times 0.009 \] \[mass_{NaCl} = 17.19478845 \mathrm{~g}\]

Step 4: Round to appropriate decimal places

Since we are working with grams, it is appropriate to round our answer to two decimal places. In this case, we have: \[mass_{NaCl} \approx 17.19 \mathrm{~g}\] So, there are approximately 17.19 grams of sodium chloride in 0.5 gallons of saline solution.

Key Concepts

Percentage Composition

When dealing with solutions like saline, understanding percentage composition is essential. Percentage composition tells us how much of a particular component is present in a solution, expressed as a percentage of the total mass. In our exercise, the saline solution contains 0.9% sodium chloride by mass.

This means that for every 100 grams of the solution, 0.9 grams are sodium chloride. To find the actual mass of sodium chloride in a specific amount of solution, you multiply the total mass of the solution by the percentage (expressed as a decimal).

• Percentage as a decimal is 0.9% = 0.009.
• By calculating: mass of solution × 0.009, you obtain the mass of sodium chloride.

Understanding percentage composition is vital in chemistry to analyze how solutions are mixed and to prepare them correctly.

Unit Conversion

Converting units is often the first step in solving many chemistry problems. It's crucial because measurements in chemistry must be consistent. In our problem, we started with gallons and needed to determine the volume in milliliters.

Here’s how unit conversion works:

• Know the conversion factor: 1 gallon = 3785.41 mL.
• Multiply the given volume by the conversion factor to change gallons to milliliters.
• For this problem: 0.5 gallons × 3785.41 mL/gal = 1892.705 mL.

Mastering unit conversion ensures you can work with the appropriate units, crucial when dealing with formulas involving mass, volume, or density.

Mass Calculation

Calculating mass using volume and density is a common task in chemistry. Once you know the volume and the density of a solution, you can find its mass easily with a simple formula:

• Mass = Volume × Density.

In our exercise, the density of the saline solution was 1.01 g/mL, and the volume was converted to 1892.705 mL.

Using the formula:

• Mass = 1892.705 mL × 1.01 g/mL = 1910.53205 g.

This formula is vital for determining how much material is present when dealing with solutions, especially in laboratory settings where precise measurements are needed. Once you have the mass, you can do further calculations, like finding out how much of a specific component is in your solution using percentage composition.