Problem 78 - Chemistry: The Central Science

Question

Automobile batteries contain sulfuric acid, which is commonly referred to as "battery acid." Calculate the number of grams of sulfuric acid in \(1.00\) gal of battery acid if the solution has a density of \(1.28 \mathrm{~g} / \mathrm{mL}\) and is \(38.1 \%\) sulfuric acid by mass.

Step-by-Step Solution

Answer

There are approximately \(1845.11 \,\mathrm{g}\) of sulfuric acid in 1.00 gal of battery acid with a density of \(1.28 \mathrm{g} / \mathrm{mL}\) and a composition of \(38.1 \%\) sulfuric acid by mass.

Step 1: 1. Convert the volume from gallons to milliliters.

To work with the density in g/mL, we first need to convert the volume of the solution from gallons to milliliters. We know that \(1 \,\text{gal} = 3.78541 \,\text{L} = 3785.41 \,\text{mL}\). Volume of solution (in milliliters): \(1.00 \text{ gal} \times 3785.41 \,\frac{\mathrm{mL}}{\mathrm{gal}} = 3785.41 \,\mathrm{mL}\)

Step 2: 2. Calculate the total mass of the solution.

Now that we have the volume of the solution, we can use the density to find the total mass of the solution using the formula: mass = volume × density Total mass of solution: \(3785.41\,\mathrm{mL} \times 1.28\,\frac{\mathrm{g}}{\mathrm{mL}} = 4845.32\,\mathrm{g}\)

Step 3: 3. Calculate the mass of sulfuric acid in the solution.

To find the mass of sulfuric acid in the solution, we use the mass percent given in the problem. We have \(38.1 \%\) sulfuric acid by mass. Mass of sulfuric acid: \(0.381 \times 4845.32\,\mathrm{g} = 1845.11 \,\mathrm{g}\) So, there are approximately \(1845.11 \,\mathrm{g}\) of sulfuric acid in the given solution.

Key Concepts

Density and Mass

Density is a crucial concept in chemistry that relates the mass of a substance to its volume. The formula for density is expressed as: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This relationship allows us to determine how much mass is present in a given volume of a substance. In our problem, sulfuric acid in battery form is the substance in question. Given that the density of the solution is 1.28 g/mL, it tells us that every milliliter of this sulfuric acid solution weighs 1.28 grams.
To calculate the total mass of the sulfuric acid solution, we simply need to know the volume of the solution and its density. By multiplying the density (1.28 g/mL) by the volume (in milliliters), we obtain the total mass of the solution. Here, after converting the volume from gallons to milliliters, we proceed with the multiplication to find the total mass of the entire solution.

Mass Percentage

The mass percentage of a component in a solution signifies how much of that component there is compared to the entire solution, expressed as a percentage. The formula used for mass percentage is:\[ \text{Mass percentage} = \left(\frac{\text{Mass of component}}{\text{Total mass of solution}}\right) \times 100 \% \]In the context of our problem, sulfuric acid makes up 38.1% of the total solution by mass. This tells us that for every 100 grams of the solution, 38.1 grams are sulfuric acid.
Thus, to find out how many grams of sulfuric acid are present, we multiply the mass percentage in decimal form (0.381) by the total mass of the solution (calculated as 4845.32 g). This multiplication yields the mass of sulfuric acid, which is the portion we are interested in for the solution provided.

Volume Conversion

Volume conversion is often necessary in chemistry to ensure calculations are consistent with the units used in other measurements, such as density. Density is typically expressed in terms of mass per milliliter (g/mL), so when working with solutions provided in different units such as gallons, we must convert these volumes into milliliters for accurate computation.
In this problem, we start with a volume of 1.00 gallon of battery acid. Knowing that 1 gallon is equivalent to 3.78541 liters, and 1 liter equals 1000 milliliters, we can set up a conversion as follows:

• 1 gallon = 3.78541 liters
• 1 liter = 1000 milliliters
• 1 gallon = 3785.41 milliliters

Using this conversion factor, we transform the volume from gallons to milliliters, making it possible to proceed with density-based calculations. Therefore, converting the volume correctly is key in determining the mass of the solution using its density.