Problem 43 - Chemistry The Molecular Science

Question

A liquid sample of lead-based paint contains 60.5 ppm lead. The density of the paint is \(10.0 \mathrm{lb} / \mathrm{gal}\). Calculate the mass of lead (in grams) that would be present in \(50 . \mathrm{gal}\) of this paint.

Step-by-Step Solution

Answer

13.73 grams of lead.

Step 1: Convert ppm to mass fraction

The concentration of lead in the paint is 60.5 ppm. Since ppm stands for 'parts per million', this means there are 60.5 parts of lead for every 1,000,000 parts of the paint. Thus, the mass fraction is \(\frac{60.5}{1,000,000}\).

Step 2: Calculate the total mass of the paint

The density of the paint is given as \(10.0\, \mathrm{lb/gal}\). For \(50\, \mathrm{gal}\) of paint, the total mass is \(50\, \mathrm{gal} \times 10.0\, \mathrm{lb/gal} = 500\, \mathrm{lb}\).

Step 3: Convert the mass from pounds to grams

There are \(453.592\, \mathrm{g}\) in a pound. Thus, \(500\, \mathrm{lb} \times 453.592\, \mathrm{g/lb} = 226796\, \mathrm{g}\).

Step 4: Calculate the mass of lead in the paint

Using the mass fraction from Step 1, the mass of lead in the paint is \(226796\, \mathrm{g} \times \frac{60.5}{1,000,000} = 13.73\, \mathrm{g}\).

Key Concepts

ppm (parts per million)

Understanding ppm, which stands for "parts per million," is crucial when dealing with concentrations, especially in fields like chemistry and environmental science. A concentration of 60.5 ppm means there are 60.5 units of the substance, here lead, for every 1,000,000 units of the total mixture or solution. This concept is important because it allows us to express very dilute concentrations in an easily understandable way. When you hear ppm, think of it as 60.5 parts of lead per million parts of paint, akin to saying that out of one million drops of water, 60.5 drops are lead.

It simplifies calculations because it uses the decimal system, making it easy to convert into different units like percentages. For example, to convert ppm to a mass fraction, you just divide the number by 1,000,000. Thus, 60.5 ppm becomes \( \frac{60.5}{1,000,000} \) in terms of a mass fraction. This fraction is significant in determining the actual amount of lead in a certain volume of paint as it directly represents the ratio of lead present.

density conversion

Density serves as a bridge between volume and mass, allowing us to convert one into the other. In our situation, the density given is \( 10.0 \mathrm{lb/gal} \), which means one gallon of the paint weighs 10 pounds.

To find out how much 50 gallons weigh, simply multiply the density by the volume: \[ 50 \text{ gal} \times 10.0 \text{ lb/gal} = 500 \text{ lb} \]. This calculates the total mass of the paint, which then becomes a starting point for other conversions.

Density is useful because it provides a quick way to determine uniform mass across varying sizes of substances, essential for tasks like this. Whether we are talking about a cup or a drum of paint, density ensures that we stay accurate with our mass calculations.

mass calculation

Once we've calculated the mass of the paint as 500 pounds, the next step is to convert this into a usable unit for scientific calculations, which is grams in most cases. Knowing that there are 453.592 grams in a pound helps us do this efficiently.

So, by multiplying, \( 500 \mathrm{lb} \times 453.592 \mathrm{g/lb} \), we find that the paint weighs \( 226796 \mathrm{g} \). This figure is critical because it sets the stage for the next and final conversion.

Having the mass of the paint in grams allows us to apply the mass fraction derived from the ppm to find the mass of lead. We multiply \( 226796 \mathrm{g} \) by the previously calculated mass fraction \( \frac{60.5}{1,000,000} \), resulting in about \( 13.73 \mathrm{g} \) of lead. This straightforward mathematical conversion is how we calculate the actual amount of a specific component in a mixture, like lead in paint.