Problem 73 - Chemistry: Principles and Reactions

Question

Show how 1 ppb (part per billion) is equivalent to 1 microgram \(/ \mathrm{kg}\). One microgram \(=10^{-6} \mathrm{~g}\).

Step-by-Step Solution

Answer

Question: Show that 1 ppb (part per billion) is equivalent to 1 microgram/kg, given that 1 microgram equals \(10^{-6}\) gram. Answer: By converting 1 ppb to a fraction, considering 1 kg of the mixture, and applying unit conversions, we can demonstrate that 1 ppb is indeed equivalent to 1 µg/kg.

Step 1: Define 1 ppb

One part per billion (1 ppb) means that there is one unit of a substance for every billion (1,000,000,000) units of the mixture. In other words, for every billion parts of the total mixture, one part is the substance in question.

Step 2: Convert 1 ppb to a fraction

We can write 1 ppb as a fraction: $$\frac{1}{1,000,000,000}$$

Step 3: Understand the unit conversion

To show that 1 ppb is equivalent to 1 microgram/kg, consider 1 kg of the mixture. Our aim is to find out how many micrograms of the substance would be present in this 1 kg mixture.

Step 4: Convert 1 ppb to micrograms per kilogram

Since we know that 1 ppb is equivalent to $$\frac{1}{1,000,000,000}$$ of the total mixture and 1 microgram equals \(10^{-6}\) gram, we can now convert the ppb unit to micrograms per kilogram: $$\frac{1~\mathrm{part}}{1,000,000,000~\mathrm{parts}} \times 1~\mathrm{kg} \times \frac{10^{-6}~\mathrm{g}}{1~\mathrm{part}} = \frac{1}{1,000}~\mathrm{g} = 1~\mathrm{\mu g/kg}$$

Step 5: Conclusion

Through the above calculations, we've shown that 1 ppb is indeed equivalent to 1 microgram per kilogram (1 µg/kg).

Key Concepts

Unit Conversion

Understanding unit conversion is essential when dealing with different quantities and measurements. It enables us to translate one type of measurement into another within the same dimension. For example, translating parts per billion (ppb) to micrograms per kilogram involves converting a dimensionless unit of concentration to a mass-to-mass ratio.

Unit conversion often requires multiplication or division by a conversion factor that relates the two units. In the case of the exercise, the conversion factor is understanding that 1 microgram is equivalent to \(10^{-6}\) grams. Recognizing these relationships makes it possible to move fluently between units, whether you're measuring ingredients in a kitchen or assessing chemical concentrations in a lab.

Mass-to-Mass Conversion

Mass-to-mass conversion comes into play when dealing with chemical measurements. The ability to convert from one mass unit to another (e.g., micrograms to kilograms) is vital for accurate scientific calculations. In the context of our exercise, we focus on determining the mass of a substance in a given mass of mixture or solution.

When we say that 1 ppb equals 1 microgram per kilogram, we're expressing that in 1 kilogram of a substance, there is 1 microgram of a specific component. This concept can be challenging to visualize, but thinking about it in terms of smaller, more manageable portions can be helpful. Imagine dividing a kilogram into a billion tiny parts; only one of those parts would be equal to the substance we're measuring. That's the heart of mass-to-mass conversion—relating different amounts of mass to each other to find a ratio or concentration.

Chemical Measurement

Chemical measurement in parts per billion (ppb) provides a way to express extremely small concentrations of substances. It's a measurement unit often utilized in environmental science, chemistry, and medicine to describe the presence of pollutants, drugs, or other minute quantities in a given sample.

The exercise showcases how crucial it is to have a keen understanding of how to measure and interpret these infinitesimal amounts accurately. We've learned that through careful unit conversion and mass-to-mass calculation, we can express chemical concentrations in ways that are both meaningful and scientifically accurate. Mastering these conversions ensures clarity and precision in scientific communication, allowing researchers to convey findings that influence public health decisions, environmental policies, and a range of scientific investigations.