Problem 92 - Chemistry A Molecular Approach

Question

Perform each unit conversion. a. \(28.9 \mathrm{nm}\) to \(\mu \mathrm{m}\) b. \(1432 \mathrm{~cm}^{3}\) to \(\mathrm{L}\) c. 1211 Tm to Gm

Step-by-Step Solution

Answer

a. 28.9 nm is 0.0289 µm, b. 1432 cm^3 is 1.432 L, c. 1211 Tm is 1211000 Gm.

Step 1: Convert nanometers to micrometers

One nanometer (nm) is equal to 0.001 micrometers (µm). To convert 28.9 nm to µm, multiply the number of nanometers by this conversion factor.

Step 2: Convert cubic centimeters to liters

One cubic centimeter (cm^3) is equal to 0.001 liters (L). To convert 1432 cm^3 to L, multiply the number of cubic centimeters by this conversion factor.

Step 3: Convert terameters to gigameters

One terameter (Tm) is equal to 1000 gigameters (Gm). To convert 1211 Tm to Gm, multiply the number of terameters by this conversion factor.

Key Concepts

Converting Nanometers to Micrometers

Understanding how to convert nanometers (nm) to micrometers (\textmu m) is essential in fields like chemistry, physics, and biology, where working with microscopic distances is common. The key is recognizing that the metric system is based on powers of 10, which simplifies this conversion process.

A nanometer is a unit of length in the metric system equal to one billionth of a meter, or expressed in scientific notation, it's \(10^{-9}\) meters. On the other hand, a micrometer is one millionth of a meter, or \(10^{-6}\) meters. Since both these units are based on the meter, converting between them is straightforward – you shift the decimal place by three positions.

To convert nanometers to micrometers, you divide the number of nanometers by 1,000 because there are 1,000 nanometers in one micrometer. Therefore, for our example of 28.9 nm, you would calculate:\[ 28.9 \, \text{nm} \times \frac{1 \, \textmu m}{1,000 \, \text{nm}} = 0.0289 \, \textmu m \].This calculation shows that 28.9 nm is equal to 0.0289 micrometers.

Converting Cubic Centimeters to Liters

Volume measurement is another common task in chemistry and other scientific disciplines. Converting cubic centimeters (cm³) to liters (L) is a frequent requirement, particularly when dealing with the volumes of liquids or gases.

A cubic centimeter is a volume that is contained in a cube with each side measuring one centimeter. In the metric system, this measurement is also a straightforward conversion because 1,000 cubic centimeters equal one liter. To convert cubic centimeters to liters, you can simply divide the number of cubic centimeters by 1,000. It follows the same power-of-ten conversion principle as length measurements.

In practice, for our problem, transforming 1,432 cm³ to liters looks like this:\[ 1432 \, \text{cm}^{3} \times \frac{1 \, \text{L}}{1,000 \, \text{cm}^{3}} = 1.432 \, \text{L} \].This result reveals that a volume of 1,432 cubic centimeters is equivalent to 1.432 liters.

Converting Terameters to Gigameters

When studying astronomy or geophysics, you might encounter incredibly vast distances measured in terameters (Tm) or gigameters (Gm). The prefix ‘tera’ indicates a factor of \(10^{12}\), whereas ‘giga’ corresponds to \(10^{9}\). Hence, one terameter equals 1,000 gigameters.

Because these units are less familiar than meters or centimeters, understanding the magnitude they represent can be challenging. However, using the power-of-ten rule simplifies the conversion. To convert from terameters to gigameters, you multiply the number of terameters by 1,000.

For the value provided in the exercise, converting 1,211 Tm to Gm, our equation is:\[ 1211 \, \text{Tm} \times \frac{1,000 \, \text{Gm}}{1 \, \text{Tm}} = 1,211,000 \, \text{Gm} \].This translates the astronomical distance of 1,211 Tm to an equally impressive 1,211,000 Gm.