Question

Perform each conversion. (a) \(5.88 \mathrm{dL}\) to liters (b) \(3.41 \times 10^{-5} \mathrm{~g}\) to micrograms (c) \(1.01 \times 10^{-8} \mathrm{~s}\) to nanoseconds (d) \(2.19 \mathrm{pm}\) to meters

Short answer

a) 0.588 liters, b) 34.1 micrograms, c) 10.1 nanoseconds, d) \(2.19 \times 10^{-12}\) meters

Step-by-step solution

Convert Deciliters to Liters

To convert deciliters to liters, use the fact that 1 deciliter (dL) is equal to 0.1 liters (L). Multiply the given value in deciliters by 0.1 to find the value in liters. \[5.88 \, \text{dL} \times 0.1 \, \frac{\text{L}}{\text{dL}} = 0.588 \, \text{L}\]

Convert Grams to Micrograms

To convert grams to micrograms, note that 1 gram (g) is equal to \(10^{6}\) micrograms (µg). Multiply the value in grams by \(10^{6}\) to get the value in micrograms. \[3.41 \times 10^{-5} \, \text{g} \times 10^{6} \, \frac{\text{µg}}{\text{g}} = 3.41 \times 10^{1} \, \text{µg}\]

Convert Seconds to Nanoseconds

For converting seconds to nanoseconds, use that 1 second is equal to \(10^{9}\) nanoseconds. Multiply the given value in seconds by \(10^{9}\) to obtain the value in nanoseconds. \[1.01 \times 10^{-8} \, \text{s} \times 10^{9} \, \frac{\text{ns}}{\text{s}} = 1.01 \times 10^{1} \, \text{ns}\]

Convert Picometers to Meters

A picometer (pm) is \(10^{-12}\) of a meter. To convert picometers to meters, multiply the given value by \(10^{-12}\). \[2.19 \, \text{pm} \times 10^{-12} \, \frac{\text{m}}{\text{pm}} = 2.19 \times 10^{-12} \, \text{m}\]

Key concepts

Deciliters to Liters

Understanding the conversion between deciliters (dL) and liters (L) is fundamental in chemistry, especially for measuring liquids. 1 deciliter is equal to 0.1 liters. This means that to convert deciliters to liters, you can simply multiply the number of deciliters by 0.1.

For example, if you have a volume of 5.88 dL, the conversion to liters is as follows: \[5.88 \, \text{dL} \times 0.1 \, \frac{\text{L}}{\text{dL}} = 0.588 \, \text{L}\]
This process is a direct scaling down by a factor of ten, reflecting that a deciliter is just one-tenth of a liter.

Grams to Micrograms

Mass is a key measurement in chemistry, and converting between grams and micrograms is a basic skill. Since 1 gram is equal to 1,000,000 micrograms, to convert grams to micrograms, you multiply the mass in grams by 1,000,000 (or \(10^{6}\)).

Take the mass of 3.41 x 10^-5 g as an example. The conversion to micrograms is done as follows:\[3.41 \times 10^{-5} \, \text{g} \times 10^{6} \, \frac{\text{\ensuremath{\mu}}g}{\text{g}} = 3.41 \times 10^{1} \, \text{\ensuremath{\mu}}g\]
This conversion is very straightforward, and it's essential for working accurately with measurements in the laboratory.

Seconds to Nanoseconds

In both physics and chemistry, time can play a critical role, and sometimes it needs to be measured in very small units like nanoseconds. There are 1 billion nanoseconds in a second, so to convert from seconds to nanoseconds, you would multiply by 1 billion, or \(10^{9}\).

For instance, to convert 1.01 x 10^-8 seconds to nanoseconds, you would calculate:\[1.01 \times 10^{-8} \, \text{s} \times 10^{9} \, \frac{\text{ns}}{\text{s}} = 1.01 \times 10^{1} \, \text{ns}\]
This type of unit conversion is crucial in fields that study phenomena occurring at very fast timescales, such as nuclear or particle physics.

Picometers to Meters

When studying atomic-scale distances or wavelengths of light in chemistry and physics, picometers to meters conversion is used. One picometer is one trillionth of a meter (\(10^{-12}\) meters). To convert a measurement in picometers to meters, multiply the picometer value by \(10^{-12}\).

For example:\[2.19 \, \text{pm} \times 10^{-12} \, \frac{\text{m}}{\text{pm}} = 2.19 \times 10^{-12} \, \text{m}\]
Understanding this conversion is essential for accurately describing and quantifying phenomena at the scale of atoms and molecules.