Question

Convert \(1.88 \times 10^{-6} \mathrm{~g}\) to each unit. (a) \(\mathrm{mg}\) (b) \(\mathrm{cg}\) (c) \(\mathrm{ng}\) (d) \(\mu g\)

Short answer

\(1.88 \times 10^{-6} \mathrm{~g}\) is equal to \(1.88 \times 10^{-3} \mathrm{~mg}\), \(1.88 \times 10^{-4} \mathrm{~cg}\), \(1.88 \times 10^{3} \mathrm{~ng}\), or \(1.88 \mu\mathrm{g}.\)

Step-by-step solution

Converting to milligrams (mg)

To convert from grams to milligrams, multiply the mass by 1,000 (since 1 gram equals 1,000 milligrams):\(1.88 \times 10^{-6} \text{ g} \times 1,000 \frac{\text{mg}}{\text{g}} = 1.88 \times 10^{-3} \text{ mg}\).

Converting to centigrams (cg)

To convert from grams to centigrams, multiply the mass by 100 (since 1 gram equals 100 centigrams):\(1.88 \times 10^{-6} \text{ g} \times 100 \frac{\text{cg}}{\text{g}} = 1.88 \times 10^{-4} \text{ cg}\).

Converting to nanograms (ng)

To convert from grams to nanograms, multiply the mass by 1,000,000,000 (since 1 gram equals 1,000,000,000 nanograms):\(1.88 \times 10^{-6} \text{ g} \times 1,000,000,000 \frac{\text{ng}}{\text{g}} = 1.88 \times 10^{3} \text{ ng}\).

Converting to micrograms (μg)

To convert from grams to micrograms, you multiply by 1,000,000 (since 1 gram equals 1,000,000 micrograms):\(1.88 \times 10^{-6} \text{ g} \times 1,000,000 \frac{\mu\text{g}}{\text{g}} = 1.88 \times 10^{-0} \mu\text{g} = 1.88 \mu\text{g}\), noting that \(10^{0} = 1\).

Key concepts

Metric System Units

Understanding the metric system is essential in chemistry where precise measurements lead to accurate results. The metric system is founded on a base unit for each type of measurement (length, mass, volume) and uses prefixes to denote multiples or fractions of these units. For mass, the base unit is the gram (g).

Prefixes like milli- (mg), centi- (cg), and micro- (μg) represent one thousandth, one hundredth, and one millionth of a gram respectively. Other common prefixes include nano- (ng), which signifies one billionth of a gram. These standardized units allow scientists to communicate their findings unambiguously, across the world. While working with units, it's crucial to know how to shift between them, as mastering this skill allows for a deeper understanding of material quantities, concentrations, and reactions.

Converting Grams to Milligrams

Converting between grams and milligrams is a fundamental skill in chemistry. Since milligrams (mg) are smaller than grams (g), to convert from grams to milligrams, you need to multiply by 1,000. This is because one gram equals one thousand milligrams.

Looking at the equation, \(1.88 \times 10^{-6} \text{ g} \times 1,000 \frac{\text{mg}}{\text{g}} = 1.88 \times 10^{-3} \text{ mg}\), it's clear that the exponent changes when converting between units. This operation is simply a way of scaling up the number of smaller units (mg) to represent an equal mass. Remembering that the milli- prefix always denotes a multiplication factor of 1,000 can help avoid conversion errors.

Grams to Micrograms Conversion

When converting grams to micrograms (μg), the principle is similar to converting to milligrams but with a different scale factor. A microgram is one millionth of a gram, hence you multiply the number of grams by 1,000,000 to convert to micrograms.

The operation \(1.88 \times 10^{-6} \text{ g} \times 1,000,000 \frac{\mu\text{g}}{\text{g}} = 1.88 \mu\text{g}\) demonstrates the conversion. Also, notice the simplification from \(1.88 \times 10^{-0} \mu\text{g}\) down to \(1.88 \mu\text{g}\), as any number raised to the 0 power equals 1. Recognizing this relationship can simplify conversions and calculations in the lab.

Scientific Notation in Chemistry

Scientific notation is a way to express very large or very small numbers in a compact form, which is highly useful in chemistry, where measurements can span many orders of magnitude. For example, \(1.88 \times 10^{-6}\) grams is written in scientific notation to emphasize its scale.

The coefficient (here, 1.88) is a number greater than or equal to 1 but less than 10, and the exponent (here, -6) indicates the power of 10 required to return to the actual number. When converting units, such as in the earlier example where grams were converted to micrograms, attention must be paid to how the exponent changes; it shifts in response to the multiplication or division by powers of 10. Grasping scientific notation streamlines the understanding of chemical quantities and allows for easier manipulation of data.