Question

a. How many kilograms are in one teragram? b. How many nanometers are in $6.50\times {10}^2$ terameters? c. How many kilograms are in 25 femtograms? d. How many liters are in $8.0$ cubic decimeters? e. How many microliters are in one milliliter? f. How many picograms are in one microgram?

Short answer

a. $1\times {10}^9$ kg b. $6.50\times {10}^{23}$ nm c. $2.5\times {10}^{-17}$ kg d. 8.0 liters e. 1000 µL f. 1000 pg

Step-by-step solution

a. Convert teragrams to kilograms

To convert from teragrams (Tg) to kilograms (kg), we need to know the conversion factor. 1 Tg equals 10^12 grams, and since there are 10^3 grams in a kilogram, we have: 1 Tg = 10^12 g = 10^12 / 10^3 kg = 10^9 kg So there are 10^9 kg in one teragram.

b. Convert terameters to nanometers

To convert from terameters (Tm) to nanometers (nm), first convert terameters to meters by multiplying by the conversion factor of 10^12, and then convert meters to nanometers by multiplying by the conversion factor of 10^9. Thus, we have: $6.50\times {10}^2$ Tm = $6.50\times {10}^2\times {10}^{12}$ m = $6.50\times {10}^{14}$ m Now, convert meters to nanometers: $6.50\times {10}^{14}$ m = $6.50\times {10}^{14}\times {10}^9$ nm = $6.50\times {10}^{23}$ nm So there are $6.50\times {10}^{23}$ nanometers in $6.50\times {10}^2$ terameters.

c. Convert femtograms to kilograms

To convert from femtograms (fg) to kilograms (kg), first convert femtograms to grams by dividing by the conversion factor of 10^15, and then convert grams to kilograms by dividing by the conversion factor of 10^3. Thus, we have: 25 fg = 25 / 10^15 g = $2.5\times {10}^{-14}$ g Now, convert grams to kilograms: $2.5\times {10}^{-14}$ g = $2.5\times {10}^{-14}/{10}^3$ kg = $2.5\times {10}^{-17}$ kg So there are $2.5\times {10}^{-17}$ kg in 25 femtograms.

d. Convert cubic decimeters to liters

Since 1 cubic decimeter (dm³) is equal to 1 liter (L), we can simply write: 8.0 cubic decimeters = 8.0 liters

e. Convert milliliters to microliters

To convert from milliliters (mL) to microliters (µL), multiply by the conversion factor of 10^3. Thus, we have: 1 mL = 1 * 10^3 µL = 1000 µL So there are 1000 microliters in one milliliter.

f. Convert micrograms to picograms

To convert from micrograms (µg) to picograms (pg), multiply by the conversion factor of 10^3. Thus, we have: 1 µg = 1 * 10^3 pg = 1000 pg So there are 1000 picograms in one microgram.

Key concepts

Metric System

The metric system is a universally accepted system of measurement used worldwide. It is based on simple unit conversions that rely on powers of ten. This makes it an intuitive and easy-to-use system, especially for scientific calculations. The metric system includes units of measure such as meters for length, grams for mass, and liters for volume. With prefixes like 'kilo', 'micro', and 'nano', each signifies a power of ten, making conversions straightforward.

For example:

• Kilo- (10³) means 1,000 units.
• Micro- (10^-6) is one-millionth of a unit.
• Nano- (10^-9) is one-billionth of a unit.

Such a structured approach allows for consistent and clear communication in scientific fields and day-to-day use.

Scientific Notation

Scientific notation is a way of writing very large or very small numbers succinctly. It is expressed as a product of a number between 1 and 10 and a power of ten. For instance, the number 6,500,000 can be written as $6.5\times {10}^6$. This notation simplifies working with extreme values by reducing the number of digits, which makes calculations easier and more reliable.

In metric conversions, scientific notation is especially handy. By using this notation, conversion tasks are simplified, such as converting meters to kilometers or nanometers, as it just involves adjusting the exponent on the power of ten.

Kilograms

A kilogram (kg) is the base unit of mass in the metric system. It is equivalent to 1,000 grams. This unit is widely used in science, engineering, and everyday life for measuring weight or mass. The prefix 'kilo-' signifies a thousandfold. Therefore, multiplying or dividing by powers of ten allows for conversions between kilograms and other units such as grams, milligrams, or even femtograms.

For instance, understanding that $1$ teragram (Tg) = ${10}^9$ kilograms is vital for larger scales, while $25$ femtograms (fg) = $2.5\times {10}^{-17}$ kilograms strengthens the comprehension of tiny masses.

Nanometers

Nanometers (nm) are a unit of length in the metric system, equal to one-billionth of a meter (1 nm = 10^-9 meters). They are commonly used in fields like physics, chemistry, and biology to measure atomic scales, wavelengths, and structures.

Conversion between different lengths, such as from terameters (Tm) to nanometers, highlights the scale difference. To convert meter to nanometer, multiply by 10⁹, making it straightforward to relate large units to the incredibly small scale measurements typically observed in advanced scientific research.

Microliters

Microliters (µL) are a unit of volume in the metric system, where 1 microliter equals one-millionth of a liter (1 µL = 10^-6 L). This unit is often used in medical and laboratory settings where precise small volume measurements are crucial.

Converting between microliters and milliliters is simple. Since 1 milliliter (mL) is equal to 1,000 microliters, you can effortlessly switch between these units by multiplying or dividing by 1,000, depending on whether you are scaling up or down, enhancing the precision and understanding in volumetric analysis.