Question

Challenge What conversion factors are needed to convert: a. nanometers to meters? b. density given in \(g / c m^{3}\) to a value in \(\mathrm{kg} / \mathrm{m}^{3} ?\)

Short answer

To convert: a. nanometers to meters: Multiply by \(10^{-9}\) since 1 nanometer = \(10^{-9}\) meters. b. density given in \(g/cm^3\) to a value in \(kg/m^3\): Multiply the mass by the mass conversion factor \(10^{-3}\) kg/g and divide the volume by the volume conversion factor \(10^{-6}\) m³/cm³, resulting in the expression \(\dfrac{mass \times10^3}{volume}\) \(\frac{kg}{m^3}\).

Step-by-step solution

1. Convert nanometers to meters

First, let's understand the measurement units. Nano refers to a scale of \(10^{-9}\), which means that 1 nanometer is equal to \(10^{-9}\) meters. To convert nanometers to meters, simply use the following relationship: 1 nanometer = \(10^{-9}\) meters So, to convert any given number of nanometers to meters, simply multiply it by \(10^{-9}\).

2. Convert density from g/cm³ to kg/m³

To convert density from \(g/cm^3\) to \(kg/m^3\), we need to handle mass and length measurements separately. First, let's convert grams (g) to kilograms (kg). We know: 1 gram = \(10^{-3}\) kg Let's remember this conversion factor and now move to the length measurements. Next, we need to convert cubic centimeters (cm³) to cubic meters (m³). Here, both length units are related through a power of 10: 1 cm = \(10^{-2}\) m Now, to find the conversion factor for cubic centimeters to cubic meters, we should cube the length conversion factor (since we have cubic measurements): (1 cm)³ = \((10^{-2} m)^{3}\) = \(10^{-6}\) m³ 1 cm³ = \(10^{-6}\) m³ Finally, combine both mass and length conversion factors to convert density from \(g/cm^3\) to \(kg/m^3\): Density = (mass in grams) / (volume in cm³) Now, multiply the mass by the mass conversion factor and divide the volume by the volume conversion factor: Density = \(\dfrac{(mass \times 10^{-3} kg)}{(volume \times 10^{-6} m^3)}\) To simplify this expression: Density = \(\dfrac{mass \times10^3}{volume}\) \(\frac{kg}{m^3}\) Now, you can use this conversion factor to convert any density given in \(g/cm^3\) to \(kg/m^3\).

Key concepts

Nanometers to Meters Conversion

When working in the field of chemistry or physics, you'll often need to convert units from one scale to another to make sense of your measurements. Particularly, converting from nanometers to meters is essential when dealing with the microscopic world and relating it to the macroscopic scales we are more familiar with. A nanometer is a unit of length in the metric system that is equal to one billionth of a meter. To convert nanometers to meters, use the conversion factor:

1 nanometer = \(10^{-9}\) meters

This conversion is based on the prefix 'nano-' which signifies a factor of \(10^{-9}\). Therefore, to convert any given value from nanometers (nm) to meters (m), you multiply that value by the conversion factor \(10^{-9}\). For example, 500 nm converted to meters is \(500 \times 10^{-9}\) meters, or \(5 \times 10^{-7}\) meters. Remembering this simple relationship helps in quickly transitioning between scales that are vastly different in size.

Density Conversion

Density is an important concept in chemistry as it relates the mass of a substance to its volume, usually expressed in terms like grams per cubic centimeter (\(g/cm^3\)) or kilograms per cubic meter (\(kg/m^3\)). You may need to convert the density from one unit to another to match the context of a problem or the standards of a specific scientific field. The conversion process involves two steps: converting mass and converting volume.

Mass Conversion

Firstly, to convert grams to kilograms, the conversion factor is:
1 gram = \(10^{-3}\) kilograms
This factor is derived from the 'kilo-' prefix, which denotes a thousandfold (\(10^3\)) increase or decrease. Thus, the mass value in grams is multiplied by \(10^{-3}\) to obtain the mass in kilograms.

Volume Conversion

Secondly, to convert cubic centimeters to cubic meters, you must remember that cubic measurements involve raising the linear conversion factor to the third power. The relationship is:
1 cm = \(10^{-2}\) meters
Therefore, when dealing with volume, you cube the conversion:
1 cm³ = \(10^{-6}\) m³
To complete the conversion of density from \(g/cm^3\) to \(kg/m^3\), you combine the mass and volume conversions:
Density in \(kg/m^3\) = (Density in \(g/cm^3\)) \(\times 10^3\)
Always make sure to keep track of your units throughout the conversion to ensure accuracy.

Mass and Volume Conversions

Mastering mass and volume conversions is crucial in various scientific calculations, as they form the basis for understanding concepts like density and concentration. Mass is commonly measured in grams (g) or kilograms (kg), and volume in liters (L), cubic centimeters (\(cm^3\)), or cubic meters (\(m^3\)).

Mass Conversion

For mass, the metric system allows for straightforward conversion between units. The basic conversion is:
1 kilogram = 1000 grams = \(10^3\) grams
To convert grams to kilograms, you divide the mass value by 1000 or multiply it by \(10^{-3}\).

Volume Conversion

Volume conversion can be a bit trickier, especially when going from a smaller unit to a larger one, as volume is a measure of three-dimensional space. The conversion factors depend on the cubic relationship of the length units:
1 liter = 1000 cubic centimeters = \(10^3\) \(cm^3\)
1 cubic meter = \(10^6\) cubic centimeters
For converting smaller units to larger ones (like \(cm^3\) to \(m^3\)), divide by the appropriate power of 10, and to convert larger units to smaller ones, multiply. Keeping these exponential relationships in mind will greatly assist in performing mass and volume conversions accurately and with confidence.