Problem 86 - General Chemistry

Question

A submicroscopic particle suspended in a solution has a volume of \(1.3 \mu \mathrm{m}^{3}\). What is this volume in liters?

Step-by-Step Solution

Answer

1.3 x 10 extsuperscript{-15} L

Step 1: Understanding the Units

The problem is to convert a volume given in cubic micrometers ( extmu m extsuperscript{3}) to liters (L). We know that 1 micrometer ( extmu m) is equal to 10 extsuperscript{-6} meters (m), so 1 cubic micrometer ( extmu m extsuperscript{3}) is equal to (10 extsuperscript{-6} m) extsuperscript{3}.

Step 2: Convert Cubic Micrometers to Cubic Meters

To convert the volume from cubic micrometers to cubic meters, use the conversion of lengths:\[1 ext{ extmu m}^{3} = (10^{-6} ext{ m})^3 = 10^{-18} ext{ m}^3\]Thus, the particle's volume in cubic meters is:\[1.3 ext{ extmu m}^{3} imes 10^{-18} ext{ m}^{3}/ ext{ extmu m}^{3} = 1.3 imes 10^{-18} ext{ m}^{3}\]

Step 3: Convert Cubic Meters to Liters

Since 1 cubic meter (m extsuperscript{3}) equals 1000 liters (L), we can convert the volume from cubic meters to liters:\[1.3 imes 10^{-18} ext{ m}^{3} imes 1000 ext{ L/m}^{3} = 1.3 imes 10^{-15} ext{ L}\]

Step 4: Final Calculation and Result

The calculation shows that the volume of the particle is\[1.3 imes 10^{-15} ext{ L}\]in liters. Therefore, the volume of the particle in liters is 1.3 x 10 extsuperscript{-15} L.

Key Concepts

Cubic Micrometers

Cubic micrometers, also commonly written as \(\mu m^3\), represent a very tiny volume measurement. A micrometer, otherwise known as a micron, is one-millionth of a meter, or \(10^{-6}\) meters. So when we cube this to measure volume, it results in \(10^{-18}\) cubic meters. Cubic micrometers are useful in describing volumes on a microscopic or nanoscopic scale, such as in biological cells or nanoparticles. When dealing with problems that require the conversion of cubic micrometers to other metric units, it's crucial to understand their tiny scale. This helps ensure accurate conversion to larger units like cubic meters or liters.

Cubic Meters

Cubic meters (\(m^3\)) are a common metric unit for measuring volume, especially in the sciences and many industrial applications. One cubic meter is equivalent to the volume of a cube with each side measuring one meter in length. This unit is often used to quantify the volume of larger objects or spaces. For example, the interior space of a room, the content of a swimming pool, or even the production capacity of industrial equipment. Converting from cubic micrometers to cubic meters, as discussed, involves understanding that \(1 \mu m^3 = 10^{-18} m^3\). This illustrates the vast difference in scale between these units, highlighting how minute the cubic micrometer measurement is.

Liters

A liter is another important metric unit of volume, often used for measuring liquids. One liter is equal to 1,000 milliliters (mL) or exactly \(0.001\) cubic meters (\(m^3\)). This means one cubic meter is quite a large volume equivalent, holding exactly 1,000 liters.Liters are typically familiar for household and everyday contexts, such as a bottle of water or the fuel capacity of a car. In scientific contexts, converting volumes into liters can provide a more tangible sense of scale for those less accustomed to cubic meters, making the measurement more accessible and relatable.

Metric Conversions

Metric conversions involve transforming one unit of measurement into another and are pivotal in science, engineering, and everyday tasks. They help in comparing and understanding different metrics. With volume, we often convert between units such as cubic micrometers, cubic meters, and liters, each serving different purposes based on context and scale.To convert volumes, one must know the relationships, like:

• \(1 \mu m^3 = 10^{-18} m^3\)
• \(1 m^3 = 1,000 L\)

Understanding these conversions allows accurate calculation and communication across varied fields. It is the backbone of global scientific endeavors that rely on standardized metric units for clarity and consistency.