Question

The volume of a certain bacterial cell is \(2.56 \mu \mathrm{m}^{3} .\) (a) What is its volume in cubic millimeters (mm \(^{3}\) )? (b) What is the volume of \(10^{5}\) cells in liters (L)?

Short answer

The volume in cubic millimeters is \(2.56 \times 10^{-9} \text{mm}^3\). The volume of \(10^5\) cells in liters is \(2.56 \times 10^{-10} \text{L}\).

Step-by-step solution

- Convert Volume to Cubic Millimeters

First, convert the volume from cubic micrometers (\textmu\text{m}^3) to cubic millimeters (mm^3). We use the conversion factor: \(1 \textmu\text{m} = 1 \times 10^{-3} \text{mm}\). Therefore, \((1 \textmu\text{m})^3 = (1 \times 10^{-3} \text{mm})^3 = 1 \times 10^{-9} \text{mm}^3\). Thus, the volume in cubic millimeters is \(2.56 \text \textmu\text{m}^3 \times 1 \times 10^{-9} \text{mm}^3/\text{\textmu\text{m}^3} = 2.56 \times 10^{-9} \text{mm}^3\).

- Calculate Volume of 10^5 Bacterial Cells in Cubic Millimeters

Next, find the total volume for \(10^5\) bacterial cells. Multiply the volume of one cell by \(10^5\): \(2.56 \times 10^{-9} \text{mm}^3 \times 10^5 = 2.56 \times 10^{-4} \text{mm}^3 \).

- Convert Volume to Liters

Finally, convert the volume from cubic millimeters (mm^3) to liters (L). Use the conversion factor: \(1 \text{mm}^3 = 1 \times 10^{-6} \text{L}\). Thus, the volume in liters is: \(2.56 \times 10^{-4} \text{mm}^3 \times 1 \times 10^{-6} \text{L}/\text{mm}^3 = 2.56 \times 10^{-10} \text{L}\).

Key concepts

Volume Conversion

Understanding volume conversion is crucial in many scientific fields, including chemistry and biology. It allows scientists to compare and analyze quantities that are presented in different units. When converting volume between units, it's essential to understand the relationship between these units.
The basic principle of volume conversion involves multiplying or dividing by specific conversion factors. For instance, converting from smaller units to larger units typically involves dividing by a conversion factor, while converting from larger units to smaller units involves multiplying.

Cubic Micrometers to Cubic Millimeters

To convert volume from cubic micrometers (\textmu\text{m}^3) to cubic millimeters (mm^3), you need to know the relationship between these two units. 1 micrometer (\textmu\text{m}) is equal to 1 x 10^-3 millimeters (mm).
When dealing with volume, you cube that relationship: \((1 \textmu\text{m})^3 = (1 \times 10^{-3} \text{mm})^3 = 1 \times 10^{-9} \text{mm}^3\)
Thus, to convert 2.56 \textmu\text{m}^3 to mm^3, we multiply the volume in micrometers by \(1 \times 10^{-9}\).
So, \(2.56 \textmu\text{m}^3 = 2.56 \times 10^{-9} \text{mm}^3\).This little volume is typical for bacterial cells, showing how tiny these units are.

Cubic Millimeters to Liters

Converting from cubic millimeters (mm^3) to liters (L) is another important volume conversion. The conversion factor between these units is: 1 mm^3 equals 1 x 10^-6 liters.
When we have the total volume of bacterial cells in cubic millimeters, we can convert it to liters for a broader perspective. For example, after finding the total volume of \(10^5\) bacterial cells is \(2.56 \times 10^{-4}\text{mm}^3\), next, we use the conversion factor:\(2.56 \times 10^{-4} \text{mm}^3 \times 1 \times 10^{-6} \text{L/mm}^3 = 2.56 \times 10^{-10} \text{L}\).Such conversions are essential, especially in fields where we deal with vastly different scales, enhancing our understanding of these measurements.