Problem 39 - Chemistry and Chemical Reactivity

Question

A red blood cell has a diameter of \(7.5 \mu \mathrm{m}\) (micrometers). What is this dimension in (a) meters, (b) nanometers, and (c) picometers?

Step-by-Step Solution

Answer

(a) \(7.5 \times 10^{-6}\) m, (b) 7500 nm, (c) \(7.5 \times 10^6\) pm.

Step 1: Understand the Conversion Relation

To convert micrometers to meters, know that one micrometer \(\mu \text{m}\) is defined as \(1 \times 10^{-6}\) meters. Therefore, to convert from micrometers to meters, multiply the micrometer value by \(10^{-6}\).

Step 2: Convert Micrometers to Meters

Given a diameter of \(7.5 \mu \text{m}\), convert this to meters as follows: \[7.5 \mu \text{m} \times 10^{-6} = 7.5 \times 10^{-6} \text{m}.\]

Step 3: Convert Micrometers to Nanometers

To convert micrometers to nanometers, use the fact that one micrometer \(\mu \text{m}\) is equal to \(1000\) nanometers (nm). Multiply \(7.5 \mu \text{m}\) by \(1000\) to get the nanometer value: \[7.5 \mu \text{m} \times 1000 = 7500 \text{ nm}.\]

Step 4: Convert Micrometers to Picometers

For converting micrometers to picometers, use the relation: 1 micrometer \(= 10^6\) picometers (pm). Therefore, multiply \(7.5 \mu \text{m}\) by \(10^6\): \[7.5 \mu \text{m} \times 10^6 = 7.5 \times 10^6 \text{ pm}.\]

Key Concepts

Micrometer to Meter Conversion

Understanding unit conversion involves knowing the basic equivalent units required for each situation. When converting from micrometers to meters, it is crucial to recognize that these conversions are based on powers of ten.
A micrometer (\(\mu\)m) is much smaller than a meter, specifically, it is one-millionth of a meter. This is expressed in scientific notation as \(1 \mu \text{m} = 1 \times 10^{-6} \text{ m}\).
It means that for every single micrometer, one must move six decimal places to the left to arrive at the corresponding meter value.
To convert, you multiply the number of micrometers you have by \(10^{-6}\).
For instance, if we take the diameter of a red blood cell which is \(7.5 \mu \text{m}\), the conversion to meters would be:
\[7.5 \mu \text{m} \times 10^{-6} = 7.5 \times 10^{-6} \text{ m}.\]
This simple multiplication allows us to understand microscopic measurements on a larger scale.

Micrometer to Nanometer Conversion

Micrometers and nanometers are both units often used to measure extremely tiny distances or sizes.
Knowing how to convert one to the other is useful, especially in fields like biology and physics.
A micrometer is equivalent to one thousand nanometers, expressed as \(1 \mu \text{m} = 1000 \text{ nm}\).
This means that there are precisely 1,000 times more nanometers in the same distance as there are micrometers. The multiplication factor makes it relatively straightforward:
Simply multiply the number of micrometers by 1000.
Taking the red blood cell example with a diameter of \(7.5 \mu \text{m}\), converting to nanometers requires calculating:
\[7.5 \mu \text{m} \times 1000 = 7500 \text{ nm}\].
It provides insight into how these very small measurements fit in a hierarchical scale of units.

Micrometer to Picometer Conversion

Picometers might seem even more complex due to their relatively rare use, but they are crucial for describing atomic-scale dimensions.
A picometer is a trillionth of a meter, expressed as one micrometer equating to one million picometers.
This makes the conversion factor for micrometers to picometers \(1 \mu \text{m} = 10^{6} \text{ pm}\).
The conversion process requires multiplying the micrometer value by \(10^{6}\), which aligns closely with the powers-of-ten methodology seen elsewhere in metric conversions.
For the red blood cell example, transforming \(7.5 \mu \text{m}\) into picometers involves:
\[7.5 \mu \text{m} \times 10^{6} = 7.5 \times 10^{6} \text{ pm}\].
Such calculations help explore minute scales and unlock a deeper understanding of the micro-world around us.