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Difficulty: MediumScientific Notation and Unit Conversions

During an environmental monitoring study of urban air quality, scientists use a cascade impactor to collect and analyze different types of airborne particulate matter. The average diameters of four distinct particulate samples were recorded as follows:

* Particle W: 3.5×102 nm3.5 \times 10^2 \text{ nm}
* Particle X: 1.2×10−6 m1.2 \times 10^{-6} \text{ m}
* Particle Y: 7.5×10−2 μm7.5 \times 10^{-2} \text{ }\mu\text{m}
* Particle Z: 4.0×10−5 cm4.0 \times 10^{-5} \text{ cm}

Based on these measurements, arrange the four particulate samples in order of their average diameters from smallest to largest.

  1. 1Particle Y (7.5×10−2 μm7.5 \times 10^{-2} \text{ }\mu\text{m})
  2. 2Particle W (3.5×102 nm3.5 \times 10^2 \text{ nm})
  3. 3Particle Z (4.0×10−5 cm4.0 \times 10^{-5} \text{ cm})
  4. 4Particle X (1.2×10−6 m1.2 \times 10^{-6} \text{ m})

Answer

The correct order from smallest to largest is Particle Y, Particle W, Particle Z, and Particle X.
By converting all measurements to meters, we find: Particle Y is 7.5×10−8 m7.5 \times 10^{-8} \text{ m}, Particle W is 3.5×10−7 m3.5 \times 10^{-7} \text{ m}, Particle Z is 4.0×10−7 m4.0 \times 10^{-7} \text{ m}, and Particle X is 1.2×10−6 m1.2 \times 10^{-6} \text{ m}. Comparing these values shows that Particle Y is the smallest, followed by Particle W, Particle Z, and Particle X as the largest.

Step-by-Step Solution

1
Convert all measurements to meters (m{\text{m}}) using standard conversion factors (1 nm=10−9 m1 \text{ nm} = 10^{-9} \text{ m}, 1 μm=10−6 m1 \text{ }\mu\text{m} = 10^{-6} \text{ m}, and 1 cm=10−2 m1 \text{ cm} = 10^{-2} \text{ m}).
Particle W: 3.5×102 nm=3.5×102×10−9 m=3.5×10−7 m3.5 \times 10^2 \text{ nm} = 3.5 \times 10^2 \times 10^{-9} \text{ m} = 3.5 \times 10^{-7} \text{ m}.
Particle X: 1.2×10−6 m1.2 \times 10^{-6} \text{ m}.
Particle Y: 7.5×10−2 μm=7.5×10−2×10−6 m=7.5×10−8 m7.5 \times 10^{-2} \text{ }\mu\text{m} = 7.5 \times 10^{-2} \times 10^{-6} \text{ m} = 7.5 \times 10^{-8} \text{ m}.
Particle Z: 4.0×10−5 cm=4.0×10−5×10−2 m=4.0×10−7 m4.0 \times 10^{-5} \text{ cm} = 4.0 \times 10^{-5} \times 10^{-2} \text{ m} = 4.0 \times 10^{-7} \text{ m}.
Expressing all values in the same base metric unit (meters) allows for a direct comparison of their scales.
2
Express all values in terms of the same exponent of 1010 (such as 10−710^{-7}) to easily compare the coefficients.
Particle Y: 0.75×10−7 m0.75 \times 10^{-7} \text{ m}.
Particle W: 3.5×10−7 m3.5 \times 10^{-7} \text{ m}.
Particle Z: 4.0×10−7 m4.0 \times 10^{-7} \text{ m}.
Particle X: 12.0×10−7 m12.0 \times 10^{-7} \text{ m}.
Aligning the exponents simplifies the comparison to just ordering the coefficient numbers.
3
Compare the coefficients from smallest to largest.
Since 0.75<3.5<4.0<12.00.75 < 3.5 < 4.0 < 12.0, the order from smallest to largest is Particle Y, Particle W, Particle Z, then Particle X.
The coefficients directly scale the common base exponent, yielding the final ordered list.

Key Concept

To compare values with different metric prefixes, convert each value to a common base unit (such as meters) and write them in scientific notation with a matching exponent.
Estimated Time:1m 30s
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Topics

Scientific Notation and Unit Conversions

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