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Difficulty: HardScientific Notation and Unit Conversions

In an environmental study monitoring air quality near an industrial site, a scientist uses a flat horizontal collector plate with an active surface area of 5.0×10−2 m25.0 \times 10^{-2}\text{ m}^2 to gather falling dust particles. Over a sampling period of 1.8×104 seconds1.8 \times 10^4\text{ seconds}, a total mass of 2.0×10−3 grams2.0 \times 10^{-3}\text{ grams} of dust is deposited on the plate. Assuming the dust deposition rate is constant, what is the average dust deposition rate in micrograms per square meter per hour (μg/(m2⋅hr)\mu\text{g}/(\text{m}^2\cdot\text{hr}))?

Answer: 8000 ug/(m^2*hr)

Answer

The average dust deposition rate is 8000 micrograms per square meter per hour.
The correct rate of 8000 micrograms per square meter per hour is found by converting mass (2.0×10−3 g=2,000 μg2.0 \times 10^{-3}\text{ g} = 2,000\text{ }\mu\text{g}) and time (1.8×104 s=5.0 hours1.8 \times 10^4\text{ s} = 5.0\text{ hours}), then dividing this mass by the product of the area (5.0×10−2 m25.0 \times 10^{-2}\text{ m}^2) and the time (5.0 hours5.0\text{ hours}).

Step-by-Step Solution

1
Convert the mass of collected dust from grams (g) to micrograms (\mu g).
2,000μg2,000 \mu g
The target unit requires mass in micrograms. Since 1 g=106 μg1\text{ g} = 10^6\text{ }\mu\text{g}, multiplying 2.0×10−3 g2.0 \times 10^{-3}\text{ g} by 10610^6 gives 2.0×103 μg=2,000 μg2.0 \times 10^3\text{ }\mu\text{g} = 2,000\text{ }\mu\text{g}.
2
Convert the collection time from seconds (s) to hours (hr).
5.0 hr
The target unit requires time in hours. Since 1 hr=3,600 s1\text{ hr} = 3,600\text{ s}, dividing the total seconds by 3,6003,600 gives 1.8×104 s÷(3.6×103 s/hr)=5.0 hr1.8 \times 10^4\text{ s} \div (3.6 \times 10^3\text{ s/hr}) = 5.0\text{ hr}.
3
Calculate the average deposition rate by dividing mass by the product of area and time.
8,000μg/(m2⋅hr)8,000 \mu g/(m^2\cdot hr)
Deposition rate is given by the formula Rate=MassArea×Time\text{Rate} = \frac{\text{Mass}}{\text{Area} \times \text{Time}}. Substituting the values: Rate=2,000 μg(5.0×10−2 m2)×5.0 hr=2,0000.25=8,000 μg/(m2⋅hr)\text{Rate} = \frac{2,000\text{ }\mu\text{g}}{(5.0 \times 10^{-2}\text{ m}^2) \times 5.0\text{ hr}} = \frac{2,000}{0.25} = 8,000\text{ }\mu\text{g}/(\text{m}^2\cdot\text{hr}).

Key Concept

Multi-step dimensional analysis and calculation with scientific notation
Estimated Time:2m 30s
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Topics

Scientific Notation and Unit Conversions

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