Final answer:
The volume of Avogadro's number of sand grains is approximately 6.022 × 10^5 km^3. The sand grains, neglecting air spaces between them, would cover a beach with a length of approximately 6.022 × 10^3 kilometers.
Step-by-step explanation:
Avogadro's number is a fundamental constant in chemistry and physics that represents the number of particles (atoms or molecules) in one mole of a substance. It is approximately equal to 6.022 × 10^23. Since each grain of sand is a cube with side length 1.0 mm, the volume of one grain is (1.0 mm)^3 = 1.0 mm^3. To find the volume of Avogadro's number of sand grains, we multiply the volume of one grain by Avogadro's number:
V = (1.0 mm^3) × (6.022 × 10^23) = 6.022 × 10^23 mm^3
To convert this volume to km^3, we can use the following conversion factors: 1 mm = 1.0 × 10^-6 km and 1 mm^3 = (1.0 × 10^-6 km)^3:
V = (6.022 × 10^23 mm^3) × (1.0 × 10^-6 km/mm)^3 = 6.022 × 10^23 × 10^-18 km^3 = 6.022 × 10^5 km^3 (rounded to 3 significant figures)
Therefore, the volume of Avogadro's number of sand grains is approximately 6.022 × 10^5 km^3.
To answer part (b), we need to calculate the total volume of all the sand grains and then convert it to kilometers. The volume of one sand grain is 1.0 mm^3, and we have Avogadro's number of grains, so the total volume is:
Total volume = (1.0 mm^3/grain) × (6.022 × 10^23 grains)
Using the conversion factor 1 mm = 1.0 × 10^-6 km, we can convert the total volume from mm^3 to km^3:
Total volume = (1.0 mm^3/grain) × (6.022 × 10^23 grains) × (1.0 × 10^-6 km/mm)^3
Total volume = (6.022 × 10^5 km^3) (rounded to 3 significant figures)
Assuming the beach has a uniform width of 100 m and a depth of 10.0 m, we can calculate the length of the beach that would be covered by the sand grains:
Length of beach = (6.022 × 10^5 km^3) / (0.1 km × 10.0 km) = 6.022 × 10^3 km
Therefore, the sand grains, neglecting air spaces between them, would cover a beach with a length of approximately 6.022 × 10^3 kilometers.