Question

How many molecules of water are necessary if scientists wanted to check the 10^31-yr estimate of proton decay within the course of one calendar year?

a) 6.022 × 10^23 molecules

b) 1.0 × 10^31 molecules

c) 1.0 × 10^52 molecules

d) 3.154 × 10^7 molecules

Answer 1

This choice aligns with the estimated proton decay duration of 10^31 years, indicating the need for one water molecule per year to assess this timeframe. The correct answer is (b) 1.0 × 10^31 molecules.

Step-by-step explanation:

The given question involves determining the number of water molecules necessary to investigate the 10^31-year estimate of proton decay within one calendar year. The fundamental unit for quantifying substances in chemistry is the mole, with Avogadro's number (6.022 × 10^23) representing the number of entities (atoms, molecules, etc.) in one mole. To find the number of molecules needed for one year, we multiply the estimated duration (10^31 years) by Avogadro's number:

`\[ \text{Number of molecules} = \text{Duration} * \text{Avogadro's number} \]`

`\[ = 10^31 * 6.022 * 10^23 \]`

`\[ = 6.022 * 10^54 \]`

This result indicates the total number of water molecules needed for the entire 10^31-year duration. However, the question specifies the requirement within one calendar year. Therefore, to find the annual requirement, we divide the total by the number of years in one calendar year
`(approximately \(3.154 * 10^7\))`:

`\[ \text{Annual requirement} = (6.022 * 10^54)/(3.154 * 10^7) \]`

`\[ \approx 1.0 * 10^31 \]`

This calculation yields the number of water molecules necessary annually to evaluate the 10^31-year estimate of proton decay. Hence the correct answer is (b) 1.0 × 10^31 molecules.