   Chapter 12, Problem 4C

Chapter
Section
Textbook Problem

If the average lifetime of the proton were 10 years, estimate the number of protons per kilogram of human body mass that would decay radioactively each year. Assume that a human being is made entirely of water molecules and that each molecule contributes 10 protons. (Hint: The molar weight of water is 18 g, and each mole of any substance contains particles.) If each proton decay released 1 MeV of energy, how much energy per kilogram would be produced in total by the decaying protons in your body in 1 year? Give the result in joules per kilogram. If the energy released is mostly in the form of kinetic energy of the positrons produced in the decay, then 1 J/kg of this type of ionizing radiation is rated at about 1 Sv (the sievert [Sv] is the SI unit of radiation dose). Compare the total exposure in sieverts of a person to the radiation produced by his/her own body to the maximum safe dose of 1 mSv/yr for the general public from all artificial sources.

To determine

The number of protons per kg of human body, energy per kilogram b decaying proton.

Explanation

Given:

Average life time, tm=1016years

Molar mass of water, M = 18g.

Calculation:

The average life time of proton is given by,

tm=1λλ=1tm

The number of protons at time tm is given by,

N(tm)=N0eλtm

N0 = number of protons initially

N(tm) = number of protons at time t,

The number of protons initially is,

N0=100018×10=555proton

The number of protons at time tm is given by,

N(tm)=N0eλtmN(tm)=555×0.37N(tm)205

Thus, number of proton decay in a year is,

N0N(tm)=555205=350

One decay proton releases 1MeV of energy

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