Question

BETA DECAY:

part 1:

given data from plots:

hydrogen-3 trendline equation: y= -0.0531x +4.4662

carbon-14 trendline equation: y= -9E5x + 4.5819

A.) What is the physical interpretation of the slopes? The y-intercept?

B.) Using ( t 1/2 = 0.693/decay constant) calculate the half life of both hydrogen-3 and carbon-14.

C.) Compare to that of online sources, calculate percent difference.

Answer 1

Radioactivity equation ...................(1)

where N_o is number of atoms at initial time , N is number of atoms at time t and is decay constant

By Taking logarithm on both sides , we get , log ( N ) = log ( N_o ) - ( t ) ...................(2)

plot of Log(N) as a function of time t gives a graph of linear equation with slope - and intercept log(N_o )

Given trendline equation for ³H is y = -0.0531 x + 4.4662

Hence decay constant of ³H is 0.0531 , Half life = ln(2) / 0.0531 = 13.05 years

Half life of tritium ³H from standard reference ( https://www.radioactivity.eu.com ) is 12.3 years

Percentage of error = { | (12.3 -13.05 ) | / 12.3 } 100 = 6 %

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Given trendline equation for ¹⁴C is y= -9E-5x + 4.5819

( slope can't be 9E5 , or 9 10⁵ , it has to be 9 10^-5 . It is assumed given number 9E5 is some typing error while posting the question )

Hence decay constant of ¹⁴C is  9 10^-5 , Half life = ln(2) / 9 10^-5 = 7701.6 years

Half life of tritium ¹⁴C from standard reference ( https://www.radioactivity.eu.com ) is 5700 years

Percentage of error = { | ( 5700 - 7701.6 ) | / 5700 } 100 = 35 %