Solving for r in Carbon-14's Half-Life

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SUMMARY

The discussion focuses on solving for the constant rate r in the context of Carbon-14's half-life, which is established as 5730 years. The differential equation dQ/dt = -rQ is correctly manipulated to yield lnQ = -rt through integration. To determine r, the initial condition Q = 1 at t = 0 and the condition Q = 1/2 at t = 5730 years are applied. The integration can also be approached as a definite integral from t=0 to t=5730, utilizing the Qf/Qi ratio of 1/2.

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  • Understanding of differential equations, specifically first-order linear equations.
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  • Familiarity with the concept of half-life in radioactive decay.
  • Basic logarithmic properties and their applications in solving equations.
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Homework Statement
The half life of carbon 14 is 5730 years long. Assume that dQ/dt = -rQ. Determine the constant rate r.

The attempt at a solution

dQ/dt = -rQ (Given)

dQ/Q = -rdt (algebra)
lnQ = -rt (integrate)
-lnQ/t = r (algebra)

How do I work in 5,730? Also, did I do the equations correctly?
 
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I think you have forgotten a constant in your inetgration if it is an indefinite integral, otherwise integration looks ok

to get r
start with Q = 1 at t = 0
then by the meaning of half life, you have Q = 1/2 at t = 5730yrs

Another option is to re-perform the integration as a definite integral from t=0 to t= 5730 knowing the Qf/Qi ratio = 1/2
 

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