Derive relationship between radioactive constant and half life

In summary, the equation for radioactive decay is A = A<sub>0</sub>e<sup>-kt</sup>, where A is the amount of radioactive material remaining, A<sub>0</sub> is the initial amount, k is the decay constant, and t is the time. The half-life of a radioactive substance is the amount of time it takes for half of the initial amount to decay, represented by t<sub>1/2</sub>. The relationship between the half-life and decay constant is t<sub>1/2</sub> = ln2/k, meaning that the half-life is inversely proportional to the decay constant. The decay constant can be calculated from the half-life using the equation k
  • #1
quantic123
1
0
Hi,

I was wondering how to derive relationship between radiocative constant and half life, which is t1/2=ln2/b, where b=decay constant.

It seemed like the it was just replaced into the equation A=A0(1/2)t/t1/2.

Thanks in advance
 
Last edited:
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  • #2
From ##A=A_0 e^{-bt}##, just manipulate the exponential: $$A=A_0 e^{-bt}=A_0 2^{\frac{-bt}{\ln(2)}} = A_0 2^{-\frac{t}{t_{1/2}}}$$
 

Related to Derive relationship between radioactive constant and half life

What is the equation for radioactive decay?

The equation for radioactive decay is A = A0e-kt, where A is the amount of radioactive material remaining, A0 is the initial amount, k is the decay constant, and t is the time.

What is the half-life of a radioactive substance?

The half-life of a radioactive substance is the amount of time it takes for half of the initial amount to decay. It is represented by the symbol t1/2.

How is the half-life related to the decay constant?

The relationship between the half-life and decay constant is t1/2 = ln2/k. This means that the half-life is inversely proportional to the decay constant.

Can the decay constant be calculated from the half-life?

Yes, the decay constant can be calculated from the half-life using the equation k = ln2/t1/2.

How does the decay constant affect the rate of radioactive decay?

A higher decay constant means a faster rate of decay, as there will be more decays occurring per unit time. Conversely, a lower decay constant results in a slower rate of decay.

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