Effective half-life multiple exponential decay

• NanakiXIII
In summary, the conversation discusses the possibility of calculating the half-life of a decay of the form A e^{-a x} + B e^{-b x} and whether it is possible to prove that a decay given by A e^{-a c x} + B e^{-b c x} has a half-life of \frac{t}{c}. While there is no clear proof, it is suggested that this relation is true based on the fact that scaling a function by a factor of c also scales the half-life by \frac{1}{c}.
NanakiXIII
Say we have a decay of the form

$$A e^{-a x} + B e^{-b x}$$.

I haven't had much luck trying to calculate the half-life of such a decay (I'm not sure it's possible, analytically), i.e. solve

$$A e^{-a x} + B e^{-b x} = \frac{A+B}{2}$$.

However, if that's not possible, I'm wondering whether there might still be a way to prove that if the above decay has half-life $$t$$, then a decay given by

$$A e^{-a c x} + B e^{-b c x}$$

has half-life $$\frac{t}{c}$$. This seems to be true empirically and would make sense, I think. Does anyone have an idea how one might prove it to be true, though?

Not really a 'proof', but I think it's rather obvious:

If f(t) is a function, then f(ct) is that function scaled to 1/c of it's size in the t-direction.

So if we plot the graph of these two functions, the point at which f(ct)=f(0)/2 will clearly be at 1/c times the point where f(t) = f(0)/2.

EDIT: I just read the above back to myself and it sounded a bit patronizing to say this was 'obvious' - sorry.

By way of a proof I suppose you could say let

$$f(x) = A e^{-a x} + B e^{-b x}$$.

and let

$$g(x) = A e^{-a c x} + B e^{-b c x}$$.

then:

$$g(x) = f(cx)$$

or equivalently:

$$g(\frac{y}{c}) = f(y)$$

So then if 't' is our half life of f, i.e. when:

$$f(t) = \frac{f(0)}{2}$$

then:

$$f(t) = g(\frac{t}{c})$$

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1. What is the concept of effective half-life in multiple exponential decay?

The effective half-life in multiple exponential decay refers to the time it takes for the activity of a radioactive substance to decrease by half, taking into account all the different decay rates of its multiple decay modes. It takes into consideration the contributions of each decay mode to the overall decay of the substance.

2. How is effective half-life different from traditional half-life?

Traditional half-life only considers the decay of a substance through one decay mode. In contrast, effective half-life takes into account the contributions of multiple decay modes, making it a more accurate measure of the overall decay rate of a substance.

3. What factors influence the effective half-life in multiple exponential decay?

The effective half-life in multiple exponential decay is influenced by the individual half-lives of each decay mode, as well as the relative contributions of each decay mode to the overall decay process. It can also be affected by factors such as temperature, pressure, and chemical environment.

4. How is effective half-life calculated?

The effective half-life in multiple exponential decay can be calculated using a mathematical formula that takes into account the individual half-lives and contributions of each decay mode. This formula can vary depending on the specific scenario and must be derived from the decay chain of the radioactive substance.

5. Why is understanding effective half-life important in radioactive decay studies?

Understanding effective half-life is important because it provides a more accurate measure of the overall decay rate of a substance, taking into account all of its decay modes. This is particularly useful in predicting the behavior of radioactive substances and assessing potential health risks associated with exposure to them.

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