Slope of N - t graph of radioactive decay

In summary, the slope of an N-t graph of radioactive decay represents the rate of decay of a radioactive substance, where N is the quantity of the substance and t is time. The graph typically shows an exponential decline, indicating that the number of undecayed nuclei decreases over time at a rate proportional to the remaining quantity. The negative slope reflects the loss of radioactive atoms as they decay, with the steepness of the slope indicating the half-life of the substance.
  • #1
songoku
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Homework Statement
Please see below
Relevant Equations
##N=N_o e^{-\lambda t}##
1713067012655.png


I am not really sure how to interpret the slope. The equation is:

$$N=N_o e^{-\lambda t}$$

If the graph is N against t, then what is the slope?

I can find the slope if the graph is log:
$$log N=log N_o -\lambda t$$

So if the graph is log N against t, then the slope is ##-\lambda##

But if the graph is N against t, I have no idea what the slope is.

Thanks
 
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  • #2
Are you familiar with derivatives?
 
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  • #3
Orodruin said:
Are you familiar with derivatives?
Yes. I understand your hint.

Thank you very much Orodruin
 

FAQ: Slope of N - t graph of radioactive decay

What does the slope of an N-t graph represent in radioactive decay?

The slope of an N-t graph, which plots the number of undecayed nuclei (N) against time (t), represents the rate of decay of the radioactive substance. Specifically, it indicates how quickly the quantity of the radioactive material decreases over time, typically expressed in terms of the decay constant (λ).

How is the slope related to the decay constant?

The slope of the N-t graph is negative and is equal to the negative decay constant (−λ). This means that as time progresses, the number of undecayed nuclei decreases exponentially, and the slope indicates the rate of this exponential decay.

What is the mathematical expression for the N-t graph?

The N-t relationship for radioactive decay can be described by the equation N(t) = N0 * e^(-λt), where N0 is the initial quantity of the substance, λ is the decay constant, and t is time. The slope of the graph at any point is given by the derivative of this equation with respect to time.

How can you determine the half-life from the slope?

The half-life (T½) of a radioactive substance can be derived from the decay constant using the formula T½ = ln(2) / λ. By knowing the slope of the N-t graph, which is related to the decay constant, you can calculate the half-life of the substance.

What factors influence the slope of the N-t graph?

The slope of the N-t graph is primarily influenced by the properties of the radioactive isotope itself, specifically its decay constant, which is a characteristic of the isotope. Environmental factors, such as temperature and pressure, do not significantly affect the decay rate of a radioactive substance, as decay is a nuclear process governed by the fundamental forces within the nucleus.

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