# Application Of Calculus in Radioactive

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24-Aug-2014Category

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Application Of Calculus in RadioactivePrepared by : Ng Seng Wei D20091035105 Nor Awaathif Bt Mohd Ghazali Lee D20091035070

Words Of The Day Radioactive Decay Law Decay Constant Half-Life

APPLICATION OF CALCULUS IN RADIOACTIVE

RADIOACTIVE DECAY LAW

N t = N 0e

t

Let us say that in a sample of radioactive material there are N nuclei which have not decayed at a certain time, t. Therefore, we can say that the number which will decay depends on total number of nuclei, N and also the length of the brief period of time. In other words the more nuclei there are the more will decay and the longer the time period the more nuclei will decay. Let us donate the number which will have decayed as dN and the small time interval as dt.

How Calculus was Applied in It?1. First we used integral calculus to figure out what was happening over a period of time by integrating what we knew could occur in a brief interval of time. - dN N.dt - dN = N.dt

dN = dt NdN = dt N N0 0Nt t

2. Secondly, we used calculus relationship of

Nt ln N 0

= t

3. Thirdly, we used the definition of logarithmsN t ln N 0 = t

Nt t =e N0Until then it will form an equation of,

N t = N 0e

t

Radioactive Decay Law

The Radioactive Decay Law tells us that the number of radioactive nuclei will decrease with time in an exponential pattern with the rate of decrease being controlled by the decay constant. The law could be shown in graphical figure as below:

The law tells us that at any time, t:

RELATIONSHIP BETWEEN RADIOACTIVE DECAY LAW AND HALF-LIFEN t = N 0eN0 Nt = 2 t

Meanwhile, the definition of half-life tells us that:

When

t =t1

2

Therefore the equation of Radioactive Decay Law could be rewrite by substituting Nt and t.

t 1 N0 2 = N 0e 2 t 1 1 2 =e 2

2

1

=e

t 1

2

1 l 2 = 1 n t

l 2 = t1 n 0.6 3 9 t1

2

= t1

2

2

2

0.6 3 9 =

0.6 3 9 = t12

These last two equations express the relationship between the decay constant and the half-life

~Lets Try This !!~

QUESTION

THE END THANK YOU