The half-life of cesium-137 is 30 years. Suppose we have a...

The half-life of cesium-137 is 30 years. Suppose we have a
100-mg sample.
(a) Find the mass that remains after t years.
(b) How much of the sample remains after 100 years?
(c) What is the rate of decay after 100 years?
(d) After how long will only 1 mg remain?

Key Concepts

Half-life

Half-life is a measure of the time required for a quantity undergoing exponential decay to reduce to half of its initial value. It is a fundamental concept in radioactive decay, providing a consistent way to describe the rate at which unstable isotopes transform.

Exponential Decay

Exponential decay describes a process where the amount of a substance decreases at a rate proportional to its current amount. This model is commonly used in radioactive decay, where the remaining mass of a radioactive substance after time t is given by an exponential function.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a particular atom will decay. It is directly related to the half-life of a substance through the relation k = ln(2) / (half-life), and it is used in the exponential decay equation to model the decay process continuously.

Rate of Decay

The rate of decay refers to the instantaneous rate at which the mass of a radioactive substance decreases. This is determined by differentiating the exponential decay function with respect to time, providing insight into how fast the material is decaying at any given moment.

Logarithms

Logarithms are used in solving equations involving exponential decay, particularly when determining the time required for a substance to decay to a certain amount. They provide a mathematical tool to isolate the variable in the exponent, such as finding the time at which only 1 mg of a substance remains.

Transcript

00:01 Now for problem number 7, let's say that m of t is the mass of cesium in milligrams, remaining after time t in years.

00:16 So m of t is equal to m of zero, the full mass of cesium at year zero, multiplied by exponential k.

00:29 So m of 0 is given to be 100 milligrams exponential kt.

00:42 Now we need to get the value of k so that we can get the mass of cesium at any year t.

00:53 So first to get the value of k, we have another giving that's a half life of cesium is 30 years.

01:02 M of 30 is actually equal to half of m of 0.

01:16 So m of 30 is equal to 100 exponential k multiplied by 30 using this term.

01:31 And the half -life cesium is half multiplied by 100.

01:39 We can terminate this with 100 with this 100 so we have e 13k equal to one half and we can take len for both sides so here we will have 30k equals to len half and len half is also equal to negative len 2 for simplicity so the k value would be negative length over 30.

02:30 And now we can say that m of t is the mass at in here is equal to 100 exponential negative length 2 over 30 k t sorry over 30 t okay okay.

02:58 And that can be simplified that m of t is equal to 200, sorry, 200 to the power of negative t over 30.

03:17 Because exponential length 2 is equal to 2.

03:23 So here we have 200.

03:26 So now we have got the main function.

03:30 So with part b, we want to know after when t is equal to 100.

03:40 How much mass remains? so we basically seeing m of 100, direct substitution is equal to 200 to the power of negative 100 over 30, which is equal to about 9 .92 milligrams...

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