
What is the definition of exponential decay?
Answer
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Hint: A quantity is subjected to exponential decay if it decreases at a rate proportional to its current value. Exponential decay is the decrease in a quantity N according to the law , where is a positive number called the decay constant of the decaying quantity. We have to derive this formula from by rearranging the terms, integrating and taking the exponents.
Complete step-by-step answer:
Let us see the definition of exponential decay. We can say that a quantity is subjected to exponential decay if it decreases at a rate proportional to its current value. The decay law calculates the number of undecayed nuclei in a given radioactive substance.
Let us see the formula for Half-Life in Exponential Decay. Half-life is the length of time it takes an exponentially decaying quantity to decrease to half its original amount. The formula for Half-Life in Exponential Decay is given as
where is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.), N(t) is the quantity that still remains and has not yet decayed after a time t and is the half-life of the decaying quantity.
We can also write the decay formula as
where is a positive number called the mean lifetime of the decaying quantity.
We mainly use the below formula for exponential decay.
Exponential decay is described by the first-order ordinary differential equation.
Let us rearrange this equation.
Let us integrate both the sides.
We know that and .
Let us take exponents on both the sides.
We know that and .
Let us denote as . This s obtained by evaluating the equation at t = 0, as is defined as being the quantity at .
where is a positive number called the decay constant of the decaying quantity.
We can also relate the parameters , and .
When comparing the above three formulas, we will get
Let us take the natural logarithms on both sides.
We know that and . Hence, the above equation can be written as
Let us cancel t from all the terms.
Let us consider . We have to take to the RHS.
We know that .
We know that .
Let us cancel the negative sign from both the sides.
Let us take to the LHS.
Now, let us consider
Similarly, when we solve the LHS, we will get
Let us take to the LHS.
From (i) and (ii), we can write
Let us consider .
We know that
Let us take to the LHS and t to the RHS.
Decay constant ( ) gives the ratio of number of radioactive atoms decayed to the initial number of atoms.
The graph of exponential decay is shown below for the function .
Therefore, exponential decay is the decrease in a quantity N according to the law .
Note: Students may get confused with exponential growth and exponential decay. xponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function’s current value, resulting in its growth with time being an exponential function. The formula for exponential growth is , where t is the time (number of periods), P(t) is the amount of some quantity at time t, is the initial amount at time , r is the the growth rate and e is the Euler’s number = 2.71828.
Complete step-by-step answer:
Let us see the definition of exponential decay. We can say that a quantity is subjected to exponential decay if it decreases at a rate proportional to its current value. The decay law calculates the number of undecayed nuclei in a given radioactive substance.
Let us see the formula for Half-Life in Exponential Decay. Half-life is the length of time it takes an exponentially decaying quantity to decrease to half its original amount. The formula for Half-Life in Exponential Decay is given as
where
We can also write the decay formula as
where
We mainly use the below formula for exponential decay.
Exponential decay is described by the first-order ordinary differential equation.
Let us rearrange this equation.
Let us integrate both the sides.
We know that
Let us take exponents on both the sides.
We know that
Let us denote
where
We can also relate the parameters
When comparing the above three formulas, we will get
Let us take the natural logarithms on both sides.
We know that
Let us cancel t from all the terms.
Let us consider
We know that
We know that
Let us cancel the negative sign from both the sides.
Let us take
Now, let us consider
Similarly, when we solve the LHS, we will get
Let us take
From (i) and (ii), we can write
Let us consider
We know that
Let us take
Decay constant (
The graph of exponential decay is shown below for the function

Therefore, exponential decay is the decrease in a quantity N according to the law
Note: Students may get confused with exponential growth and exponential decay. xponential growth occurs when the growth rate of the value of a mathematical function is proportional to the function’s current value, resulting in its growth with time being an exponential function. The formula for exponential growth is
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