Show that if a radioactive substance has a half life of T, then the corresponding exponential decay function can be written as y=y0e^kt
amountleft=akmountoriginal (1/2)^t
left/original= (1/2)^t
taking the ln of each side
ln(left/orig)=t ln(.5)=kt
now take the antilog of each side
left/orig=e^kt
left=yo*e^kt
so what is k?
if left/yo=e^kt
at 1/2= e^k*T
ln of each side
.693=kT
k= .693/T
so the equation is of the form...
y= yo e^ln.5 * t/T
To show that the exponential decay function for a radioactive substance with a half-life of T can be written as y = y0e^kt, we need to understand the concept of half-life and how it relates to exponential decay.
The half-life of a radioactive substance is the amount of time it takes for half of the substance to decay. In other words, after one half-life has passed, only half of the original substance remains.
Now, let's derive the exponential decay equation y = y0e^kt.
Assuming we start with an initial amount of y0, after one half-life T, the amount remaining would be y0/2. We can represent this mathematically as:
y(T) = y0/2
Next, we need to determine the decay constant k. The decay constant is defined as the natural logarithm of 2 divided by the half-life:
k = ln(2)/T
Substituting this value of k back into the equation, we have:
y(T) = y0e^(ln(2)/T * T)
The T in the numerator and denominator cancels out, which leaves us with:
y(T) = y0e^ln(2)
The natural logarithm of 2 (ln(2)) is approximately 0.6931. Substituting this value back into the equation, we have:
y(T) = y0e^0.6931
Since e^0.6931 is approximately 2.7183, the equation becomes:
y(T) ≈ 2.7183y0
This tells us that after one half-life, the remaining amount of the substance is approximately 2.7183 times smaller than the initial amount.
Finally, we can rewrite the equation as:
y = y0e^(kt)
Where:
- y is the amount of substance remaining at any given time t
- y0 is the initial amount of the substance
- k is the decay constant (ln(2)/T)
- t is the time elapsed
Therefore, we have shown that if a radioactive substance has a half-life of T, the corresponding exponential decay function can be written as y = y0e^(kt).