Show that if a radioactive substance has a half life of T, then the corresponding constant k in the exponential decay function is given by k=-ln(2)/T
I'm confused about how to show that that is true.
since the amount drops by 1/2 every T, the function is
f(t) = (1/2)^(t/T)
But, 1/2 = e^-ln2, so
f(t) = (e^(-ln2))^(t/T))
= e^(-ln2/T t)
= e^(kt)
To show that if a radioactive substance has a half-life of T, then the corresponding constant k in the exponential decay function is given by k = -ln(2)/T, we can use the concept of half-life.
The half-life of a substance is the time it takes for the amount of that substance to decrease by half. Let's consider a radioactive substance with an initial amount of N₀ at time t=0.
The exponential decay function is given by:
N(t) = N₀ * e^(kt),
where N(t) is the amount of the substance at time t, and e is the base of the natural logarithm. We want to find the value of k that corresponds to a half-life of T.
After a time period of T, the amount of the substance will have decreased to half its initial value:
N(T) = N₀/2.
Substituting these values into the exponential decay equation, we get:
N(T) = N₀ * e^(kT) = N₀/2.
Dividing both sides of the equation by N₀, we have:
e^(kT) = 1/2.
To solve for k, we take the natural logarithm (ln) of both sides:
ln(e^(kT)) = ln(1/2).
Using the property of logarithms, ln(e^(kT)) simplifies to kT:
kT = ln(1/2).
Finally, we solve for k by dividing both sides of the equation by T:
k = ln(1/2)/T.
Using the fact that ln(1/2) = -ln(2), we can rewrite the equation as:
k = -ln(2)/T.
Therefore, if a radioactive substance has a half-life of T, then the corresponding constant k in the exponential decay function is given by k = -ln(2)/T.
To show that if a radioactive substance has a half-life of T, then the corresponding constant k in the exponential decay function is given by k = -ln(2)/T, we can follow these steps:
Step 1: Understand the concept of half-life.
The half-life of a substance is the time it takes for half of the initial amount of the substance to decay. In other words, after one half-life, the remaining amount of the substance is half of its initial value.
Step 2: Understand the concept of exponential decay.
The decay of a radioactive substance follows an exponential decay function of the form: N(t) = N0 * e^(-kt), where N(t) represents the amount of the substance remaining at time t, N0 represents the initial amount of the substance, k is the decay constant, and e is the base of the natural logarithm.
Step 3: Derive the decay constant.
Let's start by considering the equation for half-life. Since the remaining amount after one half-life is half of the initial amount, we can write:
N(t) = N0 * 1/2
Substituting this into the exponential decay equation, we get:
N0 * 1/2 = N0 * e^(-kt)
Canceling out N0, we have:
1/2 = e^(-kt)
Step 4: Solve for the decay constant.
To find the decay constant k, we need to isolate it in the above equation. Taking the natural logarithm (ln) of both sides gives us:
ln(1/2) = ln(e^(-kt))
Using the property of logarithms, ln(a^b) = b * ln(a), we can rewrite the equation as:
ln(1/2) = -kt * ln(e)
Since ln(e) is equal to 1, the equation simplifies to:
ln(1/2) = -kt
Finally, to solve for k, we divide both sides of the equation by -t:
k = -ln(1/2)/t
Now, we need to simplify the right side of the equation. The natural logarithm of 1/2 is the same as the natural logarithm of 2 taken with the negative sign, so:
k = -ln(2)/t
And there you have it! We have shown that if a radioactive substance has a half-life of T, then the corresponding constant k in the exponential decay function is given by k = -ln(2)/T.