How do you set up this problem?
The decay constant for the radioactive element cesium 137 is .023 when time is measured in years. Find its half-life.
e^(-0.023t) = 0.5
Take natural log of both sides.
-0.023 t = -0.6932
t = 30 years
If you are using the standard decay function equation
amount = initial amount e^-kt, where k = .023 then
.5 = e^ -.023t
-.023t = ln .5
t = ln .5/-.023 = 30.14 years
check: pick 500 grams
amount = 500 e^(-.023(30.14))
= 500 e^ -.69314..
= 250
YUP!!!
To set up this problem, we need to use the equation for exponential decay:
N(t) = N₀ * e^(-λt),
where:
- N(t) is the quantity of the radioactive element remaining at time t.
- N₀ is the initial quantity of the radioactive element.
- λ is the decay constant.
- t is the time elapsed.
In this case, we know the decay constant (λ = 0.023) for cesium 137. We want to find the half-life, which is the time it takes for the quantity to decrease by half (N(t) = 0.5 * N₀).
To solve for the half-life, we can substitute N(t) = 0.5 * N₀ into the exponential decay equation:
0.5 * N₀ = N₀ * e^(-λt).
Now we can divide both sides of the equation by N₀:
0.5 = e^(-λt).
To isolate the time, we can take the natural logarithm (ln) of both sides:
ln(0.5) = -λt.
Finally, we solve for t by dividing both sides by -λ:
t = ln(0.5) / -λ.
Using the given value for λ (0.023), we can substitute it into the equation above to find the half-life of cesium 137.