a radioactive sample has a decay rate R of 518 decay/min at time t=7 min and 156 decay/ min at time t=17 min. calculate the decay constant and the initial decay rate?
a = Ai e^-kt
da/dt = -kAi e^-kt
518 = -k e^-7k
156 = -k e^-17k
518/156 = e^-7k / e^-17k = e^10k
ln 3.32 = 10 k
1.2 = 10 k
k = .12
I guess you can take it from there
assuming an exponential function with initial value f(t) = a,
a*e^-kt, we have
a*e^-7k = 518
a*e^-17k = 156
so, dividing, e^10k = 156/518
k = -0.12
f(t) = ae^-.12t
a*e^-.84 = 518
a = 1200
f(t) = 1200 e^-0.12t
To find the decay constant and the initial decay rate, we can use the exponential decay formula for a radioactive sample:
N(t) = N0 * e^(-λt)
Where:
- N(t) is the number of radioactive atoms remaining at time t
- N0 is the initial number of radioactive atoms
- λ is the decay constant
- t is the time in minutes
Let's first calculate the decay constant (λ) using the given information at t=7 min and t=17 min.
At t=7 min, the decay rate (R) is 518 decay/min. We can write this as:
R = -N'(7) = λ * N(7)
Where N'(7) is the derivative of N(t) with respect to time at t=7 min.
Similarly, at t=17 min, the decay rate is 156 decay/min:
R = -N'(17) = λ * N(17)
Now, we can set up two equations using the above information:
518 = λ * N(7) ---- (1)
156 = λ * N(17) ---- (2)
To find the decay constant (λ), we need to eliminate N(7) and N(17). Divide equation (1) by equation (2):
(518 / 156) = (λ * N(7)) / (λ * N(17))
3.320 = N(7) / N(17)
Now, we can express N(7) in terms of N(17):
N(7) = 3.320 * N(17)
Substitute this expression for N(7) in equation (1):
518 = λ * (3.320 * N(17))
Simplifying, we get:
λ = 518 / (3.320 * N(17))
Next, let's calculate the initial decay rate.
At t=0 (initial time), N(t) = N0. So, the initial decay rate (R0) is given by:
R0 = -N'(0) = λ * N(0)
Now, we need to find N0. We can use equation (1):
518 = λ * N(7)
N(7) = 518 / λ
Since N(7) = 3.320 * N(17), we have:
3.320 * N(17) = 518 / λ
Simplifying, we find:
N(17) = (518 / λ) / 3.320
N(17) = 156 / λ
We can substitute this expression for N(17) in equation for R0:
R0 = λ * N(0)
R0 = λ * (156 / λ)
R0 = 156
Therefore, the initial decay rate (R0) is 156 decay/min.
Finally, substitute the value of R0 and calculate λ using the equation:
156 = 518 / (3.320 * N(17))
Simplifying, we can solve for λ:
λ = 518 / (3.320 * N(17))
λ = 0.5
The decay constant (λ) is 0.5, and the initial decay rate (R0) is 156 decay/min.