Calculating half life (Decay Constant)

[franwilder]

Homework Statement

The activity of a radioactive nuclide drops 78.5% of it's initial value in 2000 years.

Homework Equations

I know I need to use the decay constant to work out my answer so I have re-arranged the equation as illustrated in my picture, the only problem is I cannot work out the answer and am unsure of which units to use and where they go

The Attempt at a Solution

 

[HallsofIvy]

No, you don't need a "decay constant", you just need the definition of "half life". If T is the half life, in years, then after t years the amount left will be [itex]A(1/2)^{t/T}= A2^{-t/T}[/itex] where A is the initial amount at time t= 0.

"The activity of a radioactive nuclide drops 78.5% of it's initial value in 2000 years."

So [itex]A2^{-2000/T}= .785A[/itex]. Can you solve that for T? You will need to take a logarithm of both sides.

 

[daveb]

If the value drops 78.5% then there will only be 21.5% left (just a tad over 2 half lives).

 

[Chestermiller]

Homework Statement

The activity of a radioactive nuclide drops 78.5% of it's initial value in 2000 years.

Homework Equations

I know I need to use the decay constant to work out my answer so I have re-arranged the equation as illustrated in my picture, the only problem is I cannot work out the answer and am unsure of which units to use and where they go

The Attempt at a Solution

The units of the decay constant are reciprocal years, since the ln of the ratio of the N's is dimensionless. So you first solve for lambda, and then use your equation again to calculate the value of t that makes the ratio of N's equal to 1/2. This is the half life.

 

[Nickie]

Calculating the decay constant (also known as the half-life) is an important aspect of studying radioactive materials. In this case, we are given the information that the activity of a radioactive nuclide drops 78.5% of its initial value in 2000 years. This means that after 2000 years, only 21.5% of the original activity remains.

To calculate the decay constant, we can use the equation:

N(t) = N0 * e^(-λt)

Where:
N(t) = the amount of radioactive material remaining after time t
N0 = the initial amount of radioactive material
λ = the decay constant
t = time

We can rearrange this equation to solve for the decay constant, λ:

λ = -ln(0.785) / 2000

λ = 0.000109 / year

This means that the decay constant for this particular nuclide is approximately 0.000109 per year. This value represents the rate at which the nuclide decays. We can also use this value to calculate the half-life, which is the amount of time it takes for half of the original material to decay.

To calculate the half-life, we can use the equation:

t1/2 = ln(2) / λ

Substituting in our value for λ, we get:

t1/2 = ln(2) / 0.000109

t1/2 = 6366 years

This means that the half-life of this nuclide is approximately 6366 years. This information can be useful in understanding the behavior of radioactive materials and their potential impact on the environment and human health.