I need help very soon

Question

The half-life of radium-226 is 1,600 years and the half-life of carbon-14 is 5,730 years. Mathematically prove that the decay constant is larger than the decay constant for carbon-14.

Answer 1

Lets' derive it from the start.

For exponentially decaying quantities (like the number of atoms in a sample of some radioactive isotope), the following differential equation applies

dy/dt= - ky

where k is some constant and y is the amount of sample at time t. This can be solved by simple separation of variables, and you end up with

dy/dt= - ky

dy/y=-kdt

Ln(y(t))=-kt+c

y(t)=exp(-kt+c)

y(t)=exp(c)*exp(-kt)

y(0)=exp(c)=y₀

y(t)=y₀exp(-k*t)

where y₀ is the initial quantity of atoms in your sample. This equation is general, and can be applied to any decaying isotope; all that changes are the amount in your initial sample and the decay constant k (k being an intrinsic quantity of that isotope; different isotopes have different ks, but two samples of the same isotope will have the same k).

The half-life of an element is defined as the time required for half of the sample to decay. So, at time t=t_1/2, we have

y(t_1/2)=y₀exp(-k*t_1/2)=1/2 y₀

We can solve this for k, allowing us to express k in terms of the half-life:

y₀exp(-k*t_1/2)=1/2 y₀

exp(-k*t_1/2)=1/2

-k*t_1/2=Ln(1/2)

-k*t_1/2= - Ln(2)

k*t_1/2=Ln(2)

k=Ln(2) / t_1/2

So, that's how k depends on the half-life. Remember, this expression is the same for the decay of any isotope of any element; all that changes is the half-life.

So, now for the punchline. Since k depends inversely on the half-life, increasing the half-life reduces the decay constant, since you're dividing by a larger number! That's really all you need, but you could be explicit and substitute the numbers.

k_rad-226=Ln(2)/1600=0.000433217 yr^-1

k_car-14 =Ln(2)/5730=0.000120968 yr^-1