Help I don’t understand

Question

The half life of palladium-100 is 4 days. After 16 days a sample of palladium-100 has been reduced to a mass of 3mg.

what was the initial mass(in mg) of the sample?

what is the mass 6 weeks after the start?

Answer 1

Start by using the equation for half life to determine the decay constant k.

t_1/2 = 0.693/k

Since we know the half life (4 days), we can solve for k. Plugging in 4 for t_1/2, we get k = 0.17325.

We can now use the first order decay equation to find the initial mass.

ln(N/N_O)=-kt

we plug in N=3 for the 3mg sample, 0.17325 for k, and 16 for t since our units are days. We can then solve for N_o to find the initial weight of the sample. Our equation should look like this:

ln(3/N_o) = -(0.17325)(16)

We can then use law of logs to expand the left into

ln(3) - ln(N_o) = -(0.17325)(16)

Solve for N_o knowing that to undo ln, we must use e^x. Simplifying and rearranging the above equation, we get:

ln(N_o) = 2.772 + ln(3)

ln(N_o) = 3.871

N_o = e^3.871

N_o = 47.97mg

The initial value should be about 47.97 mg to end up with 3mg after 16 days.

After 6 weeks, we can use the same decay equation above, but now we know t, the time and are solving for N. Remember that our units are in DAYS, not weeks, so 6 weeks is 42 days. The setup should look like this:

ln(N/47.97) = -(0.17325)(42)

Again, using law of logs, our simplified equation will look like:

ln(N) = -7.277 + ln(47.97)

Simplify:

ln(N) = -3.406

N = e^-3.406 = 0.033 mg

We should be approaching 0 mg, but will never reach it, so this number makes sense.