Answer :
Answer:
There will be left 80.4 of the sample in 500 years
Step-by-step explanation:
Radioactive Decay Model
Suppose N is the size of a population of radioactive atoms at a given time t, No is the size of an initial population of radioactive atoms at time t = 0, and k is the decay constant, then the equation is as follows:
[tex]N=N_o\cdot e^{-kt}[/tex]
The time required for half of the original population of radioactive atoms to decay is called the half-life Th. The relationship between the half-life and the decay constant is:
[tex]\displaystyle T_h=\frac{ln 2}{k}\approx\frac{0.693}{k}[/tex]
We know the half-life of radium-226 is about 1,590 years. We can find the value of k by solving the above equation:
[tex]\displaystyle k=\frac{ln 2}{T_h}[/tex]
[tex]\displaystyle k=\frac{ln 2}{1,590}[/tex]
k=0.000436
The initial mass of the sample is 100 gr, thus the model for the mass at time t is:
[tex]M(t)=100\cdot e^{-0.000436t}[/tex]
When t=500 years:
[tex]M(500)=100\cdot e^{-0.000436*500}[/tex]
[tex]M(500)=100\cdot 0.804[/tex]
M(500)=80.4 gr
There will be left 80.4 of the sample in 500 years