Find the half-life of the radioactive element radium, assuming that its decay constant is $\mathrm { k } = 4.332 \times 10 ^ { - 4 }$, with time measured in years.
A) 1400 years
B) 800 years
C) 1600 years
D) 2308 years

Question

Quizplus · Accepted Answer

The half-life of a radioactive element is defined as the time it takes for half of a given amount of the element to decay. 
Let T be the half-life of radium. 
Then, we know that:
- After one half-life, half of the original amount of radium will remain. 
- After two half-lives, (1/2) x (1/2) = 1/4 of the original amount of radium will remain. 
- After three half-lives, (1/2) x (1/2) x (1/2) = 1/8 of the original amount of radium will remain. 
And so on. 
We can use the decay constant to relate the rate of decay to time. 
The decay constant is defined as the fraction of radium that will decay in one year. 
So, if there are N atoms of radium at any given time, the rate of decay (dN/dt) is given by: 
dN/dt = -kN
where the negative sign indicates that the number of atoms is decreasing with time. 
Integrating both sides gives: 
ln(N/N0) = -kt
where N0 is the initial number of atoms. 
We want to find the time T that it takes for N to decrease to N0/2. 
Substituting N = N0/2 in the equation above, we get: 
ln(1/2) = -kT
Solving for T gives: 
T = ln(2)/k
Substituting the given value of k gives: 
T = ln(2)/4.332 x 10^-4 = 1602 years (rounded to the nearest hundred) 
So, the answer is C) 1600 years

Find the Half-Life of the Radioactive Element Radium, Assuming That k=4.332×10−4\mathrm { k } = 4.332 \times 10 ^ { - 4 }k=4.332×10−4

Find the Half-Life of the Radioactive Element Radium, Assuming That $\mathrm { k } = 4.332 \times 10 ^ { - 4 }$