Question

Element X is a radioactive isotope such that every 13 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 90 grams, how long would it be until the mass of the sample reached 62 grams, to the nearest tenth of a year?

Answer 1

Answer:

The mass of Element X will reach 62 grams in 7.0 years.

Step-by-step explanation:

We can find the time of decay of Element X by using the exponential decay equation:

$\frac{dm}{dt} = -\lambda t$

The solution of the above equation is:

$m_{(t)} = m_{0}e^{-\lambda t}$     (1)

Where:  

t: is the time =?

λ: is the decay constant                                                                                        

$m_{(t)}$: is the mass at time t = 62 grams

$m_{0}$: is the initial mass = 90 grams

First, we need to calculate λ

$\lambda = \frac{ln(2)}{t_{1/2}}$   (2)    

Where $t_{1/2}$ is the half-life = 13 years

By entering equation (2) into (1) and solving for "t" we have:

$\frac{m_{(t)}}{m_{0}} = e^{-\frac{ln(2)}{t_{1/2}}*t}$    

$ln(\frac{m_{(t)}}{m_{0}}) = -\frac{ln(2)}{t_{1/2}}*t$      

$t = -ln(\frac{m_{(t)}}{m_{0}})(\frac{t_{1/2}}{ln(2)}) = -ln(\frac{62}{90})(\frac{13}{ln(2)}) = 7.0 y$  

Therefore, the mass of Element X will reach 62 grams in 7.0 years.

I hope it helps you!