Answer:
32.0 min
Step-by-step explanation:
The equation that describes the decay of a radioactive isotope is:
[tex]m(t)=m_0 e^{-\lambda t}[/tex] (1)
where:
[tex]m_0[/tex] is the mass of the element at time t = 0
[tex]m(t)[/tex] is the mass of the element left at time t
[tex]\lambda[/tex] is the decay constant
The decay constant is related to the half-life of the element by
[tex]\lambda=\frac{ln 2}{t_{1/2}}[/tex]
where
[tex]t_{1/2}[/tex] is the half-life
For element X, we have
[tex]t_{1/2}=5 min[/tex]
So the decay constant is
[tex]\lambda=\frac{ln 2}{5}=0.139 min^{-1}[/tex]
We also know that for element X:
[tex]m_0 = 340 g[/tex] is the initial mass
[tex]m(t)=4 g[/tex] is the final mass
So, from eq(1) we can now find the time:
[tex]t=-\frac{ln(\frac{m(t)}{m_0})}{\lambda}=-\frac{ln(4/340)}{0.139}=32.0 min[/tex]
