Answer:
The half life of the radioisotope is 12 hours.
Explanation:
Initial mass of the radioisotope = x
Final mass of the radioisotope = [tex]\frac{1}{16}\times x[/tex] = 0.0625x
Half life of the radioisotope =[tex]t_{\frac{1}{2}}[/tex]
Age of the radioisotope = t = 48 hours
Formula used :
[tex]N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}[/tex]
where,
[tex]N_o[/tex] = initial mass of isotope
N = mass of the parent isotope left after the time, (t)
[tex]t_{\frac{1}{2}}[/tex] = half life of the isotope
[tex]\lambda[/tex] = rate constant
[tex]N=N_o\times e^{-(\frac{0.693}{t_{1/2}})\times t}[/tex]
Now put all the given values in this formula, we get
[tex]0.0625x=x\times e^{-(\frac{0.693}{t_{1/2}})\times 48 h}[/tex]
[tex]\ln(0.0625) \times \frac{1}{(-0.693\times 48 h)}=\frac{1}{t_{1/2}}[/tex]
[tex]t_{1/2} = 11.99 hours = 12 hours[/tex]
The half life of the radioisotope is 12 hours.