Question

Rachel has an unknown sample of a radioisotope listed in the table. Using a special technique, she is able to measure the mass of just the unknown isotope as 104. 8 kg at 12:02:00 P. M. At 4:11:00 P. M. On the same day, the mass of the unknown radioisotope is 13. 1 kg. Which radioisotope is in the sample? Potassium-42 Nitrogen-13 Barium-139 Radon-220.

Answer 1

The radioactive isotopes of an element are known as radioisotopes.  The radioisotope sample that is given is Barium-139.

What is radioisotope?

The radioactive isotopes of an element are known as radioisotopes. Atoms with an unstable mix of neutrons and protons, or excess energy in their nucleus, are also known as neutron-proton atoms.

As it is given that the initial mass of the sample at 12:02:00 PM is,

$m_1=104.8\ kg$

The Mass of the sample at 4:11:00 PM is given as

$m_2 = 13.1 kg$

So here since it is a sample of radioactive isotope so here we will say

$m_2=m_1e^{-\lambda t}$

Now from the above data, we know that the time taken to reduce the mass is

$\rm t = 4\ hours\ 9\ minutes = 4.15\ hours$

Further, from the above equation, we will have

$13.1 =104.8 e^{-\lambda(4.15)}\\\\\lambda(4.15)=2.08\\\\\lambda = 0.50/ hour$

So here half-life is given as

$\rm T_{\frac12}=\dfrac{ln2}{\lambda}\\\\ T_{\frac12} = 1.4\ hours$

Thus, here this isotope must be the one that will have a half-life of 1.4 hours

Therefore, it is nearly Barium-139.

Hence, the radioisotope sample that is given is Barium-139.

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