Question

What is the half-life of an isotope if 125 g of a 500 g sample of the isotope remains after 3.0 years?
a.) 3.5 years
b.) 4.5 years
c.) 1.5 years
d.) 2.5 years

Answer 1

c.) 1.5 years.
500 divided by half twice = 125, so there are two half lives. In 3.0 years, two half lives would equal 1.5 years.

Answer 2

The half-life of the radioactive isotope is 1.5 years.

The given parameters;

• Original mass of the sample, N₀ = 500 g
• Remaining mass of the sample, N = 125 g
• Time of decay, t = 3 years

The half-life of the radioactive isotope is calculated as;

$N(t) = N_0 (\frac{1}{2} )^{\frac{t}{t_{1/2}}} \\\\125 = 500 (\frac{1}{2} )^{\frac{3}{t_{1/2}}} \\\\\frac{125}{500} = (\frac{1}{2} )^{\frac{3}{t_{1/2}}} \\\\\frac{1}{4} = (\frac{1}{2} )^{\frac{3}{t_{1/2}}}\\\\2^{-2} = 2^{-1(\frac{3}{t_1_/_2})}} \\\\-2 = \frac{-3}{t_{1/2}} \\\\t_{1/2} = \frac{-3}{-2} \\\\t_{1/2} = 1.5 \ years$

Thus, the half-life of the radioactive isotope is 1.5 years.

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