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Answer :
The half-life of Radium is 1600 years, so it takes about 3700 years until only 0.2 g of radium is left. The correct option is (c) 3700 yr.
Given,
Mass of radium, m₀ = 2.0 g
Half-life of radium, t₁/₂ = 1600 years
Let the mass of radium left after time t be m(t).
Using radioactive decay law, m(t) = [tex]m₀ (1/2)^(t/t₁/₂)[/tex]
On substituting the given values, m(t) = [tex]2.0 (1/2)^(t/1600)[/tex]
When the mass of radium left is 0.2 g, m(t) = 0.2
Hence,
0.2 = 2.0 [tex](1/2)^(t/1600)[/tex]
On dividing both sides by 2.0, 0.1 = [tex](1/2)^(t/1600)[/tex]
On taking logarithms on both sides, log(0.1) = [tex]log[(1/2)^(t/1600)][/tex]⇒ [tex]log(0.1) = (t/1600) log(1/2)[/tex]⇒[tex]t = (1600 / log 2) log (1 / 0.1)[/tex]≈ 3700 years
Therefore, it takes about 3700 years until only 0.2 g of radium is left. Hence, the correct option is (c) 3700 yr.
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