A sample contains a mixture of ¹¹⁰ Ag and ¹⁰⁸ Ag isotopes, each having an initial activity of 8.0 10⁸ disintegrations per second. ¹⁰⁸ Ag is known to have a larger half-life than ¹¹⁰ Ag . The total activity A is measured as a function of time. Find the half-life of ¹¹⁰ Ag . Time (s) Activity A ( 10⁸ disintegrations s ⁻¹ ) Time (s) Activity A ( 10⁸ disintegrations s ⁻¹ ) 20 11.799 200 3.0828 40 9.1680 300 1.8899 60 7.4

Step-by-step solution

The total activity of the sample at any time t is given by the sum of the activities of the two isotopes: A(t) = A₁(t) + A₂(t) = A₀₁ e^ - ₁ t + A₀₂ e^ - ₂ t Given the initial activities are equal: A₀₁ = A₀₂ = 8.0 10^8 s ⁻¹ . Since ¹⁰⁸ Ag has a larger half-life than ¹¹⁰ Ag , its decay constant is smaller ( ₂ Using the activity values at t = 400 s and t = 500 s to find ₂ : A(500) A(400) = 0.7212 1.1671 0.6179 A₀₂ e^ -500 ₂ A₀₂ e^ -400 ₂ = e^ -100 ₂ = 0.6179 -100 ₂ = (0.6179) -0.4814 ₂ 0.004814 s ⁻¹ Now, we calculate