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, and the following data are obtained. 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

Step-by-step solution

The total activity A(t) of the mixture is the sum of the activities of the two isotopes: A(t) = A₀₁ e^ - ₁ t + A₀₂ e^ - ₂ t Given that the initial activities are equal, A₀₁ = A₀₂ = A₀ 2 , where A₀ is the total initial activity. A(t) = A₀ 2 (e^ - ₁ t + e^ - ₂ t ) A(t) A₀ = 1 2 (e^ - ₁ t + e^ - ₂ t ) Taking the natural logarithm: ( A A₀ ) = (e^ - ₁ t + e^ - ₂ t ) - 2 The slope of the curve is given by the derivative with respect to time: Slope = d dt ( A A₀ ) = - ₁ e^ - ₁ t - ₂ e^ - ₂ t e^ - ₁ t + e^ - ₂ t At t = 0 ,