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. For large values of time, the plot of (A/A₀) versus time is nearly linear. Deduce the half-life of ¹⁰⁸ Ag from this portion of the plot. Time (s) Activity A ( 10⁸ disintegration

Step-by-step solution

The total activity of the mixture is given by A = A₁₀ e^ - ₁ t + A₂₀ e^ - ₂ t , where ₁ and ₂ are the decay constants of ¹¹⁰ Ag and ¹⁰⁸ Ag respectively. Since ¹⁰⁸ Ag has a larger half-life, its decay constant ₂ is smaller than ₁ . For large values of time t , the term e^ - ₁ t decays much faster and becomes negligible compared to e^ - ₂ t . Thus, for large t , the total activity is approximately A A₂₀ e^ - ₂ t . Using the data for large times, at t₁ = 400 s , A₁ = 1.1671 10^8 s ⁻¹ and at t₂ = 500 s , A₂ = 0.7212 10