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QPages 442-444: Q43: HC Verma: Concepts of Physics - Volume 2 The Nucleus Pages 442-444: Q43 Solution: A sample contains a mixture of and isotopes each having an…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students often mistakenly assume the initial slope of the total activity plot corresponds to the longer-lived isotope, but at large times, the shorter-lived isotope decays away, leaving only the longer-lived one to define the linear asymptote. Always identify the isotope with the larger half-life as the one responsible for the tail of the decay curve.

A sample contains a mixture of 108Ag^{108}\text{Ag} and 110Ag^{110}\text{Ag} isotopes each having an activity of 8.0×1088.0 \times 10^8 disintegration per second. 110Ag^{110}\text{Ag} is known to have larger half-life than 108Ag^{108}\text{Ag}. The activity A is measured as a function of time and the following data are obtained. a) Plot ln⁡A/A0)\ln A/A_0) versus time. b) See that for large values of time, the plot is nearly linear. Deduce the half-life of 110Ag^{110}\text{Ag} from this portion of the plot. c) Use the half-life of 110Ag^{110}\text{Ag} to calculate the activity corresponding to 108Ag^{108}\text{Ag} in the first 50 s. d) Plot ln⁡A/A0)\ln A/A_0) versus time for 108Ag^{108}\text{Ag} for the first 50 s. A/A_0)versustimeforversus time for^{108}\text{Ag}forthefirst50s.e)Findthehalf−lifeoffor the first 50 s. e) Find the half-life of^{108}\text{Ag}$.

Step-by-Step Verified Solution

**Answer:** (a) Plot ln(A/A_) vs t (b) T_/_(___Ag) from linear portion (c) Activity of ___Ag in first 50 s (d) Plot for ___Ag (e) T_/_(___Ag). **Step-by-Step Explanation:** (a) Plot ln(A/A_) against time from the data. (b) At large times, the shorter-lived isotope (___Ag) has decayed away, leaving only ___Ag. The slope of the linear portion gives ____, and T_/_ = 0.693/____. (c) Subtract the contribution of ___Ag from the total activity to get the activity of ___Ag. (d) Plot ln(A___/A____) vs time for the first 50 s. (e) Slope of this plot gives ____ and hence T_/_[citation:2].

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