The count rate of nuclear radiation emitted from a radioactive sample containing ¹²⁸ I varies with time in the following manner. Time t (minute) 0 25 50 75 100 Count rate R ( 10⁹ s ⁻¹ ) 30 16 8.0 3.8 2.0 Determine the decay constant from the slope of the best straight line through the points.

Step-by-step solution

The radioactive decay equation is given by: R = R₀ e^ - t Taking the natural logarithm on both sides: R = R₀ - t This represents the equation of a straight line y = mx + c , where y = R , x = t , and the slope is m = - . Using the data points at t₁ = 25 min and t₂ = 100 min : R₁ = 16 10^9 s ⁻¹ R₂ = 2.0 10^9 s ⁻¹ The slope m is given by: m = R₂ - R₁ t₂ - t₁ = (2.0) - (16) 100 - 25 m = ( 2.0 16 ) 75 = ( 1 8 ) 75 = -3 2 75 = - 2 25 Since = -m : = 2 25 0.693 25 = 0.0277 min ⁻¹ 0.028 min ⁻¹ This is consistent with the o