In the laboratory, a more useful measurement is the decay rate R, usually measured in di...

In the laboratory, a more useful measurement is the decay rate R, usually measured in disintegrations per second, counts per minute, etc. Thus, the decay rate is defined as R = -dN/dt. Using the equation dN/dt = -λN, it is easily seen that R = λN. Furthermore, differentiating the solution with respect to t reveals that

in which is the decay rate at t = 0. That is, because R and N are proportional, they both decrease with time according to the same exponential law. Use this idea to help solve Exercise.

Jim, working with a sample of in the lab, measures the decay rate at the end of each day.

Like any modern scientist, Jim wants to use all of the data instead of only two points to estimate the constants and λ in equation (2.39). He will use the technique of regression to do so. Use the first method in the following list that your technology makes available to you to estimate λ (and at the same time). Use this estimate to approximate the half-life of .

(a) Some modern calculators and the spreadsheet Excel can do an exponential regression to directly estimate and λ.

(b) Taking the natural logarithm of both sides of equation (2.39) produces the result

Now In R is a linear function of t. Most calculators, numerical software such as MATLAB®, and computer algebra systems such as Mathematica and Maple will do a linear regression, enabling you to estimate In and λ (e.g., use the MATLAB® command polyfit).

(c) If all else fails, plotting the natural logarithm of the decay rates versus the time will produce a curve that is almost linear. Draw the straight line that in your estimation provides the best fit. The slope of this line provides an estimate of -λ.