09-20-2019, 01:57 PM

The Journal of Nuclear Medicine {vol 19, num 11} pgs. 1270-1271

Calculation of Radioactive Decay with a Pocket Calculator

JAMES S. ROBERTSON

Mayo Clinic

Rochester, Minnesota

"Radioactive decay is customarily expressed by the equation

(1) A = A₀ e^(-λt)

where t = time; λ = decay constant in t⁻¹ units; A = activity, usually μCi or mCi; A₀ activity at t = 0; and e = 2.718 …, the natural logarithm base.

However, the decay parameter most readily available is not λ, but the half-life, T. Therefore, the relationship λT = ln(2) = 0.693 … is invoked and the decay equation

becomes

(2) A = A₀ e^(-0.693t/T)

From this it would appear that the way to calculate A, given A₀, t and T, is first to determine x = -0.693 t/T and then to obtain A/A₀ from e^(x). However, a simplifying feature that is overlooked in this procedure is that e^(-0.693) = …½, and the decay equation may therefore be expressed as

(3) A = A₀ ½^(t/T)

Thus, if a pocket calculator having a y^x function is used, A/A₀ may be calculated simply by entering 0.5 as y, calculating t/T as x, and calling y^x. This saves several steps when compared with using Eq. 2. For negative values of t, the same procedure works, but alternatively the number 2 may be entered as y and the absolute value of t used.

These procedures are similar to the slide-rule method of setting T against 0.5 or 2.0 on a log-log scale and reading A/A₀ against t, a method now in danger of being forgotten with the proliferation of inexpensive pocket calculators."

'Ahh, memories'

BEST!

SlideRule

Calculation of Radioactive Decay with a Pocket Calculator

JAMES S. ROBERTSON

Mayo Clinic

Rochester, Minnesota

"Radioactive decay is customarily expressed by the equation

(1) A = A₀ e^(-λt)

where t = time; λ = decay constant in t⁻¹ units; A = activity, usually μCi or mCi; A₀ activity at t = 0; and e = 2.718 …, the natural logarithm base.

However, the decay parameter most readily available is not λ, but the half-life, T. Therefore, the relationship λT = ln(2) = 0.693 … is invoked and the decay equation

becomes

(2) A = A₀ e^(-0.693t/T)

From this it would appear that the way to calculate A, given A₀, t and T, is first to determine x = -0.693 t/T and then to obtain A/A₀ from e^(x). However, a simplifying feature that is overlooked in this procedure is that e^(-0.693) = …½, and the decay equation may therefore be expressed as

(3) A = A₀ ½^(t/T)

Thus, if a pocket calculator having a y^x function is used, A/A₀ may be calculated simply by entering 0.5 as y, calculating t/T as x, and calling y^x. This saves several steps when compared with using Eq. 2. For negative values of t, the same procedure works, but alternatively the number 2 may be entered as y and the absolute value of t used.

These procedures are similar to the slide-rule method of setting T against 0.5 or 2.0 on a log-log scale and reading A/A₀ against t, a method now in danger of being forgotten with the proliferation of inexpensive pocket calculators."

'Ahh, memories'

BEST!

SlideRule