What is lambda in radioactive decay in terms of half-life?

What is lambda in radioactive decay in terms of half-life?

• A. λ = 0.693 / t
• B. λ = 0.693 / half-life
• C. λ = ln(2) × half-life
• D. λ = half-life / 0.693

Answer: (B)

The decay constant represents the rate at which nuclei decay per unit time, and it connects time to the exponential decay law N(t) = N0 e^{-λ t}. To find how long it takes for a sample to drop to half its initial amount—the half-life t1/2—you set N(t1/2) = N0/2. This gives e^{-λ t1/2} = 1/2, so -λ t1/2 = ln(1/2) = -ln 2, and therefore λ t1/2 = ln 2. Solving for λ gives λ = ln 2 / t1/2. Since ln 2 is approximately 0.693, this becomes λ = 0.693 / t1/2. The units work out as 1/time, as expected for a decay constant. Other forms either mix up the factors or produce incorrect units. For example, multiplying by the half-life would give a quantity with time units, which isn’t correct for a rate constant. Writing λ as 0.693 divided by a general elapsed time t would only be correct if that t happened to be the half-life, not in general. So the standard and correct relation is λ = 0.693 / t1/2.

The decay constant represents the rate at which nuclei decay per unit time, and it connects time to the exponential decay law N(t) = N0 e^{-λ t}. To find how long it takes for a sample to drop to half its initial amount—the half-life t1/2—you set N(t1/2) = N0/2. This gives e^{-λ t1/2} = 1/2, so -λ t1/2 = ln(1/2) = -ln 2, and therefore λ t1/2 = ln 2. Solving for λ gives λ = ln 2 / t1/2. Since ln 2 is approximately 0.693, this becomes λ = 0.693 / t1/2. The units work out as 1/time, as expected for a decay constant.

Other forms either mix up the factors or produce incorrect units. For example, multiplying by the half-life would give a quantity with time units, which isn’t correct for a rate constant. Writing λ as 0.693 divided by a general elapsed time t would only be correct if that t happened to be the half-life, not in general. So the standard and correct relation is λ = 0.693 / t1/2.