How does the half-life t1/2 relate to the decay of the number of undecayed nuclei N?

• A. N = N0 e^(−λ t)
• B. N = N0 (1/3)^(t/t1/3)
• C. N = N0 / (1 + t/t1/2)
• D. N = N0 (1/2)^(t/t1/2)

Answer: (D)

Radium decay is exponential in time, with the number of undecayed nuclei dropping by a constant factor in each interval equal to the half-life. After each interval t1/2, the remaining amount is halved, so the amount left after time t is the original amount times the factor (1/2) raised to the number of half-lives that have passed, which is t divided by t1/2. That gives N = N0 (1/2)^(t/t1/2).

This form makes the meaning of the half-life clear: when t = t1/2, N = N0/2; when t = 2 t1/2, N = N0/4, and so on. It’s also worth noting this is equivalent to the common decay law N = N0 e^(−λ t) with λ = ln 2 / t1/2, but using (1/2)^(t/t1/2) directly encodes the idea of halving every half-life. The other algebraic forms don’t reflect that steady halving behavior.