Write the basic decay law for the number of undecayed nuclei N as a function of time.

Write the basic decay law for the number of undecayed nuclei N as a function of time.

• A. N0 e^(−λ t)
• B. N(t) = N0 e^(−λ t)
• C. N0 e^(λ t)
• D. N = N0 e^(−λ t)

Answer: (B)

N decreases exponentially with time because the rate of decay is proportional to how many undecayed nuclei remain. This comes from the equation dN/dt = −λN, with λ > 0. Solving gives N(t) = N0 e^(−λ t), where N0 is the number at time zero. This form explicitly expresses N as a function of time, shows the correct initial value N(0) = N0, and shows the population decaying as time progresses. The other forms either omit the explicit dependence on time on the left or use a positive exponent, which would imply growth rather than decay.

N decreases exponentially with time because the rate of decay is proportional to how many undecayed nuclei remain. This comes from the equation dN/dt = −λN, with λ > 0. Solving gives N(t) = N0 e^(−λ t), where N0 is the number at time zero. This form explicitly expresses N as a function of time, shows the correct initial value N(0) = N0, and shows the population decaying as time progresses. The other forms either omit the explicit dependence on time on the left or use a positive exponent, which would imply growth rather than decay.