The half-life of radium-226 is 1,600 years and the half-life of carbon-14 is 5,730 years. In one to sentences, mathematically prove that the decay constant for radium-226 is larger than the decay constant for carbon-14.
The decay constant (λ) is directly proportional to the inverse of the half-life (t1/2), i.e., λ ∝ 1/t1/2. Thus, since the half-life of radium-226 is shorter than that of carbon-14, the decay constant for radium-226 must be larger than that of carbon-14.
To mathematically prove that the decay constant for radium-226 is larger than the decay constant for carbon-14, we can compare their respective equations for exponential decay, which are given as N(t) = N₀ * exp(-λt), where N(t) is the quantity remaining at time t, N₀ is the initial quantity, λ is the decay constant, and t is the time elapsed. By comparing the exponents, it can be observed that the decay constant is inversely proportional to the half-life, so a smaller half-life implies a larger decay constant.
