The half-life and decay constant are inversely proportional; a higher decay constant means a shorter half-life, and a lower decay constant corresponds to a longer half-life.
The relationship between half-life and decay constant is that they are inversely proportional. The half-life (t₁/₂) is the time required for the quantity of a substance undergoing decay to decrease to half its initial value. On the other hand, the decay constant (k) is the proportionality constant in the radioactive decay equation, representing the probability of decay per unit time.
In mathematical terms, the relationship is expressed as t₁/₂ = 0.693/k, where 0.693 is a constant resulting from the natural logarithm of 2. This indicates that a higher decay constant will result in a shorter half-life, and conversely, a lower decay constant means a longer half-life. This makes sense as the decay constant reflects how quickly a substance undergoes radioactive decay, and thus a larger value indicates more rapid decay, leading to a shorter time for half the substance to decay.
to answer the initial question: a. Half-life and decay constant are inversely proportional.