A radioactive substance decays at a rate proportional to the...

A radioactive substance decays at a rate proportional to the quantity, $Q$, present at the time, $t$. The constant of proportionality is $k$. (a) Write a differential equation satisfied by $Q$. (b) Find the half-life as a function of $k$. (c) Is the half-life an increasing or decreasing function of $k ?$

A radioactive substance decays at a rate proportional to the quantity, $Q$, present at the time, $t$. The constant of proportionality is $k$.
(a) Write a differential equation satisfied by $Q$.
(b) Find the half-life as a function of $k$.
(c) Is the half-life an increasing or decreasing function of $k ?$

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Key Concepts

Half-Life

Half-life is a measure of the time required for a substance undergoing exponential decay to reduce to half of its initial amount. It can be derived from the decay constant by solving the equation that sets the remaining quantity to half of the original, typically resulting in the expression T½ = ln(2)/k. This demonstrates how the half-life is inversely related to the decay constant.

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They are used to model real-world phenomena where a quantity changes continuously over time. In the context of exponential decay, the differential equation expresses that the rate of change of a quantity is directly proportional to the current amount of that quantity.

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. This results in a decay model where the solution to the differential equation is an exponential function that models how the quantity diminishes over time.

Decay Constant

The decay constant is a parameter that quantifies the rate at which a substance decays in an exponential decay process. It appears in the differential equation and determines how fast the substance loses its quantity relative to time.

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