A radioactive substance decays according to the equation dN/dt = -kN. If the initial amount is N₀ and half-life is T, express N(t) in terms of N₀, T, and t.

Step 1: Solve the differential equation

The given differential equation describes the rate of decay of a radioactive substance. To solve for $N(t)$, we separate the variables by moving all terms involving $N$ to one side and all terms involving $t$ to the other side.

Step 2: Integrate both sides

Integrating both sides of the separated equation yields the natural logarithm of $N$ on the left and $-kt$ plus an integration constant $C$ on the right. This is a standard integration step for this type of differential equation.

Step 3: Solve for N(t) using initial condition

We can rewrite the equation as $N(t) = e^{-kt+C}$. Using the initial condition $N(0) = N_0$, we find that $N_0 = e^C$, so the constant $e^C$ is equal to $N_0$. This gives us the general form of exponential decay.

Step 4: Use half-life to find k

The half-life $T$ is the time it takes for the substance to decay to half its initial amount. We substitute $N(T) = N_0/2$ into the decay equation to solve for the decay constant $k$ in terms of $T$.

Step 5: Substitute k back into N(t)

Finally, we substitute the expression for $k$ back into the equation for $N(t)$. Using the logarithm property $e^{\ln(x)} = x$, we can simplify the expression to a more common form involving base 2.

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