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.
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Step-by-Step Solution
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)=N0, we find that N0=eC, so the constant eC is equal to N0. 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)=N0/2 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 eln(x)=x, we can simplify the expression to a more common form involving base 2.