What is half-life? Derive the formula for the number of un-decayed atoms. Draw a graph for decaying atoms with time.

The half-life of a substance is the time required for half of the atoms of a radioactive substance to decay. It is a characteristic property of the substance and is used to measure the rate of radioactive decay.

Deriving the formula for the number of undecayed atoms:

1. Radioactive decay is exponential: The number of undecayed atoms N(t) at any time $t$ follows an exponential decay law, which can be expressed as:

   N(t)=N0e^−λt

   where:

   • $(t)$ is the number of undecayed atoms at time $t$.
   • N0 is the initial number of atoms at time t=0.
   • λ is the decay constant (which is related to the half-life).
   • t is the time that has passed.

2. Relating decay constant to half-life: The half-life t_1/2 is the time it takes for the number of atoms to reduce by half. Therefore, at t=t_1/2​, we have:

   N(t_1/2)=N0/2​​

   Substituting into the exponential decay equation:

   N0/2=N0e^−λt_1/2​

   Dividing both sides by N0N_0N0​ and solving for $\lambda$:

   1/2=e^−λt_1/2            ​

   Taking the natural logarithm (ln) of both sides:

   ln⁡(1/2)=−λt_1/2​

   Since ln⁡(1/2)=−ln⁡(2)\ln\left(\frac{1}{2}\right) = -\ln(2)ln(21​)=−ln(2), we get:

   −ln⁡(2)=−λt_1/2​

   Therefore, solving for λ:

   λ=ln⁡(2)/t_1/2​

3. Final formula: Now that we have the decay constant λ, we can substitute it back into the original exponential decay equation:

   N(t)=N0e−ln⁡(2)/t_1/2.t

   This is the general formula for the number of undecayed atoms at any time $t$, where t_1/2 is the half-life of the substance.