The formula for half-life depends on the type of decay process being considered, but the most common scenario involves exponential decay, such as in radioactive decay. The general formula is:

Half-life formula:

$T_{1/2} = \frac{\ln(2)}{\lambda}$

Where:

• $T_{1/2}$ = the half-life (time required for half of the substance to decay)
• $\lambda$ = the decay constant, which describes the rate of decay
• $\ln(2) \approx 0.693$

Alternate form (exponential decay):

The amount of substance remaining, $N(t)$, at time $t$ can also be expressed as: $N(t) = N_0 \cdot e^{-\lambda t}$

Where:

• $N_0$ = the initial quantity of the substance
• $e$ = the base of the natural logarithm ($\approx 2.718$)
• $t$ = time elapsed
• $\lambda$ = decay constant

Half-life derived from this form:

At half-life $t = T_{1/2}$, the remaining substance $N(t) = \frac{N_0}{2}$. Substituting this into the exponential decay equation yields the half-life formula.

Would you like detailed derivations or examples?

Related Questions:

1. How is the decay constant $\lambda$ calculated experimentally?
2. How does the half-life formula apply to biological systems, like drug metabolism?
3. Can you provide an example of using this formula to solve a problem?
4. What is the relationship between half-life and the mean lifetime of a substance?
5. How do factors like temperature or pressure affect radioactive half-life?

Tip:

For problems involving half-life, memorizing the relationship $T_{1/2} = 0.693 / \lambda$ can save you significant time in calculations!