Understanding Average Life

Average life, often referred to as mean lifetime, is a concept used in various fields such as physics, chemistry, and engineering to describe the characteristic time that an unstable particle or system exists before decaying. This concept is particularly important in the study of radioactive decay and in the analysis of systems that follow exponential decay laws.

Definitions and Formulas

The average life (τ) of a particle or system is defined as the mean time it takes for a large number of identical and independent unstable particles or systems to decay. It is related to the decay constant (λ), which is the probability per unit time that a particle will decay.

The average life can be calculated using the following formula:

$$ \tau = \frac{1}{\lambda} $$

Where:

• τ (tau) is the average life
• λ (lambda) is the decay constant

The decay constant is related to the half-life (T_1/2), which is the time required for half of the unstable particles or systems to decay. The relationship between half-life and decay constant is given by:

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

Where:

• T_1/2 is the half-life
• ln(2) is the natural logarithm of 2 (approximately 0.693)

Exponential Decay Law

The number of particles N(t) remaining at time t is described by the exponential decay law:

$$ N(t) = N_0 e^{-\lambda t} $$

Where:

• N(t) is the number of particles at time t
• N_0 is the initial number of particles
• e is the base of the natural logarithm (approximately 2.71828)

Average Life vs. Half-Life

Here is a table comparing average life and half-life:

Property	Average Life (τ)	Half-Life (T_1/2)
Definition	Mean time for a particle to decay	Time for half the particles to decay
Relationship	τ = 1/λ	T_1/2 = ln(2)/λ
Dependence	Inversely proportional to decay constant	Inversely proportional to decay constant
Calculation	Integral of the survival probability over time	Specific point where N(t) = N_0/2

Examples

Example 1: Calculating Average Life

Suppose a radioactive isotope has a half-life of 10 years. We can calculate its average life as follows:

Given: T_1/2 = 10 years

First, calculate the decay constant:

$$ \lambda = \frac{\ln(2)}{T_{1/2}} = \frac{\ln(2)}{10} \approx 0.0693 \text{ year}^{-1} $$

Now, calculate the average life:

$$ \tau = \frac{1}{\lambda} = \frac{1}{0.0693} \approx 14.4 \text{ years} $$

Example 2: Using Average Life in Exponential Decay

Consider a sample of 1000 unstable particles with an average life of 5 seconds. The number of particles remaining after 10 seconds can be calculated as follows:

Given: τ = 5 seconds, N_0 = 1000, t = 10 seconds

First, calculate the decay constant:

$$ \lambda = \frac{1}{\tau} = \frac{1}{5} = 0.2 \text{ s}^{-1} $$

Now, use the exponential decay law:

$$ N(t) = N_0 e^{-\lambda t} = 1000 e^{-0.2 \times 10} \approx 1000 e^{-2} \approx 1000 \times 0.1353 \approx 135 $$

Therefore, approximately 135 particles remain after 10 seconds.

Conclusion

Understanding average life is crucial for analyzing the behavior of unstable particles and systems undergoing exponential decay. It provides a measure of the expected lifetime of a particle or system and is closely related to the decay constant and half-life. By using the formulas and concepts discussed, one can predict the behavior of decaying systems in various scientific and engineering applications.