The Age of Earth - Biology

1. Fundamental Concepts

Definition: The Earth is estimated to be approximately 4.54 billion years old, based on various scientific methods.
Radiometric Dating: A method used to determine the age of rocks and fossils by measuring the decay of radioactive isotopes.
Fossil Record: Provides evidence for the sequence of life forms that have existed over time, supporting the theory of evolution.

2. Key Concepts

Half-Life Calculation: $t_{\text{{half}}} = \frac{\ln(2)}{\lambda}$

Carbon Dating: $N_t = N_0 e^{-\lambda t}$

Application: Used in geology and archaeology to date materials up to about 50,000 years old

3. Examples

Example 1 (Basic)

Problem: Calculate the half-life of a radioactive isotope with a decay constant of $$\lambda = 1.21 \times 10^{-4}$$ per year.

Step-by-Step Solution:

Use the formula for half-life: $t_{\text{{half}}} = \frac{\ln(2)}{\lambda}$
Substitute the given decay constant: $t_{\text{{half}}} = \frac{\ln(2)}{1.21 \times 10^{-4}}$
Calculate the value: $t_{\text{{half}}} \approx 5730$ years

Validation: Using the calculated half-life, verify the consistency with known values for carbon-14 dating.

Example 2 (Intermediate)

Problem: If a sample originally contained 100 grams of a radioactive isotope and now contains 25 grams, how long has it been since the sample was formed if the half-life is 5730 years?

Step-by-Step Solution:

Identify the initial amount ($N_0 = 100$ grams) and the remaining amount ($N_t = 25$ grams).
Use the decay formula: $N_t = N_0 e^{-\lambda t}$
Rearrange to solve for time: $t = \frac{\ln(\frac{N_0}{N_t})}{\lambda}$
Substitute the values: $t = \frac{\ln(\frac{100}{25})}{\lambda}$
Given $\lambda = \frac{\ln(2)}{5730}$, calculate $t$: $t \approx 11460$ years

Validation: Substitute back into the decay equation to ensure the calculated time matches the observed data.

4. Problem-Solving Techniques

Visual Strategy: Use timelines to visualize the progression of time and decay.
Error-Proofing: Double-check calculations using a calculator and recheck units.
Concept Reinforcement: Relate the concept of half-life to real-world examples like carbon dating in archaeological findings.