Understanding Activity
* Activity (A): The activity of a radioactive sample is the rate at which its nuclei decay. It's measured in Becquerels (Bq), where 1 Bq represents one decay per second.
* Decay Constant (λ): This is a characteristic property of a particular radioactive isotope and represents the probability of a nucleus decaying per unit time. It's expressed in units of inverse seconds (s⁻¹).
* Number of Radioactive Nuclei (N): This is the total number of radioactive nuclei present in the sample.
Key Formulas
* Activity (A) = λN
* This formula directly relates activity to the decay constant and the number of radioactive nuclei.
* Half-Life (t₁/₂): The half-life of a radioactive isotope is the time it takes for half of the radioactive nuclei to decay. It's related to the decay constant by:
* t₁/₂ = ln(2) / λ
Steps to Calculate Activity
1. Identify the Isotope: You need to know the specific radioactive isotope you're working with. This is essential because each isotope has a unique decay constant (λ).
2. Determine the Number of Nuclei (N):
* If you know the mass of the sample:
* Calculate the number of moles of the isotope (moles = mass / molar mass).
* Convert moles to the number of nuclei (N = moles * Avogadro's number).
* If you know the number of atoms of the isotope:
* You can directly use this number as 'N' in your calculations.
3. Find the Decay Constant (λ):
* You can often look up the decay constant for a given isotope in a reference table or online database.
* Alternatively, if you know the half-life (t₁/₂), you can calculate the decay constant using the formula: λ = ln(2) / t₁/₂
4. Calculate Activity (A):
* Once you have λ and N, use the formula A = λN to calculate the activity of the sample in Becquerels (Bq).
Example
Let's say you have a 100-gram sample of Carbon-14 (¹⁴C), which has a half-life of 5730 years.
1. Identify the isotope: ¹⁴C
2. Determine the number of nuclei (N):
* Moles of ¹⁴C = 100 g / 14 g/mol ≈ 7.14 mol
* N = 7.14 mol * 6.022 x 10²³ nuclei/mol ≈ 4.3 x 10²⁴ nuclei
3. Find the decay constant (λ):
* t₁/₂ = 5730 years = 1.81 x 10¹¹ seconds
* λ = ln(2) / t₁/₂ ≈ 3.83 x 10⁻¹² s⁻¹
4. Calculate activity (A):
* A = λN ≈ (3.83 x 10⁻¹² s⁻¹) * (4.3 x 10²⁴ nuclei) ≈ 1.64 x 10¹³ Bq
Important Considerations
* Units: Be consistent with units. If you're using half-life in years, make sure the decay constant is calculated in units of inverse years.
* Time: Activity changes over time as radioactive nuclei decay. The formulas above give you the activity at a specific point in time.
* Safety: Radioactive materials can be dangerous. Always handle them with proper precautions and consult safety guidelines.
Let me know if you have any more questions or would like to work through another example.
