The atomic mass number of radium is A = 226 it`s half life is 1622 years What is the activity of 1g radium?

To determine the activity of 1 gram of radium-226, we need to understand a few key concepts related to radioactivity, including the half-life of the isotope and how to calculate activity using the decay constant.

Understanding Activity

Activity, measured in becquerels (Bq), refers to the number of decays that occur in a sample per second. The activity can be calculated using the formula:

A = λN

Where:

• A = activity (in Bq)
• λ = decay constant (in s^-1)
• N = number of radioactive atoms present in the sample

Calculating the Decay Constant

The decay constant (λ) can be derived from the half-life (T_1/2) using the formula:

λ = ln(2) / T_1/2

For radium-226, the half-life is given as 1622 years. First, we need to convert this into seconds:

1622 years × 365.25 days/year × 24 hours/day × 3600 seconds/hour = 5.12 × 10¹⁰ seconds

Now, substituting this value into the decay constant formula:

λ = ln(2) / (5.12 × 10¹⁰)

Calculating this gives:

λ ≈ 1.35 × 10^-11 s^-1

Finding the Number of Atoms in 1 Gram of Radium

Next, we need to find out how many atoms are in 1 gram of radium-226. The molar mass of radium-226 is approximately 226 g/mol. Using Avogadro's number (6.022 × 10²³ atoms/mol), we can calculate the number of atoms:

N = (1 g / 226 g/mol) × (6.022 × 10²³ atoms/mol)

This results in:

N ≈ 2.66 × 10²⁴ atoms

Calculating the Activity

Now that we have both λ and N, we can calculate the activity:

A = λN

Substituting the values we found:

A ≈ (1.35 × 10^-11 s^-1) × (2.66 × 10²⁴ atoms)

This gives:

A ≈ 3.59 × 10¹³ Bq

Final Thoughts

Thus, the activity of 1 gram of radium-226 is approximately 35.9 trillion becquerels. This high level of activity reflects the intense radioactivity of radium, which is why it has significant historical importance in both science and medicine.