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, we first need to understand a few key concepts: the half-life of a radioactive substance, the decay constant, and how to calculate activity using these values. Let's break this down step by step.

Understanding Half-Life and Decay Constant

The half-life of a radioactive isotope is the time it takes for half of a sample to decay. For radium, this is 1622 years. The decay constant (λ) is a measure of the probability of decay per unit time and can be calculated using the half-life with the formula:

λ = ln(2) / t_1/2

Where ln(2) is the natural logarithm of 2, approximately equal to 0.693, and t_1/2 is the half-life.

Calculating the Decay Constant

Substituting the half-life of radium into the formula gives us:

λ = 0.693 / 1622 years

Calculating this yields:

λ ≈ 0.000427 year^-1

Determining the Activity

Activity (A) is defined as the number of decays per unit time and can be calculated using the formula:

A = λN

Where N is the number of radioactive atoms present. To find N, we need to know the number of moles in 1 gram of radium and then convert that to atoms.

Finding the Number of Atoms

The molar mass of radium is approximately 226 g/mol. Therefore, the number of moles in 1 gram of radium is:

Number of moles = mass / molar mass = 1 g / 226 g/mol ≈ 0.00442 mol

Next, we convert moles to atoms using Avogadro's number (approximately 6.022 x 10²³ atoms/mol):

N = number of moles × Avogadro's number

N ≈ 0.00442 mol × 6.022 x 10²³ atoms/mol ≈ 2.66 x 10²¹ atoms

Final Calculation of Activity

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

A = λN ≈ 0.000427 year^-1 × 2.66 x 10²¹ atoms

Calculating this gives:

A ≈ 1.14 x 10¹⁸ decays per year

Converting to More Common Units

To express this activity in more commonly used units, we can convert years to seconds (1 year ≈ 3.15 x 10⁷ seconds):

A ≈ 1.14 x 10¹⁸ decays/year ÷ 3.15 x 10⁷ seconds/year ≈ 3.62 x 10¹⁰ decays/second

This means that the activity of 1 gram of radium is approximately 36.2 billion decays per second, which is a significant amount of radioactivity. This high level of activity is one reason why radium was historically used in luminous paints and other applications, although its health risks are now well understood.