To calculate the activity of 1 gram of radium, we need to use the formula:
Activity = λ * N
where λ is the decay constant and N is the number of radioactive nuclei.
The decay constant (λ) can be determined using the formula:
λ = ln(2) / t1/2
where ln(2) is the natural logarithm of 2 and t1/2 is the half-life of the radionuclide.
Given that the half-life of radium is 1590 years, we can calculate the decay constant as follows:
λ = ln(2) / t1/2 = ln(2) / 1590
Now, we need to determine the number of radioactive nuclei (N) in 1 gram of radium. To do this, we can use Avogadro's number (Na) to convert grams to moles, and then multiply by the number of atoms in one mole of radium.
The number of atoms (N) in 1 gram of radium can be calculated as follows:
N = (1 g / M) * Na
where M is the molar mass of radium.
Given that the molar mass of radium (M) is 226 g/mol and Na is 6.023 × 10^23 mol^-1, we can calculate N as follows:
N = (1 g / 226 g/mol) * (6.023 × 10^23 mol^-1)
Now we can calculate the activity:
Activity = λ * N
To find the number of alpha particles emitted during half the half-life, we need to consider that during radioactive decay, one alpha particle is emitted per decay event. So, the number of alpha particles emitted during half the half-life would be equal to half the number of radioactive nuclei (N) in 1 gram of radium.
Let's calculate the values:
λ = ln(2) / 1590 N = (1 g / 226 g/mol) * (6.023 × 10^23 mol^-1) Activity = λ * N Number of alpha particles emitted during half the half-life = N / 2
Now we can plug in the values and calculate the results.