A one gram if 226 Ra has an activity of nearly 1Cl.determine the half life of 226Ra. The half life of 238U is 4.5×109year . what is the decay constant? Find the activity of 1g,of 238U
To determine the half life of 226 Ra, we use the formula:
A = A0 * (1/2)^(t/T)
where A = activity, A0 = initial activity, t = time elapsed, and T = half life.
We know that A0 = 1 Ci (we convert to Ci from mCi for convenience), and A = 0.5 Ci (since the activity is halved after one half life). We are given that we have 1 gram of 226 Ra, which has a molar mass of 226 g/mol. Therefore, we have 1/226 moles of 226 Ra.
Using the formula for activity:
A = λ * N
where N = number of radioactive atoms and λ = decay constant, we can solve for λ:
λ = A/N = (0.5 Ci) / [(6.022 x 10^23 atoms/mol) * (1/226 mol)] = 1.19 x 10^-8 s^-1
We use the half life formula again for 238 U:
T = 4.5 x 10^9 years = (ln 2) / λ
Solving for λ:
λ = (ln 2) / T = 1.54 x 10^-10 s^-1
Finally, we can find the activity of 1 gram of 238 U:
N = (1 g) / (238 g/mol) * (6.022 x 10^23 atoms/mol) = 2.54 x 10^21 atoms
A = λ * N = (1.54 x 10^-10 s^-1) * (2.54 x 10^21 atoms) = 0.40 Ci
To determine the half-life of 226Ra, we can use its activity. The activity of a radioactive substance is defined as the number of radioactive decays that occur per unit time. In this case, we know that a one-gram sample of 226Ra has an activity of nearly 1 Ci (curie).
One curie is equal to 3.7 x 10^10 disintegrations per second (dps). Therefore, the activity of 1 gram of 226Ra is approximately 3.7 x 10^10 dps.
The decay constant (λ) is a measure of the likelihood of a radioactive decay occurring per unit time. It is related to the half-life (T1/2) by the equation λ = ln(2) / T1/2.
Given that the half-life of 238U is 4.5 x 10^9 years, we can calculate the decay constant as follows:
λ = ln(2) / T1/2
= ln(2) / (4.5 x 10^9 years)
≈ 0.693 / (4.5 x 10^9 years)
≈ 1.54 x 10^-10 year^-1
Therefore, the decay constant for 238U is approximately 1.54 x 10^-10 year^-1.
To find the activity of 1 gram of 238U, we can use the decay constant and the relationship between activity and decay constant. The activity (A) can be calculated using the equation A = λN, where N is the number of radioactive atoms.
Since the sample has a mass of 1 gram, we can convert this into the number of atoms using Avogadro's number (6.022 x 10^23 atoms/mole). The molar mass of 238U is approximately 238 grams/mole.
N = (1 gram / 238 grams/mole) * (6.022 x 10^23 atoms/mole)
≈ 2.53 x 10^21 atoms
Using the decay constant λ = 1.54 x 10^-10 year^-1 and N = 2.53 x 10^21 atoms, we can calculate the activity of 1 gram of 238U as follows:
A = λN
≈ (1.54 x 10^-10 year^-1) * (2.53 x 10^21 atoms)
≈ 3.90 x 10^11 decays per second
≈ 390 GBq (gigabecquerels)
Therefore, the activity of 1 gram of 238U is approximately 390 GBq.