A 50.6 mg sample of sodium perchlorate contains radioactive chlorine-36 (whose atomic mass is 36.0 amu). If 29.6% of the chlorine atoms in the sample are chlorine-36 and the remainder is naturally occurring nonradioactive chlorine atoms, how many disintegrations per second are produced by this sample? The half-life of chlorine-36 is 3.5*10^5 yr.
moles NaClO4 = grams/molar mass
moles Cl atoms = moles NaClO4.
# Cl atoms = moles Cl atoms x 6.02E23 atoms/mol.
Multiply by 0.296 to find the number that are radioactive.
rate = k*No where No = number Cl atoms
k can be determined from k = 0.693/t1/2 and the units are Yr^-2 since the half life is in years.
Therefore, the rate you calculate will be the decays per year. Convert that to dps.
To calculate the number of disintegrations per second produced by the sample, we need to determine the activity of the radioactive chlorine-36.
The activity of a radioactive sample is given by the equation:
A = λ*N
Where:
A = Activity (disintegrations per second)
λ = Decay constant (disintegrations per second per number of radioactive atoms)
N = Number of radioactive atoms in the sample
First, let's calculate the number of chlorine-36 atoms in the sample:
Number of chlorine-36 atoms = 29.6% * Total number of chlorine atoms
Given that the sample contains 50.6 mg of sodium perchlorate, we need to convert this to the number of chlorine atoms:
Number of chlorine atoms = (50.6 mg) / (atomic mass of chlorine)
Next, we calculate the number of chlorine-36 atoms:
Number of chlorine-36 atoms = (29.6% * Total number of chlorine atoms)
Now, let's calculate the activity:
Activity = λ * N
The decay constant (λ) is related to the half-life (t1/2) by the equation:
λ = ln(2) / t1/2
Given that the half-life of chlorine-36 is 3.5 * 10^5 years, we can calculate λ:
λ = ln(2) / (3.5 * 10^5 years)
Finally, we can calculate the activity:
Activity = λ * N
Note: To convert years to seconds, we need to multiply by the number of seconds in a year (365 days/year * 24 hours/day * 60 minutes/hour * 60 seconds/minute).
Let's plug in the values and calculate the number of disintegrations per second produced by the sample.
To calculate the number of disintegrations per second produced by the sample, we need to take into account the number of radioactive chlorine-36 atoms in the sample and the half-life of chlorine-36.
Here are the steps to calculate the number of disintegrations per second:
1. Determine the number of moles of sodium perchlorate:
- Convert the mass of the sample to grams: 50.6 mg = 0.0506 g
- Use the molar mass of sodium perchlorate to convert grams to moles. The molar mass of sodium perchlorate is the sum of the atomic masses of its constituent elements: Na (sodium) + Cl (chlorine) + 4O (oxygen):
Molar mass of NaClO4 = (23.0 g/mol) + (35.5 g/mol) + (16.0 g/mol x 4) = 122.5 g/mol
- Moles of NaClO4 = mass (grams) / molar mass = 0.0506 g / 122.5 g/mol = 0.000412 mol
2. Determine the number of radioactive chlorine-36 atoms:
- Since 29.6% of the chlorine atoms in the sample are chlorine-36, we can calculate the moles of chlorine-36 by multiplying the moles of sodium perchlorate by the percentage of chlorine-36:
Moles of chlorine-36 = Moles of NaClO4 x 29.6% = 0.000412 mol x 0.296 = 0.000122 mol
- To convert the moles of chlorine-36 to the number of atoms, multiply by Avogadro's number:
Number of chlorine-36 atoms = Moles of chlorine-36 x Avogadro's number = 0.000122 mol x 6.022 x 10^23 atoms/mol = 7.33 x 10^19 atoms
3. Determine the number of disintegrations per second:
- The half-life of chlorine-36 is given as 3.5 x 10^5 years. Since the half-life is the time it takes for half of the radioactive atoms to decay, we can use it to calculate the decay constant (λ):
λ = ln(2) / half-life = 0.693 / (3.5 x 10^5 years) = 1.98 x 10^-9 per year
- To convert the decay constant to decay constant per second, we divide by the number of seconds in a year (3.1536 x 10^7 seconds):
Decay constant (in per second) = Decay constant (per year) / seconds per year = 1.98 x 10^-9 per year / 3.1536 x 10^7 seconds/year = 6.28 x 10^-17 per second
- Now, we can determine the number of disintegrations per second by multiplying the number of radioactive chlorine-36 atoms by the decay constant:
Disintegrations per second = Number of chlorine-36 atoms x Decay constant = 7.33 x 10^19 atoms x 6.28 x 10^-17 per second = 4.6 x 10^3 disintegrations per second
Therefore, the sample of sodium perchlorate containing radioactive chlorine-36 produces approximately 4.6 x 10^3 disintegrations per second.