A radioactive source of a single nuclide emits 2.4-MeV neutrons at the rate of 9200 neutrons per second. The number of atoms in the source is 4.0 ×109. The activity of the source, in nCi, is
To find the activity of the radioactive source in nanoCuries (nCi), we need to use the following relation:
Activity (A) = λ × N
Where:
A = Activity of the source in Curie (Ci)
λ = Decay constant of the nuclide (per second)
N = Number of radioactive atoms present
First, let's convert the given number of neutrons emitted per second to the decay constant (λ):
Given:
Neutron energy = 2.4 MeV
Number of neutrons emitted per second = 9200
The energy per neutron can be converted to the decay constant (λ) using the equation:
E = m × c^2
Where:
E = Energy of a single neutron (in Joules)
m = Mass of a single neutron (in kg)
c = Speed of light (in m/s)
The mass of a neutron is approximately 1.67 x 10^-27 kg.
Using the conversion factor 1 MeV = 1.602 x 10^-13 Joules, we can calculate the energy (E) of a single neutron:
E = 2.4 MeV × 1.602 x 10^-13 Joules/1 MeV
Next, we can calculate the decay constant (λ) using the equation:
λ = Number of neutrons emitted per second / (Number of atoms × E)
Substituting the given values into the equation:
λ = 9200 neutrons/s / (4.0 × 10^9 atoms × E)
Now, we can find the activity (A) in Curie (Ci) using the equation mentioned earlier:
A = λ × N
Substituting the values of λ and N:
A = λ × N = (9200 neutrons/s / (4.0 × 10^9 atoms × E)) × 4.0 × 10^9 atoms
Finally, we can convert the activity from Curie (Ci) to nanoCurie (nCi) by multiplying by 10^9:
Activity in nCi = A × 10^9
By following these steps, you should be able to calculate the activity of the source in nanoCurie (nCi).