a) Potassium-40 undergoes three different modes of radioactive decay (Beta decay, Positron decay, and Electron Capture decay). Write balanced nuclear reactions for each of these decay modes. Are the products stable (not radioactive)?
b) Potassium-40 has a natural abundance of 0.0118%. How many atoms of potassium-40 are present in a 1.35 g sample of KCl?
a) To write balanced nuclear reactions for each of the decay modes of Potassium-40, we need to understand the decay processes involved.
1. Beta decay:
In beta decay, a neutron in the nucleus turns into a proton, emitting an electron (beta particle) and an antineutrino. The general beta decay equation for Potassium-40 is:
^40K -> ^40Ar + e- + ν
2. Positron decay:
In positron decay, a proton in the nucleus turns into a neutron, emitting a positron (positive electron) and a neutrino. The general positron decay equation for Potassium-40 is:
^40K -> ^40Ar + e+ + ν
3. Electron capture decay:
In electron capture decay, an inner shell electron combines with a proton in the nucleus, converting it into a neutron. This process releases a neutrino. The general electron capture decay equation for Potassium-40 is:
^40K + e- -> ^40Ar + ν
Now, regarding the stability of the products:
- ^40Ar (Argon-40) is stable and not radioactive in all three decay modes.
- The beta particle (e-) and the positron (e+) are not stable and will eventually undergo further decay or interactions.
- The neutrinos (ν) are very weakly interacting and essentially have no effect on stability.
b) To calculate the number of atoms of potassium-40 present in a 1.35 g sample of KCl, we need to use the Avogadro constant and the molar mass of KCl.
1. Find the molar mass of KCl:
The molar mass of KCl is calculated by adding the atomic masses of potassium (K) and chlorine (Cl):
K: atomic mass = 39.10 g/mol
Cl: atomic mass = 35.45 g/mol
Molar mass of KCl = 39.10 g/mol + 35.45 g/mol = 74.55 g/mol
2. Calculate the moles of KCl in the given sample:
Moles = mass / molar mass
Moles = 1.35 g / 74.55 g/mol
3. Calculate the number of moles of potassium-40 present in the sample:
Since the natural abundance of potassium-40 is given as 0.0118%, we need to multiply the moles of KCl by this percentage:
Moles of K-40 = (0.0118 / 100) * (moles of KCl)
4. Calculate the number of atoms:
Number of atoms = Moles of K-40 * Avogadro constant
Now you can plug in the values into these equations to find the number of potassium-40 atoms present in the given sample.