1)7Be decays with a half-life of about 53 d. It is produced in the upper atmosphere, and filters down onto the Earth's surface. If a plant leaf is detected to have 300 decays/s of 7Be, how many days do we have to wait for the decay rate to drop to 13.6/s?

2) What is the energy of the alpha particle emitted in the decay 232U ==> 228Th + alpha ?

To find the answer to these questions, we will use the concept of radioactive decay and the decay equation.

1) To calculate the time it takes for the decay rate to drop to a certain value, we can use the equation for radioactive decay:

N(t) = N₀ * (1/2)^(t/t₁/₂)

Where:
N(t) is the number of radioactive atoms remaining at time t
N₀ is the initial number of radioactive atoms
t is the time that has passed
t₁/₂ is the half-life of the radioactive isotope

In this case, we are given that the initial decay rate is 300 decays/s and we want to know how long it takes for the decay rate to drop to 13.6 decays/s. So we can set up the equation:

13.6 = 300 * (1/2)^(t/53)

To solve for t, we need to rearrange the equation:

(1/2)^(t/53) = 13.6/300

Now we can take the logarithm of both sides:

log(base 1/2) [(1/2)^(t/53)] = log(base 1/2) (13.6/300)

t/53 = log(base 1/2) (13.6/300)

Finally, we can solve for t by multiplying both sides by 53:

t = 53 * log(base 1/2) (13.6/300)

2) To find the energy of the alpha particle emitted in the decay, we can use the conservation of energy. The energy released in a nuclear decay can be calculated using Einstein's famous equation E = mc².

In this decay process, we know the masses of the reactants and products:

Mass of 232U = 232 atomic mass units (AMU)
Mass of 228Th = 228 atomic mass units (AMU)

To find the mass of the alpha particle, we need to subtract the mass of uranium-232 and thorium-228 from each other:

Mass of alpha particle = (Mass of 232U - Mass of 228Th - Mass of neutrino)

The neutrino is also produced in the decay but does not contribute significantly to the alpha particle's mass or energy.

Finally, we can use Einstein's equation to calculate the energy:

E = Mass of alpha particle * c²

Where c is the speed of light.

It is important to note that the exact values of masses and energy could vary depending on the source used for calculation, so it is always recommended to cross-reference with reliable scientific databases or literature.