How Is Initial Power Calculated in a Plutonium-238 Powered Pacemaker?

[erinec]

Homework Statement

Plutonium-238 with a half-life of 90 years, emits and alpha particle of energy 5.10 MeV. 180mg of this radioisotope is used to power a pacemaker. Calculate the initial power in mW, supplied by Plutonium-238, assuming a 100% efficiency for the conversion of decay energy into usable power.

Homework Equations

I tried using:
N = N_o(1/2)^(t/T), where T is the half-life
P = W/t

The Attempt at a Solution

I do not know what "time" to use in order to calculate the power. What does it mean by "initial"?
Help would be very appreciated.

 

[mgb_phys]

Initial means assuming that none of it has decayed yet
Watts are joules/second so you need to work out how many decays in the first second

 

[nlightner]

I would respond to this content by saying that the initial power refers to the power output at the very beginning of the pacemaker's use, when all 180mg of Plutonium-238 is still present and decaying. To calculate this initial power, we can use the equation for radioactive decay, N = No(1/2)^(t/T), where N is the current amount of Plutonium-238, No is the initial amount, t is the time since the initial amount, and T is the half-life.

Since we are assuming 100% efficiency for the conversion of decay energy into usable power, we can also use the equation P = W/t, where P is the power, W is the energy released per decay (in this case, 5.10 MeV), and t is the time for one decay (which is equal to the half-life, T).

Combining these two equations, we get:

P = W * (No/No)(1/2)^(t/T) * (1/t)

Since we are looking for the initial power, we can plug in the initial amount (No = 180mg) and the half-life (T = 90 years) into the equation, and use the given energy released per decay (W = 5.10 MeV).

P = (5.10 MeV) * (180mg/180mg)(1/2)^(t/90 years) * (1/90 years)

P = (5.10 MeV) * (1/2)^(t/90 years) * (1/90 years)

We can now choose a specific time to calculate the initial power. For example, if we choose t = 0 years (the beginning of the pacemaker's use), the equation simplifies to:

P = (5.10 MeV) * (1/2)^(0/90 years) * (1/90 years)

P = (5.10 MeV) * (1/1) * (1/90 years)

P = 0.0567 mW

Therefore, the initial power supplied by Plutonium-238 to the pacemaker would be 0.0567 milliwatts.