The power supply for a pacemaker is a small amount of radioactive ^238 Pu. This nuclide decays

The power supply for a pacemaker is a small amount of...

The power supply for a pacemaker is a small amount of radioactive ${ }^{238} \mathrm{Pu}$. This nuclide decays by alpha decay with a half-life of 86 yr. The pacemaker is typically replaced every $10.0$ yr. (a) By what percentage does the activity of the ${ }^{238}$ Pu source decrease in $10 \mathrm{yr} ?$ (b) The energy of the alpha particles emitted is $5.6 \mathrm{MeV}$. Assume an efficiency of $100 \%-$ all of the alphaparticle energy is used to run the pacemaker. If the pacemaker starts with $1.0 \mathrm{mg}$ of ${ }^{238} \mathrm{Pu}$, what is the power output initially and after $10.0$ yr?

Key Concepts

Power Output Calculation

Power output in the context of radioactive decay is calculated by multiplying the energy released per decay by the activity (the number of decays per unit time). This allows for quantifying the rate at which energy is produced and is essential for designing systems that utilize nuclear decay as an energy source.

Energy Conversion in Nuclear Decay

In nuclear decay processes, a specific amount of energy is released with each decay event. This energy, often given in electronvolts (such as MeV), can be converted into other forms of energy. In practical applications, the released energy may be harnessed, for example, to generate electrical power.

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, consisting of two protons and two neutrons. This process results in a new nucleus with a mass number reduced by four and an atomic number reduced by two, and releases a fixed amount of energy.

Activity

Activity refers to the number of radioactive decays occurring per unit time, typically measured in units such as Becquerels or Curies. It is directly proportional to the number of undecayed nuclei and the decay constant, providing a measure of the intensity of the radioactive emission.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This process occurs spontaneously and is described statistically, leading to a decrease in the number of undecayed nuclei over time.

Exponential Decay Law

The exponential decay law mathematically describes the decrease in the number of radioactive nuclei over time. It shows that the number of remaining nuclei decreases exponentially according to the formula N(t) = N0 exp(-?t), where N0 is the initial quantity and ? is the decay constant.

Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. It is an intrinsic property of the nuclide and is used to quantify the rate of decay, independent of the initial amount of the substance.

Decay Constant

The decay constant is a parameter that characterizes the probability per unit time that a single nucleus will decay. It is directly related to the half-life by the equation ? = ln(2)/t1/2, linking the microscopic behavior of individual nuclei to the macroscopic decay rate.

Transcript

00:01 So for this question we're looking at the decay of plutonium 23a in a pacemaker.

00:08 So it's actually used as a source of power for a pacemaker via its alpha decay.

00:18 And we are looking at what is the percentage decrease in activity, right, after a certain period of time.

00:27 This time you're given is 10 years.

00:30 So we know that the activity at any given point time t is equal to the original amount activity times half to the power of t over the half life and the fraction, right, we can find the fraction of activity left is equals to half to power t over t half.

01:01 And if we were to look at 10 years, right, there will be half, the power of 10 over 86.

01:10 Right, we're given that the t half is 86.

01:14 So we can put that into our calculator, there will be 0 .92.

01:23 So this is the fraction of activity that is left.

01:28 Right? and in terms of percentage, this is actually 92%.

01:31 So 92 % of the original activity is left after 10 years.

01:37 And we want to find out what is the percentage decrease.

01:40 So the percentage decrease, we simply take total of 100 % subtracted by this.

01:47 92 gets 8%.

01:51 So the decrease is 8%.

01:55 Now next, we're looking at what is the power output.

01:59 Right? at power output initially, which is p0, b at zero time.