Question

The half Life \(T_{1 \Omega}\) is not the average lifetime rof a radioactive nucleus. We find the average lifetime by multiplying 8 by the probability per unit tinie \(P(t)\) that the nucleus will "live" that long, then integrating over all time. (a) Show that \(P(t)\) should be given by \(\lambda e^{-A \text { 1. }}\). Hint: What must be the total probability?) (b) Show that \(r=T_{1 \Omega} / \ln 2\)

Short answer

The probability distribution function of a radioactive nucleus is given by: \(P(t) = \lambda e^{-\lambda t}\) where \(\lambda\) is the decay constant. The average lifetime of a radioactive nucleus is related to its half-life using the formula: \(T_{1/2} = \tau \ln 2\).

Step-by-step solution

Derivation of the Probability Distribution function

The decay of radioactive nuclei is a random process and the probability that a nucleus will 'live' up to time t is given by the product of the decay constant \(\lambda\) and the exponential of the negative decay constant multiplied by time \(e^{-(\lambda)t}\). This is because the total probability of decay occurring over all time must equal 1. So the Probability Distribution function \(P(t)\) is given by: \(P(t) = \lambda e^{-\lambda t}\).

Verification of Total Probability

Now we need to confirm that the total probability equals 1. Let's integrate over all time from t=0 to t=\infty. \(\int_{0}^{\infty} P(t) dt = \int_{0}^{\infty} \lambda e^{-\lambda t} dt\). On solving this integration we get 1. Thus, the total probability is equal to 1 which verifies the formula for Probability Distribution function.

Derivation of average lifetime

The average lifetime (or mean lifetime), denoted by \(\tau\), is obtained by summing the product of time and the probability at that time for all times. This gives us the formula: \(\tau = \int_{0}^{\infty} t\lambda e^{-\lambda t} dt\). On solving this integral, we get \(\tau = \frac{1}{\lambda}\).

Relating the average lifetime to half-life

The half-life of a substance, denoted by \(T_{1/2}\), is the time taken for half of the radioactive nuclei to decay. It is related to the decay constant by: \(T_{1/2} = \frac{0.693}{\lambda}\). Substituting the average lifetime in place of the reciprocal of decay constant we get, \(T_{1/2} = \tau \ln 2\)

Key concepts

Half-Life

Understanding the half-life of a radioactive substance is critical for interpreting its decay over time. The half-life, often denoted as \(T_{1/2}\), is defined as the duration required for half of the radioactive nuclei within a sample to undergo decay. This time parameter is crucial because it provides a measure of how quickly a radioactive substance transforms into another element or isotope.

It can be surprising to learn that half-life is not the same as the average lifetime of a radioactive nucleus; rather, it is a statistical measure with broad implications in fields such as geology, archaeology, and medicine where radiometric dating or treatments involve radioisotopes. As we saw in the exercise, by understanding the relationship between half-life and the decay constant \(\lambda\), we can unlock further insights into the behavior of radioactive materials over time. Specifically, the half-life is inversely related to the decay constant as expressed in the equation \(T_{1/2} = \frac{0.693}{\lambda}\).
It's important for students to remember that this equation is a cornerstone of nuclear chemistry and physics, allowing us to predict the longevity of substances undergoing radioactive decay.

Decay Constant

The decay constant, symbolized as \(\lambda\), is an intrinsic property of each radioactive substance, representing the probability of decay of a nucleus per unit time. The notion that each type of radioactive material has its unique decay constant is essential, as it reflects how rapidly or slowly a material decays regardless of the quantity present.

In the solution, we explored how the decay constant plays a pivotal role in gauging the probability of an atom's decay at a given instant, and is used to formulate the probability distribution function for the decay process. As evidenced by the solution, the decay constant is directly incorporated into the exponential factor of the probability distribution function, emphasizing its significance in nuclear decay dynamics. Moreover, knowing the decay constant not only aids in calculating half-life but also in determining the average lifetime of radioactive atoms, fostering a deeper understanding of the temporal aspects of radioactive decay.

Average Lifetime

The average lifetime, or mean lifetime, represented by \(\tau\), is a measure that indicates the typical length of time a single nucleus of a radioactive substance exists before it decays. This concept differs from half-life in that it provides an average value over all nuclei, instead of focusing on the time required for a specific proportion of nuclei to decay.

Calculating the average lifetime involves the integration of time multiplied by the probability distribution function over all possible times, essentially finding a weighted 'average' of all decay events. As shown in the exercise, solving this integral yields the average lifetime as \(\tau = \frac{1}{\lambda}\), which inversely relates the average lifetime to the decay constant. This inverse relationship enlightens us to the fact that a high decay constant indicates a short average lifetime, while a low decay constant signifies a longer average lifespan of the atoms before decay. Understanding average lifetime helps students in various scientific inquiries, such as in estimating the duration of radiation exposure from isotopes.

Probability Distribution Function

The probability distribution function, denoted as \(P(t)\), is a statistical function that describes the likelihood of a radioactive nucleus decaying over time. It is a key concept that aids in understanding the random nature of radioactive decay.

As dissected in the exercise, this probability for any given time 't' is expressed as \(P(t) = \lambda e^{-\lambda t}\), where \(\lambda\) is the decay constant. The exponential nature of this function reflects the continuous and spontaneous process of decay that characterizes radioactive substances. Moreover, the requirement that the total probability of decay occurring at any time from now to infinity add up to 1 assures us of the function's legitimacy in predicting decay behaviors.
By grasping this distribution, students can model and predict how likely it is for decay to occur at any given moment, which is invaluable in many practical applications such as medical radiotherapy planning or the safe handling and storage of radioactive materials.