# How Long Until a Radioactive Daughter Isotope Reaches 97% of Its Equilibrium?

• Winzer
In summary, the parent nucleus with a half-life of \tau_\frac{1}{2}=\delta decays through a series of daughters to a final stable isotope. One of the daughter particles has the greatest half life of \tau_\frac{1}{2}=\alpha-- the others are less than a year. At t=0, there are N_0 nuclei of the parent present and no daughters. To calculate the time it takes for the population with the greatest half life to reach 97% of its equilibrium value, we need to solve the differential equation \frac{dN}{dt}=e^{-\lambda t}, taking into account the decay rate of the daughter nuclei. The equilibrium is reached when the decay rates
Winzer

## Homework Statement

A parent isotope has $$\tau_\frac{1}{2}=\delta$$. Its decays through a series of daughters to a final stable isotope. One of the daughter particles has the greatest half life of $$\tau_\frac{1}{2}=\alpha$$-- the others are less then a year. At t=0 the parent nuclei has $$N_0$$ nuclei, no daughters are present.

How long does it take for the population with the greatest half life to reach 97% its equilibrium value?
At some t, how many nuclei of the isotope with the greatest half life are present, assume no branching.

## Homework Equations

$$\frac{dN}{dt}=e^{-\lambda t}$$

## The Attempt at a Solution

So for the first one:
Its just solving the diff eq above right? The daughter is in its eq. value or do we have to worry about decay from the other daughters?

the second one:
Basically plugging in t right for the solved diff eq with initial nuclei right?

Just checking, I feel like I'm missing something.

Hi there,

You have the right equation: $$\frac{dN}{dt} = e^{-\lambda t}$$ But don't forget that the daughter nuclei also decay at a certain rate. Therefore, you need to consider the same equation for the long life daughter nucleus.

By the way, just a further comment, typically what half-life are you talking about here? Because, daughter nuclei with half-life of more than a few split second are normally considered into the decay chain.

Cheers

the halflife(longest) for the daughter is 20yr. The parent is 10^4 yr.
So for the daughter nuclei(20 yr):
$$\frac{dN}{dt} = e^{-\lambda_1 t}- e^{-\lambda_2 t}$$
Where 2 is the daughter. Should 1 be the half life of the 1yr daughter?

Hi there,

When the equilibrium is reach, the decay rate of the parent nuclei is the same as the decay rate of the daughter nuclei, and it is independant of the daughters formed in the process. Therefore, you would have: $$\frac{dN_1}{dt} = \frac{dN_2}{dt}$$

If you solve this simple equation, you have the time needed to reach equilibrium.

Cheers

Hi there,

Your question really caught my attention, and with the half lives you gave me, I find that the system will reach equilibrium after 138.2 years.

Cheers

## 1. What is a population half life?

A population half life refers to the time it takes for a population to decrease by half of its initial size. It is commonly used in the study of populations, such as animal or plant populations, and can also be applied to human populations.

## 2. How is population half life calculated?

To calculate population half life, the initial population size and the rate of decrease must be known. The population half life can be calculated using the following formula: t1/2 = ln(2)/r, where t1/2 is the population half life, ln is the natural logarithm, and r is the rate of decrease.

## 3. What factors can affect population half life?

There are several factors that can affect population half life, including environmental conditions, availability of resources, competition for resources, and predation. Human activities, such as pollution and habitat destruction, can also have a significant impact on population half life.

## 4. Why is population half life important?

Population half life is important because it provides valuable information about the dynamics of a population. It can help scientists understand the health and sustainability of a population, as well as predict future trends. It can also inform conservation efforts and management strategies for endangered species.

## 5. Can population half life be applied to all types of populations?

Yes, population half life can be applied to all types of populations, including animals, plants, and human populations. However, the specific factors and calculations may vary depending on the type of population being studied.

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