How Do You Calculate Mean Life and Half-Life of Radioactive Nuclei?

[Apollo14LMP]

Homework Statement

Radioactive nuclei are produced in an irradiated sample at the rate of 10 s-1. If the number in the sample builds up to a maximum of 1000, calculate the mean life and the half-life of the radioactive nuclei

Can anyone advise on this one not sure how mean life is found or if half life is right .. thanks

Homework Equations

Using dN / dt = P - λ N

The Attempt at a Solution

1000 nuclei / 10s-1 = 100 nuclei s-1

P - λ N = 100 nuclei s-1

(In2) / λ = (In2) / (10 s-1) = 6.93 x10^-2 s-1

is Half life = 6.93 x10^-2 s-1 ??

 

[Orodruin]

You were given the production rate P and the N for the steadystate. What is true for the steady state?

 

[Apollo14LMP]

Is the steady state is the difference between the rate of production 10s-1 and the rate of decay ?

 

[Orodruin]

The steady state is when the number of nuclei no longer changes.
It is the stable solution that N will tend to at large times.

 

[Apollo14LMP]

So the steady state is the 1000 nuclei

So I would just need to use that amount and apply the decay constant equation ?? I am confused and going round in a circle ...

 

[Orodruin]

So how does your differential equation look like in the steady state?
What are the different terms?

 

[Apollo14LMP]

1000 nuclei - 10s-1 = 990 nuclei s-1

(In2) / λ = (In2) / (990s-1) = 7.00 x 10^-4 s-1

 

[Orodruin]

No, your units do not make sense now. What is the meaning of each of the terms? What do they represent?

Alternatively, you can try to solve the differential equation and go from there, but it really is not necessary.

 

[Orodruin]

Homework Equations

Using dN / dt = P - λ N

So to be specific

• What is dN/dt?
• What is P?
• What is N?

(Describe them in words, do not try to put numbers at this point)

What then are their values in the steady state?

 

[Apollo14LMP]

There is a total of 1000 nuclei in the solution, the rate of increase (decrease) is 10s-1

If there are 1000 nuclei and the rate of decay is 10 per second, then the difference is 990 nuclei

Divided by by the decay constant 0.693 ..

 

[Apollo14LMP]

dN is the total number of nuclei 1000
dt is the time the rate of production = 10s-1

Production rate 10 s-1
N is the total number of nuclei after I have deducted 10 nuclei from the total amount 990 nuclei

 

[Orodruin]

No, answer the questions in my previous post each in turn without attaching numbers.

You cannot subtract a rate from a number - the result will simply be nonsense.

 

[Apollo14LMP]

So to be specific

• What is dN/dt?
• What is P?
• What is N?

(Describe them in words, do not try to put numbers at this point)

What then are their values in the steady state?

dn / dt is the total number divided by time ...

P is the production rate

N is the total of dn/dt - production rate

 

[Orodruin]

Ok, we have identified some misunderstandings.

The differential equation describes how the total number of nuclei will vary with time. The left-hand side is the rate of change of the number N. dN/dt is therefore the rate of change in N.

Now, what may influence this rate? Well, we have production - we are producing a certain number of nuclei per time unit P. But the total number will also change due to decays. In a small time, the number of nuclei that decay will be proportional to the number of available nuclei, that is N. We call the constant of proportionality λ and thus the nuclei will decay with a rate λN. The total rate of change in the number of particles, i.e., dN/dt, is the rate at which nuclei are produced minus the rate at which they decay, thus
$$
\frac{dN}{dt}= P - \lambda N.
$$

Given the above, what would you say that dN/dt, P, and N are in the steady state solution? (Noting that it is the solution where the number of nuclei does not change with time.)

 

[Apollo14LMP]

dn/dt = 1000 - 10 = 990
p = 10
n = 1000

mind block is here ...

 

[Orodruin]

No, again you are trying to subtract something with units s^-1 from a number.

The steady state is when the number of nuclei does not change with time. What is the rate of change for something that does not change with time?

Always write out the units. If you do not, your equations will not make sense.

 

[Apollo14LMP]

dN / dt = 1000n /10/ 1 = 100
P - λ N = 10 – λ 1000N
(In2) / λ = (In2) / 1000) = 6.93x10^-4 seconds

 

[Orodruin]

Let me repeat the question:

What is the rate of change for something that does not change with time?

I am sorry, I do not know how to give you a stronger hint without giving you the answer.

It seems you are randomly throwing numbers into your equations without really understanding what they represent.

What condition between the production rate and decay rate must be fulfilled if the total number is not changing?

Or would you prefer trying to solve the differential equation?

 

[Apollo14LMP]

Please don't be sorry - I have been struggling with this for a while. I am going to sleep on this, I am sure I am not too far off the mark. I am so grateful to you for steering me along .. thank you very much.

I must work through these issues and find my own way and solutions. Thank you once again for your sharing your expertise ...

 

[Apollo14LMP]

dN / dt = 10 /1000 = 0.01t - the rate at which nuclei are produced minus the rate at which they decay

P - is the production rate = 10 s-1

N - number of nuclei = 1000

So, N(t) = N0 exp(-λt),

Where N0 = number of radioactive nuclei at t = 0.

N(t) = N0 exp (.01t)

N0 = N0 /2 and t = t1/2 So, t1/2 = (ln2) /.01t = 69.3 seconds

The half life is 69.3 seconds

 

[Orodruin]

P - is the production rate = 10 s-1

N - number of nuclei = 1000

These two things are true in equilibrium. Disregard everything else that you have written except the differential equation
$$
\frac{dN}{dt} = P - \lambda N.
$$
What is this differential equation with your numbers inserted? Do you know how to solve this differential equation?

 

[Apollo14LMP]

p = 10, n = 1000 nuclei

 

[Orodruin]

No, N = 1000 is just the stable limit. In general, the number of nuclei will follow the differential equation
$$
\frac{dN}{dt} = 10 - \lambda N.
$$
Now, if you are in the stationary state, by definition the number of nuclei does not change with time. So in the stationary state, N = 1000 and dN/dt is equal to what?

 

[Apollo14LMP]

difference in number of nuclei over difference in time ...

 

[Curious3141]

difference in number of nuclei over difference in time ...

Not really, $\frac{dN}{dt}$ is the rate of change in the number of nuclei.

The number of nuclei increases till it hits 1000, then it *stops* increasing. The total number of nuclei doesn't change from that point on.

When something *stops* changing, the rate of change is now? What?

Answer that, and you'll go a long way toward solving this question.

 

[Orodruin]

difference in number of nuclei over difference in time ...

Which is what in the stationary state when the number is no longer changing?