Radiation Decay Homework: Linearizing Data

[jumbogala]

Homework Statement

dN/dt, is proprotional to the number of nuclei present, N.

How would you linearize data that you collected?

Homework Equations

The Attempt at a Solution

I'm confused about what the data would look like. Would I be given something like N = 4, or N=4t?

To graph it so it comes out as a straight line, would I take the derivative of N and plot that against dN/dt? Or do I plot N vs t?

 

[Astronuc]

Write the first order differential equation according to the information given. One is given, dN(t)/dt, N(t) and constant of proportionality, λ. Note that N(t) is decreasing.


With each half-life, the number of atoms is approximately equal the number of atoms at the beginning of the half-life period.

 

[Andrew Mason]

Homework Statement

The time rate of decrease in the number of radioactive nuclei, dN/dt, is proprotional to the number of nuclei present, N. The constant of proportionality is λ.

If you are given data that is the number of nuclei (N) as a function of time, how can you determine the value of λ and the initial number of nuclei present? (Hint: graph the data as a straight line).

Homework Equations

The Attempt at a Solution

I'm confused about what the data would look like. Would I be given something like N = 4, or N=4t?

To graph it so it comes out as a straight line, would I take the derivative of N and plot that against dN/dt? Or do I plot N vs t?

The solution to this differential equation:

$$\dfrac{dN}{dt} = \lambda N$$

is the function:

$$N = N_0e^{\lambda t}$$

So plot the data for N(t) on the y-axis and time, t, on the x axis, and take the slope at different points. Then plot the data for N(t) (y axis) vs. the slope of this first graph (x axis) on another graph. What kind of a graph is the second graph? How do you determine $\lambda$ from the second graph?

AM

 

[jumbogala]

The second graph would be linear, and λ is the slope?

Just wondering, could I also do it this way? If I solve the differential equation to get ln(N) - ln(No) = -λt, and graph t vs ln(N)?

 

[jumbogala]

Also, how do you get the equation -dN/dt = λN into the form ln(N) - ln(No) = -λt?

I know I need to solve the differential equation but I'm getting -ln(N) = -λt. Where does the ln(No) come from and why is ln(N) positive?

EDIT: nevermind, I figured it out =)