Chapter 45, Problem 1PLA

To determine

Show that equation 3 in laboratory can be expressed as $\ln \frac{A_0}{A}=\lambda t$.

Answer to Problem 1PLA

It is proved that the equation 3 in laboratory can be expressed as $\ln \frac{A_0}{A}=\lambda t$.

Explanation of Solution

Write equation 3 from the text

    $A=A_0{e}^{-\lambda t}$

Here, $A$ is the activity of the radioactive sample at time $t$, $A_0$ is the activity of the radioactive sample at $t=0$, $t$ is the time and $\lambda$ is the decay constant.

Divide by $A_0$ on both sides in the above equation

$\frac{A}{A_0}=e^{-\lambda t}$

Inverting the terms on both sides

$\frac{A_0}{A}=e^{\lambda t}$

Taking ln on both sides,

$\ln \left(\frac{A_0}{A}\right)=\lambda t$

Hence it is proved.

Conclusion:

It is proved that the equation 3 in laboratory can be expressed as $\ln \frac{A_0}{A}=\lambda t$.