Chapter 30, Problem 39P

To determine

Fraction of Ge that will remain after 2.5 years.

Answer to Problem 39P

Solution:

10%

Explanation of Solution

Given Info:

Half-life of Germanium is 9 months.

Formula Used:

The exponential decay law for radioactive decay can be described as

$N=N_0{e}^{-\lambda t}$

Where N₀ is the initial number of particles and N is number of particles at time t, λ is decay constant.

The relationship between concentration, decay constant and half-life can be given by

$\ln \left(\frac{\left[A_0\right]}{\left[A_t\right]}\right)=\lambda t$

Calculation:

Decay constant can be calculated as

$\begin{array}{l}\text{9}\text{months= 9/12 =0}\text{.75 years}\\ \text{λ=}\frac{\text{0}\text{.693}}{{\text{T}}_{\text{1/2}}}\text{=}\frac{\text{0}\text{.693}}{\text{0}\text{.75}}\text{=0}\text{.924}{\text{years}}^{\text{-1}}\end{array}$

Using the above equation we can find the fraction of Germanium remaining after 2.5 years

$\begin{array}{l}\text{ln}\left(\frac{\left[A_0\right]}{\left[A_t\right]}\right)=\lambda t\\ 0.924\times 2.5=\ln \left(\frac{\left[A_0\right]}{\left[A_t\right]}\right)\\ 2.31=\ln \left(\frac{\left[A_0\right]}{\left[A_t\right]}\right)\\ \left(\frac{\left[A_t\right]}{\left[A_0\right]}\right)=0.0992\approx 0.1\end{array}$