Chapter 30.2, Problem 28PP

To determine

The ratio of brightness of fluorescent glow of titanium watches after $6\text{ years}$ to present.

Answer to Problem 28PP

The ratio of brightness of fluorescent glow of titanium watches after $6\text{ years}$ to present is $0.7132$

Explanation of Solution

Given:

Half-life, $t_{1/2}=12.3\text{ years}$

Formula used:

The formula of variation of activity with time is:

  $A(t)=A_0{e}^{\left(-\lambda t\right)}$

Where, $A_0$ = initial activity

  $A(t)$ = activity at time $t$

  $\lambda$ = rate of disintegration

Calculation:

The rate of disintegration is calculated as,

  $\begin{array}{l}\lambda =\frac{0.693}{t_{1/2}}\\ \lambda =\frac{0.693}{12.3}\\ \lambda =0.05634{\text{ years}}^{-1}\end{array}$

The brightness of fluorescent glow is proportion to activity of tritium.

The activity of tritium after 6 years is,

  $\begin{array}{l}A(t)=A_0{e}^{\left(-\lambda t\right)}\\ \frac{A(t)}{A_0}=e^{\left(-0.05634\times 6\right)}\\ \frac{A(t=6\text{ yeras})}{A_0}=0.7132\end{array}$

Conclusion:

The value of ratio is $0.7132$ .