Nuclear physics problem No.1

Chemistry Level 2

A sample of rock contains two isotopes of uranium $\displaystyle ^{235}_{92}U$ and $\displaystyle ^{238}_{92}U$ , with the ratio of $\displaystyle ^{235}_{92}U$ particles and $\displaystyle ^{238}_{92}U$ particles is $\displaystyle R = \frac{3}{100}$ . Calculate how long it will take so that $\displaystyle R' = \frac{7}{1000}$ , know that the half life periods of two isotopes $\displaystyle ^{235}_{92}U$ and $\displaystyle ^{238}_{92}U$ is $\displaystyle T_{235}=0.7 \times 10^{9}$ years and $\displaystyle T_{235}=4.5 \times 10^{9}$ years.

Answer comes in billion years , rounds up to 3 decimal places.

The answer is 1.740404571.

1 solution

F Zs
Jun 11, 2021

The amount of not-decayed particles is given by the

$n \times 2^{-\frac t{t_h}}$

exponential equation, where $n$ is the amount of particles at $t = 0$ , $t$ is the time passed and $t_h$ is the halving time of the particles.

By substituting data into this formula, we get to functions, one for each isotope:

$f_{235}(t) = 3 \times 2^{-\frac t{0.7 \cdot 10^9}}$ $f_{238}(t) = 100 \times 2^{-\frac t{4.5 \cdot 10^9}}$

From that, we can calculate the ratio of the two isotopes at any time point:

$R(t) = \frac{f_{235}(t)}{f_{238}(t)}$

All we need to do now is substitute $\frac7{1000}$ in the place of $R$ .

By solving the equation, we get that

$\boxed{t = 1740404571.50}$

Here's this explanation visualized on GeoGebra

Nuclear physics problem No.1

The answer is 1.740404571.

1 solution

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