Question

Question: The three naturally occurring isotopes of uranium are ${}^{\mathbf{234}}\mathbf{U}$ (half-life $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{years}$), ${}^{\mathbf{235}}\mathbf{U}$(half-life$\mathbf{7}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}\mathbf{}}\mathbf{years}$) and ${}^{\mathbf{238}}\mathbf{U}$( half-life$\mathbf{4}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{9}}\mathbf{}\mathbf{years}$). As time passes, will the average atomic mass of the uranium in a sample taken from nature increase, decrease, or remain constant?

Short answer

The average atomic mass increases with time.

Step-by-step solution

Calculation of decay constant of U234,U and U238235

The decay constant of ${}^{234}\mathrm{U}$is calculated below:

$\begin{array}{rcl}{\mathrm{k}}_{{}^{234}\mathrm{U}}& =& \frac{0.693}{{\mathrm{t}}_{1/2}}\\ & =& \frac{0.693}{2.5\times {10}^5}{\mathrm{years}}^{-1}\\ & =& 2.7\times {10}^{-6}{\mathrm{years}}^{-1}\end{array}$

The decay constant of ${}^{235}\mathrm{U}$is calculated below:

$\begin{array}{rcl}{\mathrm{k}}_{{}^{235}\mathrm{U}}& =& \frac{0.693}{{\mathrm{t}}_{1/2}}\\ & =& \frac{0.693}{7.0\times {10}^8}{\mathrm{years}}^{-1}\\ & =& 9.9\times {10}^{-10}{\mathrm{years}}^{-1}\end{array}$

The decay constant of ${}^{238}\mathrm{U}$is calculated below:

localid="1663343788570" $\begin{array}{rcl}{\mathrm{k}}_{{}^{238}\mathrm{U}}& =& \frac{0.693}{{\mathrm{t}}_{1/2}}\\ & =& \frac{0.693}{4.5\times {10}^5}{\mathrm{years}}^{-1}\\ & =& 1.54\times {10}^{-10}{\mathrm{years}}^{-1}\end{array}$

Therefore the decay constant of the three isotopes of uranium follows the order:

${\mathrm{k}}_{{}^{234}\mathrm{U}}>{\mathrm{k}}_{{}^{235}\mathrm{U}}>{\mathrm{k}}_{{}^{238}\mathrm{U}}$

Predicting the change in the average atomic mass

As we can see, that the decay constant is highest for the ${}^{234}\mathrm{U}$isotope and is lowest for ${}^{238}\mathrm{U}$isotope. Thus, we can say that the ${}^{234}\mathrm{U}$isotope will decay at a much faster rate than the ${}^{238}\mathrm{U}$isotope. Hence, with time, the average atomic mass should increase as the isotopes with lower atomic mass are decaying at a faster rate than isotopes of higher atomic masses.