24.28 P - Chemistry: Molecular Nature Of Matter And Change

Question

Question: In a natural decay series, how many $\mathit{\alpha }$ and ${\mathit{\beta }}^{\mathbf{-}}$ emissions per atom of uranium-235 result in an atom of lead-207?

Step-by-Step Solution

Answer

The number of alpha decays is seven.

The number of beta decays is four.

Step 1: alpha, and beta-decay

When a nuclide loses doubly ionised ${}_{\mathbf{2}}\mathbf{He}^{\mathbf{4}}$ nuclide then it is called alpha-decay.

When it loses ${}_{\mathbf{-}\mathbf{1}}{\mathbf{e}}^{\mathbf{0}}$ the atomic number of the element increases by one and it is called beta-decay.

Step 2: Calculation of the number of alpha decays.

We have a natural decay series of uranium-235 $\left({}_{92}{}^{235}\text{U}\right)$, with the final product being lead-207 $\left({}_{82}{}^{207} \text{Pb}\right)$.

The amount of alpha and beta emissions in this sequence must also be determined.

${}_2{}^4\text{He}$ is the alpha particle.

 ${}_1{}^{0}e$ for the beta particle

We know that atomic mass falls by 4 and atomic number decreases by 2, whereas atomic mass remains constant and atomic number increases by 1 in each alpha decay.

We'll start by calculating the number of alpha decays because atomic mass changes exclusively in alpha decay.

$\begin{array}{rcl}\text{No alpha decays}& =& \frac{\left(\text{ atomic mass U}\right)-\left(\text{ atomic mass Pb}\right)}{\text{ atomic mass alpha particle }}\\ & =& \frac{235-207}{4}\\ & =& 7\end{array}$

Therefore, number of alpha decays is 7.

Step 3: Calculation of the number of beta decays

After 7 alpha decays of uranium-235, we must now determine the atomic number of a product.

 $\begin{array}{rcl}\text{Z}& =& \text{ atomic number of} \text{U}-7\times \text{ atomic number alpha particle}\\ & =& 92-7\times 2\\ & =& 78\end{array}$

 $\begin{array}{rcl}\text{No beta decays}& =& \frac{\text{Z}-\text{ atomic number Pb}}{\text{ atomic number beta particle }}\\ & =& \frac{78-82}{-1}\\ & =& 4\end{array}$

Therefore, the number of beta decay is 4.