Question

A radioactive nucleus decays as follows: \[ X \xrightarrow{\alpha} X_1 \xrightarrow{\beta^-} X_2 \xrightarrow{\alpha} X_3 \xrightarrow{\gamma} X_4 \] If the mass number and atomic number of \( X_4 \) are 172 and 69 respectively, then the atomic number and mass number of \( X \) are

• A. 72, 180
• B. 69, 170
• C. 68, 172
• D. 70, 172

Hint:

Decay problems: \begin{itemize} \item Work backward carefully. \item Track A and Z separately. \end{itemize}

Answer 1

The Correct Option is A

Concept: Decay rules: \begin{itemize} \item \( \alpha \): \( A-4,\; Z-2 \) \item \( \beta^- \): \( Z+1 \) \item \( \gamma \): no change \end{itemize} Step 1: {\color{red}Reverse steps from \( X_4 \).} Given: \[ A_4 = 172,\quad Z_4 = 69 \] Step 2: {\color{red}Undo gamma.} No change: \[ X_3: (172,69) \] Step 3: {\color{red}Undo alpha.} Add 4 and 2: \[ X_2: (176,71) \] Step 4: {\color{red}Undo beta minus.} Decrease Z by 1: \[ X_1: (176,70) \] Step 5: {\color{red}Undo alpha again.} \[ X: (180,72) \]