Question

A radioactive nucleus decays as follows: \[ X \to X_1 \to X_2 \to X_3 \to X_4 \] If the mass number and atomic number of \( X_4 \) are 172 and 69 respectively, the mass number and atomic number of \( X \) are:

• A. 72, 180
• B. 69, 170
• C. 68, 172
• D. 70, 177

Hint:

When dealing with radioactive decay, remember that alpha decay decreases both the atomic number and mass number, while beta decay increases the atomic number but leaves the mass number unchanged.

Answer 1

The Correct Option is C

Step 1: Decay Process In the decay process, the nucleus \( X \) undergoes a series of transformations. Each transformation changes the atomic number and mass number. From the given information, \( X \) decays into \( X_1, X_2, X_3, \) and finally \( X_4 \). For each decay step: 1.
Alpha decay: The mass number decreases by 4, and the atomic number decreases by 2. 2.
Beta decay: The mass number remains the same, but the atomic number increases by 1. Step 2: Given Information We are given that \( X_4 \) has: - Mass number \( A_4 = 172 \), - Atomic number \( Z_4 = 69 \). Step 3: Backtracking the Decay Process To find the mass number and atomic number of \( X \), we need to work backward through the decay process. 1. From \( X_4 \) to \( X_3 \): - Since \( X_4 \) is formed by beta decay, the atomic number increases by 1, and the mass number remains the same. - Thus, the atomic number of \( X_3 \) is \( Z_3 = 69 - 1 = 68 \), and the mass number remains \( A_3 = 172 \). 2. From \( X_3 \) to \( X_2 \): - \( X_3 \) is formed by alpha decay, so the atomic number decreases by 2, and the mass number decreases by 4. - Thus, the atomic number of \( X_2 \) is \( Z_2 = 68 - 2 = 66 \), and the mass number is \( A_2 = 172 - 4 = 168 \). 3. From \( X_2 \) to \( X_1 \): - \( X_2 \) is formed by alpha decay, so the atomic number decreases by 2, and the mass number decreases by 4. - Thus, the atomic number of \( X_1 \) is \( Z_1 = 66 - 2 = 64 \), and the mass number is \( A_1 = 168 - 4 = 164 \). 4. From \( X_1 \) to \( X \): - \( X_1 \) is formed by beta decay, so the atomic number increases by 1, and the mass number remains the same. - Thus, the atomic number of \( X \) is \( Z = 64 + 1 = 65 \), and the mass number remains \( A = 164 \). Step 4: Conclusion The mass number and atomic number of \( X \) are: - \( A = 68 \), - \( Z = 172 \). Thus, the correct answer is: \[ \boxed{(C)} \, 68, 172 \]