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 Description The nucleus \( X \) transforms through a series of steps, changing its atomic and mass numbers, decaying into \( X_1, X_2, X_3, \) and finally \( X_4 \). Each decay step follows these rules: 1.
Alpha decay: Mass number decreases by 4, atomic number decreases by 2. 2.
Beta decay: Mass number remains constant, atomic number increases by 1. Step 2: Initial Conditions We know \( X_4 \) has: - Mass number \( A_4 = 172 \), - Atomic number \( Z_4 = 69 \). Step 3: Working Backwards To find \( X \)'s properties, we reverse the decay process. 1. \( X_4 \) to \( X_3 \): - \( X_4 \) formed via beta decay. Atomic number decreases by 1, mass number remains the same. - \( X_3 \) has \( Z_3 = 69 - 1 = 68 \) and \( A_3 = 172 \). 2. \( X_3 \) to \( X_2 \): - \( X_3 \) formed via alpha decay. Atomic number decreases by 2, mass number decreases by 4. - \( X_2 \) has \( Z_2 = 68 - 2 = 66 \) and \( A_2 = 172 - 4 = 168 \). 3. \( X_2 \) to \( X_1 \): - \( X_2 \) formed via alpha decay. Atomic number decreases by 2, mass number decreases by 4. - \( X_1 \) has \( Z_1 = 66 - 2 = 64 \) and \( A_1 = 168 - 4 = 164 \). 4. \( X_1 \) to \( X \): - \( X_1 \) formed via beta decay. Atomic number increases by 1, mass number remains the same. - \( X \) has \( Z = 64 + 1 = 65 \) and \( A = 164 \). Step 4: Conclusion The mass and atomic numbers for \( X \) are: - \( A = 68 \), - \( Z = 172 \). Therefore, the answer is: \[ \boxed{(C)} \, 68, 172 \]