If a radioactive nucleus decay according to the following reaction
`._72X^180 oversetalphato X_1 oversetbetato X_2 oversetalphatoX_3 oversetgammato X_4` then the mass number and atomic number of `X_4` will be respectively

Options

A. 172,70
B. 172,69
C. 172,68
D. 171,69

Answer

To solve the problem, we need to analyze the decay process step by step, starting from the parent nucleus \( _{72}X^{180} \).

### Step 1: Initial Nucleus
The initial nucleus is given as \( _{72}X^{180} \). Here, the atomic number (Z) is 72 and the mass number (A) is 180.

### Step 2: First Decay - Alpha Decay
In the first decay, the nucleus undergoes alpha decay. In alpha decay:
- The mass number decreases by 4.
- The atomic number decreases by 2.

Calculating the new values:
- New mass number: \( 180 - 4 = 176 \)
- New atomic number: \( 72 - 2 = 70 \)

Thus, after the first decay, we have:
\[ _{70}X^{176} \]

### Step 3: Second Decay - Beta Decay
Next, the nucleus \( _{70}X^{176} \) undergoes beta decay. In beta decay:
- The mass number remains the same.
- The atomic number increases by 1.

Calculating the new values:
- Mass number remains: \( 176 \)
- New atomic number: \( 70 + 1 = 71 \)

Thus, after the second decay, we have:
\[ _{71}X^{176} \]

### Step 4: Third Decay - Alpha Decay
Now, the nucleus \( _{71}X^{176} \) undergoes another alpha decay. Again, in alpha decay:
- The mass number decreases by 4.
- The atomic number decreases by 2.

Calculating the new values:
- New mass number: \( 176 - 4 = 172 \)
- New atomic number: \( 71 - 2 = 69 \)

Thus, after the third decay, we have:
\[ _{69}X^{172} \]

### Step 5: Fourth Decay - Gamma Decay
Finally, the nucleus \( _{69}X^{172} \) undergoes gamma decay. In gamma decay:
- The mass number remains the same.
- The atomic number remains the same.

Thus, after the fourth decay, we have:
\[ _{69}X^{172} \]

### Conclusion
The final nucleus \( X_4 \) has:
- Mass number \( A = 172 \)
- Atomic number \( Z = 69 \)

Therefore, the mass number and atomic number of \( X_4 \) are respectively \( 172 \) and \( 69 \).

### Final Answer
The mass number and atomic number of \( X_4 \) will be respectively \( 172 \) and \( 69 \).

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