A nucleus undergoes a series of decay according to the scheme `A overset(alpha)(rarr) B overset(beta)(rarr)C overset(alpha)(rarr)D overset(gamma)(rarr) E`
Atomic number and mass number of E are 69 and 172

Options

A. Atomic number of A is 72
B. Mass number of B is 176
C. Atomic number of D is 69
D. Atomic number of C is 69

Answer

To solve the problem step by step, we need to analyze the decay chain and the changes in atomic number (Z) and mass number (A) at each step.

### Step 1: Understand the decay process
The decay process is as follows:
1. Nucleus A decays into nucleus B by emitting an alpha particle.
2. Nucleus B decays into nucleus C by emitting a beta particle.
3. Nucleus C decays into nucleus D by emitting another alpha particle.
4. Nucleus D emits a gamma particle to become nucleus E.

### Step 2: Define the initial parameters
Let:
- Atomic number of A = Z
- Mass number of A = A

### Step 3: First decay (A to B)
In the first decay (A to B), an alpha particle (which has an atomic number of 2 and mass number of 4) is emitted:
- Atomic number of B = Z - 2
- Mass number of B = A - 4

### Step 4: Second decay (B to C)
In the second decay (B to C), a beta particle is emitted. In beta decay, the atomic number increases by 1 while the mass number remains unchanged:
- Atomic number of C = (Z - 2) + 1 = Z - 1
- Mass number of C = A - 4

### Step 5: Third decay (C to D)
In the third decay (C to D), another alpha particle is emitted:
- Atomic number of D = (Z - 1) - 2 = Z - 3
- Mass number of D = (A - 4) - 4 = A - 8

### Step 6: Fourth decay (D to E)
In the fourth decay (D to E), a gamma particle is emitted. Gamma decay does not change the atomic number or mass number:
- Atomic number of E = Z - 3
- Mass number of E = A - 8

### Step 7: Given information about E
We know that:
- Atomic number of E = 69
- Mass number of E = 172

### Step 8: Set up equations
From the information about E, we can set up the following equations:
1. \( Z - 3 = 69 \)
2. \( A - 8 = 172 \)

### Step 9: Solve for Z and A
From equation 1:
- \( Z = 69 + 3 = 72 \)

From equation 2:
- \( A = 172 + 8 = 180 \)

### Step 10: Summary of results
Now we have:
- Atomic number of A = 72
- Mass number of A = 180

### Step 11: Determine properties of B, C, and D
Using the values of Z and A, we can find the properties of B, C, and D:
- For B:
- Atomic number = 72 - 2 = 70
- Mass number = 180 - 4 = 176
- For C:
- Atomic number = 70 + 1 = 71
- Mass number = 176 (remains unchanged)
- For D:
- Atomic number = 71 - 2 = 69
- Mass number = 176 - 4 = 172

### Step 12: Conclusion
Now we can summarize the findings:
- Atomic number of A = 72
- Atomic number of B = 70
- Atomic number of C = 71
- Atomic number of D = 69
- Atomic number of E = 69
- Mass number of A = 180
- Mass number of B = 176
- Mass number of C = 176
- Mass number of D = 172
- Mass number of E = 172

### Final Answer
The correct options based on the findings are:
1. Atomic number of A is 72 (Correct)
2. Atomic mass of B is 176 (Correct)
3. Atomic number of D is 69 (Correct)
4. Atomic number of C is 71 (Incorrect)