Which of the following statements are correct?

Options

A. If angular momentum the Earth due to its motion around the Sun were quantized according to Bohr's ralation `L = nh//2 pi`, then the quantum number`n` would be the order of `10^(74)`
B. If element with principal quantum number `gt 4` were not allowed in nature , then the number number of possible element would be `64`.
C. Rydberg's constant varies with mass number of the elment.
D. The ratio of the wave number of `H_(a)` lines of Balmer series for hydrogen and that of `H_(a)` line of Balmer series for singly ionized helium is excatly`4`

Explanation

To determine which of the statements are correct, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement A**: "If the angular momentum of the earth due to its motion around the sun were quantized according to Bohr's relation, L = nH/2π, the quantum number would have been of the order of 10^74."
- We know from Bohr's model that angular momentum (L) can be expressed as L = nH/2π.
- For Earth, we can express angular momentum as L = mvr, where m is mass, v is velocity, and r is the radius of the orbit.
- By substituting the values of m, v, and r into the equation and solving for n, we find that n is approximately 10^74.
- **Conclusion**: This statement is **True**.

2. **Statement B**: "If the element with principal quantum number greater than 4 were allowed in nature, then the number of possible elements would be 64."
- The maximum number of electrons in a shell is given by the formula 2n².
- For n = 4, the maximum number of electrons is 2(4²) = 32.
- Therefore, if we consider n = 1, 2, 3, and 4, the total number of electrons would be:
- n=1: 2(1²) = 2
- n=2: 2(2²) = 8
- n=3: 2(3²) = 18
- n=4: 2(4²) = 32
- Total = 2 + 8 + 18 + 32 = 60.
- **Conclusion**: This statement is **False**.

3. **Statement C**: "Rydberg constant varies with the mass number of the element."
- The Rydberg constant (R) can be expressed in terms of the reduced mass (μ) of the electron and nucleus system.
- The formula for Rydberg constant involves the reduced mass, which depends on the mass of the nucleus and the electron.
- Therefore, as the mass number changes, the reduced mass changes, which in turn affects the Rydberg constant.
- **Conclusion**: This statement is **True**.

4. **Statement D**: "The ratio of the wave number of H-alpha lines for the Balmer series for hydrogen and H-alpha for the Balmer series of a singly ionized helium is exactly 4."
- The wave number (1/λ) is proportional to the Rydberg constant.
- Since the Rydberg constant changes with the mass of the nucleus, the ratio of the wave numbers will not be exactly 4 due to the different masses of hydrogen and helium.
- **Conclusion**: This statement is **False**.

### Final Summary of Statements:
- Statement A: True
- Statement B: False
- Statement C: True
- Statement D: False