In hydrogen and hydrogen-like atom , the ratio of `E_(4 n) - E_(2 n) and E_(2 n) - E_(n)` varies with atomic nimber `z` and principal quantum number`n` as

Options

A. `(Z^(2))/(N^(2))`
B. `(Z^(4))/(n^(4))`
C. `(Z)/(n)`
D. None of these

Answer

To solve the problem, we need to find the ratio of \( E_{4n} - E_{2n} \) and \( E_{2n} - E_n \) for hydrogen and hydrogen-like atoms. The energy levels of hydrogen-like atoms are given by the formula:

\[ E_n = -\frac{E_1}{n^2} \]

where \( E_1 \) is the energy of the ground state.

### Step 1: Calculate \( E_{4n} \), \( E_{2n} \), and \( E_n \)

Using the formula for energy levels:

- For \( n = 4n \):
\[ E_{4n} = -\frac{E_1}{(4n)^2} = -\frac{E_1}{16n^2} \]

- For \( n = 2n \):
\[ E_{2n} = -\frac{E_1}{(2n)^2} = -\frac{E_1}{4n^2} \]

- For \( n = n \):
\[ E_n = -\frac{E_1}{n^2} \]

### Step 2: Compute \( E_{4n} - E_{2n} \)

Now we can find \( E_{4n} - E_{2n} \):

\[ E_{4n} - E_{2n} = \left(-\frac{E_1}{16n^2}\right) - \left(-\frac{E_1}{4n^2}\right) \]

This simplifies to:

\[ E_{4n} - E_{2n} = -\frac{E_1}{16n^2} + \frac{E_1}{4n^2} \]

Finding a common denominator (which is \( 16n^2 \)):

\[ E_{4n} - E_{2n} = -\frac{E_1}{16n^2} + \frac{4E_1}{16n^2} = \frac{3E_1}{16n^2} \]

### Step 3: Compute \( E_{2n} - E_n \)

Next, we calculate \( E_{2n} - E_n \):

\[ E_{2n} - E_n = \left(-\frac{E_1}{4n^2}\right) - \left(-\frac{E_1}{n^2}\right) \]

This simplifies to:

\[ E_{2n} - E_n = -\frac{E_1}{4n^2} + \frac{E_1}{n^2} \]

Finding a common denominator (which is \( 4n^2 \)):

\[ E_{2n} - E_n = -\frac{E_1}{4n^2} + \frac{4E_1}{4n^2} = \frac{3E_1}{4n^2} \]

### Step 4: Calculate the ratio

Now we can find the ratio:

\[ \frac{E_{4n} - E_{2n}}{E_{2n} - E_n} = \frac{\frac{3E_1}{16n^2}}{\frac{3E_1}{4n^2}} \]

The \( 3E_1 \) and \( n^2 \) terms cancel out:

\[ \frac{E_{4n} - E_{2n}}{E_{2n} - E_n} = \frac{1/16}{1/4} = \frac{1}{16} \times \frac{4}{1} = \frac{4}{16} = \frac{1}{4} \]

### Conclusion

The ratio \( \frac{E_{4n} - E_{2n}}{E_{2n} - E_n} \) is a constant value of \( \frac{1}{4} \), which does not depend on the atomic number \( Z \) or the principal quantum number \( n \). Therefore, the correct answer is:

**Option D: None of this, that is no relation between z and n.**