In a mixture of sample of H-atoms and He ions, electrons in all the H-atoms and `He^+` ions are present in n = `4^"th"` state. Then, find maximum number of different spectral lines obtained when all the electrons make transition from n=4 upto ground state

Answer

To solve the problem of finding the maximum number of different spectral lines obtained when all the electrons in a mixture of hydrogen atoms and helium ions make transitions from the n=4 state to the ground state (n=1), we can follow these steps:

### Step 1: Understand the transitions
Electrons can transition from a higher energy level (n=4) to lower energy levels (n=3, n=2, n=1). Each transition corresponds to the emission of a spectral line.

### Step 2: Use the formula for spectral lines
The formula to calculate the maximum number of different spectral lines when electrons transition from a higher energy level (n_initial) to a lower energy level (n_final) is given by:

\[ \text{Number of spectral lines} = \frac{(n_f - n_i)(n_f - n_i + 1)}{2} \]

Where:
- \( n_f \) = final energy level (ground state, n=1)
- \( n_i \) = initial energy level (n=4)

### Step 3: Substitute the values into the formula
In this case, we have:
- \( n_i = 4 \)
- \( n_f = 1 \)

Substituting these values into the formula:

\[ \text{Number of spectral lines} = \frac{(4 - 1)(4 - 1 + 1)}{2} \]

### Step 4: Simplify the expression
Calculating the values inside the parentheses:

\[ = \frac{(3)(4)}{2} = \frac{12}{2} = 6 \]

This means that for hydrogen, there are 6 different spectral lines.

### Step 5: Consider both hydrogen and helium ions
Since the question states that there is a mixture of hydrogen atoms and helium ions (He⁺), and both will have the same transitions from n=4 to n=1, we need to calculate the total spectral lines.

For helium ions (He⁺), the same calculation applies, yielding another 6 spectral lines.

### Step 6: Calculate the total number of spectral lines
Now we add the spectral lines from both hydrogen and helium:

\[ \text{Total spectral lines} = 6 \text{ (from H)} + 6 \text{ (from He⁺)} = 12 \]

### Final Answer
Thus, the maximum number of different spectral lines obtained when all the electrons make transitions from n=4 to the ground state is **12**.

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