In which of the following system will the wavelength corresponding to `n = 2 to n= 1`be minimum ?

Options

A. Hydrogen atom
B. Deuterium atom
C. single ionized helium
D. Doubly ionized lithium

Answer

To solve the problem of determining in which system the wavelength corresponding to the transition from n = 2 to n = 1 will be minimum, we can follow these steps:

### Step 1: Understand the formula for wavelength
The wavelength (λ) of the emitted photon during a transition between energy levels in an atom can be expressed using the formula:

\[ \frac{1}{\lambda} = R \cdot Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

where:
- \( R \) is the Rydberg constant (approximately \( 1.097 \times 10^7 \, \text{m}^{-1} \)),
- \( Z \) is the atomic number,
- \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and upper energy levels, respectively.

### Step 2: Identify the values of n
In this case, we have:
- \( n_1 = 1 \)
- \( n_2 = 2 \)

### Step 3: Substitute the values into the formula
Substituting \( n_1 \) and \( n_2 \) into the formula, we get:

\[ \frac{1}{\lambda} = R \cdot Z^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

This simplifies to:

\[ \frac{1}{\lambda} = R \cdot Z^2 \left( 1 - \frac{1}{4} \right) = R \cdot Z^2 \cdot \frac{3}{4} \]

### Step 4: Express λ in terms of Z
Rearranging the equation gives us:

\[ \lambda = \frac{4}{3R \cdot Z^2} \]

### Step 5: Analyze the relationship between λ and Z
From the equation, we can see that λ is inversely proportional to \( Z^2 \). This means that as \( Z \) increases, \( \lambda \) decreases. Therefore, to minimize λ, we need to maximize \( Z \).

### Step 6: Determine the maximum Z from the options
If we have multiple systems (options) with different atomic numbers \( Z \), we should select the one with the highest \( Z \) value.

### Conclusion
Thus, the system with the highest atomic number \( Z \) will yield the minimum wavelength for the transition from \( n = 2 \) to \( n = 1 \). According to the problem, if option D has \( Z = 3 \) and is the highest among the given options, then the correct answer is:

**Option D.**

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