Consider a spectral line resulting `n = 5` to `n = 1`, in the atom and loss given below. The shortest wavelength is produced by

Options

A. helium atom
B. deuterium atom
C. singly ionization helium
D. ten times ionized sodium atom

Explanation

To solve the problem of determining which atom produces the shortest wavelength for the transition from \( n = 5 \) to \( n = 1 \), we can follow these steps:

### Step 1: Understand the Transition
The transition from \( n = 5 \) to \( n = 1 \) indicates that an electron is moving from a higher energy level (5) to a lower energy level (1). This transition will emit a photon, and the wavelength of that photon can be calculated using the Rydberg formula.

### Step 2: Use the Rydberg Formula
The Rydberg formula for the wavelength (\( \lambda \)) of the emitted photon during a transition is given by:

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

where:
- \( R \) is the Rydberg constant,
- \( Z \) is the atomic number of the atom,
- \( n_1 \) is the lower energy level (1 in this case),
- \( n_2 \) is the higher energy level (5 in this case).

### Step 3: Identify the Variables
In our case:
- \( n_1 = 1 \)
- \( n_2 = 5 \)

### Step 4: Calculate the Wavelength
Substituting the values into the Rydberg formula gives:

\[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{1^2} - \frac{1}{5^2} \right) = RZ^2 \left( 1 - \frac{1}{25} \right) = RZ^2 \left( \frac{24}{25} \right) \]

### Step 5: Determine the Effect of Atomic Number \( Z \)
The wavelength \( \lambda \) is inversely proportional to \( Z^2 \). This means that as \( Z \) increases, \( \lambda \) decreases, leading to shorter wavelengths.

### Step 6: Identify the Atom with Maximum \( Z \)
To find the shortest wavelength, we need to choose the atom with the highest atomic number \( Z \). In this case, the problem indicates that we are considering a 10 times ionized sodium atom, which has a maximum \( Z \).

### Conclusion
Thus, the atom that produces the shortest wavelength for the transition from \( n = 5 \) to \( n = 1 \) is the 10 times ionized sodium atom.

### Final Answer
The shortest wavelength is produced by the 10 times ionized sodium atom.

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