The transition from the state `n = 4` to `n = 3` in a hydrogen-like atom results in ultraviolet radiation. Infared radiation will be obtained in the transition

Options

A. `2 rarr 1`
B. `3 rarr 2`
C. `4 rarr 2`
D. `5 rarr 4`

Explanation

To solve the question regarding the transitions in a hydrogen-like atom and the types of radiation produced, we will follow these steps:

### Step 1: Understand the Energy Levels
In a hydrogen-like atom, the energy levels are quantized and are given by the formula:
\[ E_n = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{n^2} \]
where \( Z \) is the atomic number and \( n \) is the principal quantum number.

### Step 2: Identify the Transition
The transition mentioned is from \( n = 4 \) to \( n = 3 \). We need to calculate the energy difference between these two levels to determine the type of radiation emitted.

### Step 3: Calculate Energy for the Transition
Using the energy level formula:
- For \( n = 4 \):
\[ E_4 = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{4^2} = -\frac{Z^2 \cdot 13.6}{16} \]
- For \( n = 3 \):
\[ E_3 = -\frac{Z^2 \cdot 13.6 \, \text{eV}}{3^2} = -\frac{Z^2 \cdot 13.6}{9} \]

Now, calculate the energy difference \( \Delta E \):
\[ \Delta E = E_3 - E_4 = \left(-\frac{Z^2 \cdot 13.6}{9}\right) - \left(-\frac{Z^2 \cdot 13.6}{16}\right) \]
\[ \Delta E = Z^2 \cdot 13.6 \left( \frac{1}{16} - \frac{1}{9} \right) \]
Finding a common denominator (144):
\[ \Delta E = Z^2 \cdot 13.6 \left( \frac{9 - 16}{144} \right) = Z^2 \cdot 13.6 \left( -\frac{7}{144} \right) \]

### Step 4: Determine the Type of Radiation
The energy difference \( \Delta E \) corresponds to the energy of the emitted photon. The type of radiation can be determined by the wavelength or frequency of the emitted photon using the relation:
\[ E = h \cdot f = \frac{h \cdot c}{\lambda} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

- **Ultraviolet Radiation**: Typically corresponds to higher energy transitions (e.g., transitions to lower energy levels).
- **Infrared Radiation**: Corresponds to lower energy transitions (e.g., transitions from \( n = 3 \) to \( n = 2 \) or \( n = 2 \) to \( n = 1 \)).

### Step 5: Identify the Infrared Transition
For infrared radiation, we look for transitions that involve lower energy differences. The transition from \( n = 3 \) to \( n = 2 \) is an example of such a transition that would emit infrared radiation.

### Conclusion
Thus, the transition from \( n = 3 \) to \( n = 2 \) will result in infrared radiation.

### Final Answer
- Ultraviolet radiation is produced in the transition from \( n = 4 \) to \( n = 3 \).
- Infrared radiation is produced in the transition from \( n = 3 \) to \( n = 2 \).