Question

The transition the state \(\mathrm{n}=4\) to \(\mathrm{n}=1\) in a hydrogen like atom results in ultraviolet radiation. Infrared radiation will be obtained in the transition form (A) \(3 \rightarrow 2\) (B) \(5 \rightarrow 4\) (C) \(4 \rightarrow 2\) (D) \(2 \rightarrow 1\)

Short answer

The correct transition that results in infrared radiation is (B) \(5 \rightarrow 4\), as it has a wavelength of approximately \(1875 \, \mathrm{nm}\), which falls within the infrared spectrum range.

Step-by-step solution

The Rydberg formula is given by \( \frac{1}{\lambda}=R_{\text{H}} \left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right) \), where \(R_{\text{H}}\) is the Rydberg constant for hydrogen (\(1.097373\times10^7\, \text{m}^{-1}\)), \(n_1\) and \(n_2\) are the principal quantum numbers of the energy levels involved in the transition, and \( \lambda \) is the wavelength of the emitted radiation. The lower energy level has quantum number \(n_1\), and the higher energy level has quantum number \(n_2\). #Step 2: Calculate the wavelength for each transition#

We will now calculate the wavelength for each given transition using the Rydberg formula: (A) For the transition: \(3 \rightarrow 2\) \[ \frac{1}{\lambda_{A}} = R_{\text{H}} \left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right) \] (B) For the transition: \(5 \rightarrow 4\) \[ \frac{1}{\lambda_{B}} = R_{\text{H}} \left(\frac{1}{4^{2}}-\frac{1}{5^{2}}\right) \] (C) For the transition: \(4 \rightarrow 2\) \[ \frac{1}{\lambda_{C}} = R_{\text{H}} \left(\frac{1}{2^{2}}-\frac{1}{4^{2}}\right) \] (D) For the transition: \(2 \rightarrow 1\) \[ \frac{1}{\lambda_{D}} = R_{\text{H}} \left(\frac{1}{1^{2}}-\frac{1}{2^{2}}\right) \] #Step 3: Determine the wavelengths and check the infrared spectrum range#

Infrared radiation has wavelengths in the range of \(700 \, \mathrm{nm}\) to \(1 \, \mathrm{mm}\). After solving the equations in Step 2 for the wavelength, we can compare the values to the infrared range: (A) \( \lambda_{A} \approx 656 \, \mathrm{nm} \) (this is in the visible light spectrum) (B) \( \lambda_{B} \approx 1875 \, \mathrm{nm} \) (this is in the infrared spectrum) (C) \( \lambda_{C} \approx 486 \, \mathrm{nm} \) (this is in the visible light spectrum) (D) \( \lambda_{D} \approx 121 \, \mathrm{nm} \) (this is in the ultraviolet spectrum) #Step 4: Choose the correct option#

Among the given transitions, only the transition from \(5 \rightarrow 4\) results in a wavelength in the infrared spectrum. Therefore, the correct option is (B).