Question

Given below are two statements:
Statement I: The Balmer spectral line for H atom with lowest energy is located at \( \frac{5}{36} R_H \, \text{cm}^{-1} \).
(\( R_H \) = Rydberg constant)
Statement II: When the temperature of blackbody increases, the maxima of the curve (intensity and wavelength) shifts to shorter wavelength.
In the light of the above statements, choose the correct answer from the options given below:

• A. Statement I is true but Statement II is false
• B. Statement I is false but Statement II is true
• C. Both Statement I and Statement II are true
• D. Both Statement I and Statement II are false

Answer 1

The Correct Option is C

Statement I: The Balmer series corresponds to electronic transitions where the electron falls to the n=2 energy level. The lowest energy transition in the Balmer series is from n=3 to n=2. The wavenumber ($\tilde{\nu}$) for this transition can be calculated using the Rydberg formula:
\[ \tilde{\nu} = R_\text{H} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For the lowest energy Balmer transition ($n_1$=2, $n_2$=3):
\[ \tilde{\nu} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{9} \right) = R_H \left( \frac{9 - 4}{36} \right) = \frac{5}{36} R_H \]
The wavenumber is indeed $\frac{5}{36} R_H$ cm$^{-1}$. Thus, statement I is true.
Statement II: Wien's displacement law states that the wavelength of maximum intensity for blackbody radiation is inversely proportional to its temperature. As the temperature increases, the wavelength of maximum intensity shifts to shorter wavelengths. Thus, statement II is true.