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 involves electronic transitions to the n=2 energy level. The lowest energy transition within this series occurs from n=3 to n=2. The wavenumber ($\tilde{u}$) for this transition is calculated using the Rydberg formula:
\[ \tilde{u} = R_\text{H} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
For the lowest energy Balmer transition ($n_1$=2, $n_2$=3):
\[ \tilde{u} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R_H \left( \frac{1}{4} - \frac{1}{9} \right) = R_H \left( \frac{9 - 4}{36} \right) = \frac{5}{36} R_H \]
The calculated wavenumber is $\frac{5}{36} R_H$ cm$^{-1}$, confirming statement I is true.
Statement II: Wien's displacement law establishes an inverse relationship between the wavelength of maximum intensity of blackbody radiation and its temperature. Consequently, an increase in temperature leads to a shift in the wavelength of maximum intensity towards shorter wavelengths. Thus, statement II is true.