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

Let's analyze the given statements one by one to determine their validity: 

1. Statement I: The Balmer spectral line for the H atom with the lowest energy is located at \(\frac{5}{36} R_H \, \text{cm}^{-1}\).
   • The Balmer series describes the spectral line emissions of the hydrogen atom as an electron transitions to the second energy level (\(n=2\)) from a higher level (\(n>2\)).
   • The wavelength of the spectral lines is given by the Rydberg formula: 

\[\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{n^2} \right)\]

• For the lowest energy transition in the Balmer series, \(n=3\). Substituting the values we get: 

\[\frac{1}{\lambda} = R_H \left( \frac{1}{4} - \frac{1}{9} \right) = R_H \left( \frac{5}{36} \right)\]

• This calculation confirms that Statement I is true.

1. Statement II: When the temperature of a blackbody increases, the maxima of the curve (intensity and wavelength) shifts to shorter wavelength.
   • This phenomenon is described by Wien's Displacement Law, which states that the wavelength at which the emission of a blackbody spectrum is maximized is inversely proportional to the temperature (\(\lambda_{\text{max}} \propto \frac{1}{T}\)).
   • As the temperature of the blackbody increases, the peak of the emission curve shifts to shorter wavelengths (higher frequency), supporting the claim made in Statement II.
   • This reasoning confirms that Statement II is true.

Therefore, based on the analysis above, both Statement I and Statement II are true.