Question

Given below are two statements: 
Statement (I) : The dimensions of Planck’s constant and angular momentum are same. 
Statement (II) : In Bohr’s model, electron revolves around the nucleus in those orbits for which angular momentum is an integral multiple of Planck’s constant. 
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both Statement I and Statement II are correct
• B. Statement I is incorrect but Statement II is correct
• C. Statement I is correct but Statement II is incorrect
• D. Both Statement I and Statement II are incorrect

Hint:

Planck’s constant \( h \) and angular momentum \( L \) have the same dimensional formula. Also, in Bohr’s model, angular momentum is quantized in integer multiples of \( h/2\pi \).

Answer 1

The Correct Option is C

This problem is solved by analyzing each statement individually:

Statement I: "The dimensions of Planck’s constant and angular momentum are identical."

Planck's constant \( h \) has the dimensional formula \([M^1L^2T^{-1}]\). Angular momentum \( L \), calculated as \( mvr \) (where \( m \) is mass, \( v \) is velocity, and \( r \) is radius), also has the dimensional formula \([M^1L^2T^{-1}]\).

Therefore, Planck's constant and angular momentum share the same dimensions. Statement I is accurate.

Statement II: "In Bohr’s model, electrons orbit the nucleus in specific orbits where their angular momentum is an integral multiple of Planck's constant."

Bohr’s model states that the electron's angular momentum \( L \) is quantized according to the formula:

\(L = n\frac{h}{2\pi}\) where \( n \) is a positive integer.

This equation indicates that angular momentum is an integral multiple of \(\frac{h}{2\pi}\) (the reduced Planck's constant \( \hbar \)), not Planck's constant \( h \) itself. Consequently, Statement II is inaccurate because it omits the \(\frac{1}{2\pi}\) factor when relating angular momentum to Planck's constant.

Based on this analysis:

• Statement I is correct.
• Statement II is incorrect.

The accurate conclusion is: Statement I is correct, but Statement II is incorrect.