Unit 15: Modern Physics
Weightage : 15–18%
According to the Bohr theory of the hydrogen atom, electrons starting in the 4th energy level and eventually ending up in the ground state could produce a total of how many lines in the hydrogen spectrum?
Correct Answer: \( \boxed{\mathrm{D}} \)
In the Bohr model, each spectral line corresponds to a transition between two different energy levels. The total number of possible emission lines from an electron initially in the \(n^{\mathrm{th}}\) energy level is
\( N=\dfrac{n(n-1)}{2} \)
For \(n=4\),
\( N=\dfrac{4(4-1)}{2}=\dfrac{12}{2}=6 \)
The six possible transitions are:
\(4\rightarrow3,\;4\rightarrow2,\;4\rightarrow1,\;3\rightarrow2,\;3\rightarrow1,\;2\rightarrow1\)
Each transition emits a photon of a different energy, producing six distinct spectral lines.
Therefore, the correct answer is (D).
According to the Bohr theory of the hydrogen atom, electrons starting in the 4th energy level and eventually ending up in the ground state could produce a total of how many lines in the hydrogen spectrum?
Correct Answer: \( \boxed{\mathrm{D}} \)
In the Bohr model, each transition between two energy levels produces one unique spectral line.
The total number of possible emission lines from an electron initially in the \(n^{\mathrm{th}}\) energy level is
\( N=\dfrac{n(n-1)}{2} \)
For \(n=4\),
\( N=\dfrac{4(4-1)}{2}=\dfrac{12}{2}=6 \)
The six possible transitions are:
\(4\rightarrow3,\;4\rightarrow2,\;4\rightarrow1,\;3\rightarrow2,\;3\rightarrow1,\;2\rightarrow1\)
These transitions produce six distinct photon energies and therefore six spectral lines.
Therefore, the correct answer is (D).
In the Bohr model of the atom, the postulate stating that the orbital angular momentum of the electron is quantized can be interpreted in which of the following ways?
Correct Answer: \( \boxed{\mathrm{A}} \)
In the Bohr model, the electron’s orbital angular momentum is quantized:
\(L=mvr=n\hbar=\dfrac{nh}{2\pi}\)
where \(n=1,2,3,\ldots\) is the principal quantum number.
According to de Broglie’s hypothesis, electrons have wave-like properties with wavelength
\(\lambda=\dfrac{h}{mv}\)
For a stable orbit, the circumference of the orbit must contain an integer number of wavelengths:
\(2\pi r=n\lambda\)
This condition produces constructive interference of the electron wave and leads directly to the quantization of angular momentum.
Therefore, the quantization postulate can be interpreted as requiring an integral number of electron wavelengths to fit around the circular orbit.
Therefore, the correct answer is (A).
