Question

A hydrogen atom in its ground state absorbs 10.2 eV of energy. What is the orbital angular momentum increased by?

• A. 2\sqrt{3}\hbar
• B. 4\sqrt{3}\hbar
• C. \sqrt{5}\hbar
• D. \sqrt{3}\hbar

Answer 1

None of the above. The question is flawed.

Answer 2

1. Energy Calculation: A hydrogen atom in its ground state (n=1) has an energy of $E_1 = -13.6$ eV. Upon absorbing 10.2 eV, its final energy becomes $E_f = -13.6 + 10.2 = -3.4$ eV. This corresponds to the n=2 energy level ($E_2 = -13.6/2^2 = -3.4$ eV). So, the electron transitions from n=1 to n=2.

2. Orbital Angular Momentum:

   • In the ground state (n=1), the only possible orbital angular momentum quantum number is l=0. The orbital angular momentum is $L_1 = \sqrt{l(l+1)}\hbar = \sqrt{0(0+1)}\hbar = 0$.
   • In the n=2 state, possible orbital angular momentum quantum numbers are l=0 (2s orbital) and l=1 (2p orbital).
     • If the electron transitions to a 2s state (l=0), $L_{2s} = \sqrt{0(0+1)}\hbar = 0$. The increase in angular momentum is $0 - 0 = 0$.
     • If the electron transitions to a 2p state (l=1), $L_{2p} = \sqrt{1(1+1)}\hbar = \sqrt{2}\hbar$. The increase in angular momentum is $\sqrt{2}\hbar - 0 = \sqrt{2}\hbar$.

3. Comparison with Options: The calculated increases are 0 or $\sqrt{2}\hbar$. None of the given options ($2\sqrt{3}\hbar$, $4\sqrt{3}\hbar$, $\sqrt{5}\hbar$, $\sqrt{3}\hbar$) match these values. Furthermore, some options like $\sqrt{3}\hbar$ and $\sqrt{5}\hbar$ do not correspond to valid orbital angular momentum magnitudes $\sqrt{l(l+1)}\hbar$ for any integer value of $l$. The option $2\sqrt{3}\hbar$ corresponds to $l=3$, which is not possible for an electron in the n=2 state ($l_{max} = n-1 = 2-1 = 1$).

4. Conclusion: Based on standard quantum mechanics, this question's options are inconsistent with the problem statement. The question is flawed.

However, if we consider the approach from a similar question where the Bohr model's angular momentum quantization ($L_n = n\hbar$) was used:

• $L_1 = 1\hbar$
• $L_2 = 2\hbar$
• Increase $\Delta L = L_2 - L_1 = 2\hbar - 1\hbar = \hbar$.

Even with the Bohr model, $\hbar$ is not among the given options.

Given the mandatory requirement to select an option, and the logical inconsistencies, this question is problematic. Without further clarification or context, selecting a correct option is not possible through derivation from standard physics principles.