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Chapter 10: Problem 28

Explain why a hydrogen atom with its electron in the ground state cannot absorb a photon of just any energy when making a transition to the second excited state \((n=3)\).

Short Answer

Expert verified
Answer: A hydrogen atom cannot absorb a photon of just any energy because the energy levels of a hydrogen atom are quantized, meaning they can only take specific discrete values. The energy conservation principle states that the energy of a photon absorbed by an electron must exactly match the energy difference between the initial and final energy levels of the electron. In this case, the electron can only absorb a photon with an energy equal to the energy difference between the ground state (n=1) and the second excited state (n=3), which is 12.09 eV. Only a photon with this energy can be absorbed by the electron to cause the transition to the second excited state.

Step by step solution

01

Recall the energy levels of a hydrogen atom

We need to recall that the energy levels of a hydrogen atom are given by the formula: \(E_n = -\frac{13.6~\text{eV}}{n^2}\) Where \(E_n\) is the energy of the \(n^{th}\) energy level (in electron-volts, eV), and \(n\) is the principal quantum number.
02

Calculate the energy difference between the ground state and the second excited state

Using the formula above, we can calculate the energy difference between the ground state \((n=1)\) and the second excited state \((n=3)\): \(E_1 = -\frac{13.6~\text{eV}}{1^2} = -13.6~\text{eV}\) (Ground State) \(E_3 = -\frac{13.6~\text{eV}}{3^2} = -\frac{13.6~\text{eV}}{9} = -1.51~\text{eV}\) (Second Excited State) Now, we can find the energy difference between these two states: \(\Delta E = E_3 - E_1 = (-1.51~\text{eV}) - (-13.6~\text{eV}) = 12.09~\text{eV}\)
03

Discuss the energy conservation principle and quantized nature of energy levels

According to the energy conservation principle, the energy of a photon absorbed by an electron must exactly match the energy difference between the initial and final energy levels of the electron. In other words, for an electron to transition from the ground state \((n=1)\) to the second excited state \((n=3)\), the photon must have an energy of \(12.09~\text{eV}\). The energy levels of a hydrogen atom, like other quantum systems, are quantized, meaning that they can only take specific discrete values. This implies that an electron in a hydrogen atom can only absorb or emit photons with energies that correspond exactly to the energy difference between the quantized energy levels.
04

Explain why a hydrogen atom cannot absorb any photon energy

Given the quantized nature of the energy levels of a hydrogen atom and the energy conservation principle, a hydrogen atom with its electron in the ground state \((n=1)\) cannot absorb a photon of just any energy when making a transition to the second excited state \((n=3)\). The electron can only absorb a photon with an energy equal to the energy difference between the two states, in this case, \(12.09~\text{eV}\). So, only a photon with this energy can be absorbed by the electron to cause the transition to the second excited state.

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