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Chapter 18: Problem 2532

If the binding energy of electron in a hydrogen atom is \(3.6 \mathrm{eV}\), the energy required to remove the electron from the second excited state of \(\mathrm{Li}^{\mathrm{H}+}\) is A) \(13.6 \mathrm{eV}\) (B) \(3.4 \mathrm{eV}\) C) \(30.6 \mathrm{eV}\) (D) \(122.4 \mathrm{eV}\)

Short Answer

Expert verified
The energy required to remove the electron from the second excited state of LiH⁺ is approximately 3.4 eV. The correct answer is (B) \(3.4 \mathrm{eV}\).

Step by step solution

01

Understand the energy levels formula for the hydrogen atom

The energy levels of a hydrogen atom can be calculated using the following formula: \[E_n = -\frac{13.6 \text{ eV}}{n^2}\] where \(E_n\) is the energy of the nth level and \(n\) is the principal quantum number.
02

Find the energy level of the hydrogen atom

The binding energy of the electron in the hydrogen atom is given as 3.6 eV. We can use the energy levels formula to find the principal quantum number n: \[3.6 = -\frac{13.6}{n^2}\] By solving for n, we get: \[n^2 = \frac{13.6}{3.6}\] \[n^2 \approx 3.78\] Since \(n\) must be an integer, we can conclude that \(n = 2\).
03

Identify the second excited state of LiH⁺

The second excited state of LiH⁺ corresponds to the third energy level (n = 3), because the ground state is n = 1, and the first excited state is n = 2.
04

Calculate the energy difference between the initial and final energy levels

We'll now find the energy difference between the initial (n = 2) and final (n = 3) energy levels using the energy levels formula: \[\Delta E = E_3 - E_2\] \[\Delta E = -\frac{13.6}{3^2} + \frac{13.6}{2^2}\] \[\Delta E = -\frac{13.6}{9} + \frac{13.6}{4}\] \[\Delta E \approx 3.4 \text{ eV}\]
05

Find the energy required to remove the electron

The energy required to remove the electron from the second excited state of LiH⁺ is equal to the energy difference between the initial and final energy levels. Therefore, the energy required is approximately 3.4 eV. The correct answer is (B) \(3.4 \mathrm{eV}\).

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