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Chapter 4: Problem 71

What would happen to the electron in a groundstate hydrogen atom if the atom were given \(5.1 \mathrm{eV}\) of energy?

Short Answer

Expert verified
Since 5.1 eV of energy is not sufficient to excite the electron from the ground state to the first excited state (which requires 10.2 eV), the electron will remain in the ground state. The excess energy will be emitted in the form of radiation.

Step by step solution

01

Recall the energy levels of hydrogen atom

The energy levels of hydrogen atom can be calculated using the following formula: \[E_n = -\dfrac{13.6}{n^2} eV\] where \(E_n\) is the energy level of hydrogen atom when the electron is in the nth orbit, and n is the principal quantum number, which takes integer values from 1 to infinity. For ground state hydrogen atom, n = 1 and the electron is in the lowest energy level.
02

Calculate the energy levels for n = 1 and n = 2

Now, let's calculate the energy levels for n = 1 and n = 2: - For ground state (n = 1): \[E_1 = -\dfrac{13.6}{1^2} eV = -13.6 eV\] - For first excited state (n = 2): \[E_2 = -\dfrac{13.6}{2^2} eV = -3.4 eV\]
03

Calculate the energy difference between the two levels

Let's find the energy difference between the ground state and the first excited state: \[\Delta E = E_2 - E_1 = -3.4 eV - (-13.6 eV) = 10.2 eV\]
04

Compare the energy difference with the given energy

The energy difference between the ground state and the first excited state is 10.2 eV. The given energy is 5.1 eV, which is less than the energy difference between these two levels.
05

Conclusion

Since the given energy (5.1 eV) is not sufficient to excite the electron from the ground state to the first excited state, the electron will remain in the ground state. The excess energy will be emitted in the form of radiation, and the atom will return to its ground state.

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Most popular questions from this chapter

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