Question

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

Short answer

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

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.

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): \[E_2 = -\dfrac{13.6}{2^2} eV = -3.4 eV\]

Calculate the energy difference between the two levels

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

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.

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.

Key concepts

Ground State

In the context of a hydrogen atom, the "ground state" refers to the lowest energy level where an electron resides, characterized by the principal quantum number, n, equal to 1. At this energy level, the electron is most stable, and it naturally tends to remain there unless persuaded to move by an external source of energy.
The energy associated with this ground state can be calculated using the formula:

• For n = 1, the energy, \(E_1\), is \(-\frac{13.6}{1^2} \, \text{eV} = -13.6 \, \text{eV}\).

This negative energy value implies that the electron is bound to the nucleus.
The ground state is significant because it represents the most energetically favorable position for an electron, where no energy is needed to maintain it there.
Furthermore, any energy added to the system must overcome this stability to elevate the electron to a higher energy state.

Excited State

An electron in a hydrogen atom can be moved to an "excited state" when it absorbs energy equivalent to the difference between its current energy level and a higher one. Once moved, the electron occupies a position with a principal quantum number greater than 1, with the first excited state corresponding to n = 2.
This shift in energy levels can be calculated:

• For n = 2, the energy, \(E_2\), is \(-\frac{13.6}{2^2} \, \text{eV} = -3.4 \, \text{eV}\).

Moving the electron to this state involves overcoming the bound energy difference. Although it's higher than the ground state, it is still not entirely free of the nucleus. Such shifts are temporary, and the electron consistently seeks a return to its ground state when left undisturbed.
During its return, energy in the form of radiation (often as light) is emitted, revealing fundamental atomic interactions.

Energy Difference

The "energy difference" between the ground and an excited state plays a crucial role in understanding electron transitions. In a hydrogen atom, moving an electron from the ground state (n=1) to the first excited state (n=2) requires overcoming a specific energy gap.
To find this difference, calculate:

• \(\Delta E = E_2 - E_1 = -3.4 \, \text{eV} - (-13.6 \, \text{eV}) = 10.2 \, \text{eV}\)

This 10.2 eV is the precise energy needed to promote the electron from its baseline position to the higher level.
In practical terms, if an electron does not receive enough energy—like the 5.1 eV mentioned in the exercise—it cannot make this transition and remains in its original position.
Such considerations are essential in fields like spectroscopy, where understanding these transitions can help identify elements and their electronic configurations.