Question

How much energy is needed for an electron revolving in the second orbit of \(\mathrm{He}^{+}\) ion, in order double its angular momentum? (a) \(40.8 \mathrm{eV}\) (b) \(2.55 \mathrm{eV}\) (c) \(10.2 \mathrm{eV}\) (d) \(12.09 \mathrm{eV}\)

Short answer

10.2 eV

Step-by-step solution

Understanding the Problem

We are tasked with finding the energy required to double the angular momentum of an electron in the second orbit of a helium ion (He+). We make use of Bohr's model for quantized angular momentum, which states that the angular momentum of an electron in orbit is given by L = n*(h/2π), where n is the orbit number and h is Planck's constant. To double the angular momentum, we need to find the energy difference between the original state (n=2) and the final state with doubled angular momentum.

Calculate Initial Energy

The initial energy of the electron in the second orbit (n=2) can be calculated using the formula for the energy of an electron in a Bohr orbit: E_n = -13.6*Z^2/n^2 eV, where Z is the atomic number of the helium ion (Z=2 for He+). Hence, E_2 = -13.6*2^2/2^2 = -13.6 eV.

Calculate Final Energy

Doubling the angular momentum means the electron moves to an orbit with n=4 (since angular momentum is proportional to n). So, E_4 = -13.6*2^2/4^2 = -3.4 eV.

Find the Energy Difference

The energy needed to double the angular momentum is the difference between the final and initial energy states, ΔE = E_4 - E_2. Therefore, ΔE = -3.4 - (-13.6) = 10.2 eV.

Key concepts

Quantized Angular Momentum

In the realm of atomic physics, the concept of quantized angular momentum is a cornerstone introduced by Bohr's model. This principle asserts that unlike what classical mechanics suggests, the angular momentum of an electron orbiting a nucleus can only take on certain discrete values, rather than any value.
According to Bohr's theory, the angular momentum (\(L\)) of an electron in a particular orbit is quantized and given by the equation \[ L = n\left(\frac{h}{2\pi}\right) \] where \(n\) is the principal quantum number, corresponding to the orbit the electron occupies, and \(h\) represents Planck's constant (∼6.62607015 × 10^-34 Js).
With this quantization, the concept of an electron jumping between orbits comes into play. For an electron to move from an orbit with quantum number \(n_1\) to another with quantum number \(n_2\), there must be a change in energy corresponding to these discrete angular momentum states. This shift is significant when computing the electronic transitions, such as doubling the angular momentum as in our original exercise.

Bohr Orbit Energy

The energy of an electron in a Bohr orbit is another fundamental concept pioneered by Niels Bohr, reflecting a significant departure from classical physics. An electron bound to an atom has specific energy levels, which are dependent on the orbit in which the electron resides.
The energy \(E_n\) associated with the \(n\)-th orbit (where \(n\) is the principal quantum number) is given by the equation \[ E_n = \frac{-13.6 \cdot Z^2}{n^2}\text{ eV} \] where \(Z\) is the atomic number of the nucleus. For a helium ion (\(\text{He}^{+}\)), \(Z\) would be 2. This formula indicates that as \(n\) increases, the absolute value of energy decreases, meaning electrons in higher orbits are less tightly bound and have higher potential energy.
In the instance of our exercise, to find the energy needed to double the electron's angular momentum, we calculated the energy in the second orbit and compared it to the fourth orbit, elucidating the energy change required for such a transition.

Helium Ion Electron Transitions

Electron transitions within a helium ion (\(\text{He}^+\)) illustrate the practical application of both the quantized angular momentum and Bohr orbit energy concepts. In a helium ion, which has lost one electron, only one electron remains that can transition between energy levels.
Such a transition involves the absorption or emission of energy, which causes the electron to move to a higher-energy orbit (absorption) or to a lower-energy orbit (emission). In our related exercise, the transition required doubling the angular momentum of the electron, implying a move from the second to the fourth orbit.
Using Bohr's equations, we were able to determine the specific energy values for each orbit and calculate the difference. The electron absorbs energy equivalent to the calculated difference in energy levels only when it moves to a higher orbit—10.2 eV in our case. This simple yet profound model explains the spectral lines observed in hydrogen and hydrogen-like ions, such as \(\text{He}^+\), and forms the basis for much of quantum mechanics.