Question

Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? (a) from \(n=4\) to \(n=2\) , (b) from an orbit of radius 2.12 A to one of radius \(8.46 \hat{A},(\mathbf{c})\) an electron adds to the \(\mathrm{H}^{+}\) ion and ends up in the \(n=3\) shell?

Short answer

(a) Energy is emitted during the transition from n=4 to n=2. (b) Energy is absorbed during the transition from 2.12 Å to 8.46 Å. (c) Energy is emitted when an electron adds to the H+ ion and ends up in the n=3 shell.

Step-by-step solution

(a) Transition from n=4 to n=2

Here, we need to determine if energy is emitted or absorbed during a transition from n=4 to n=2. First, let's observe their energy levels using the relation mentioned above: \(E_4 = -\cfrac{13.6 \text{ eV}}{4^2} = -0.85 \text{ eV}\) \(E_2 = -\cfrac{13.6 \text{ eV}}{2^2} = -3.40 \text{ eV}\) Since the electron transitions from a higher energy level (n=4) to a lower energy level (n=2), energy is emitted.

(b) Transition from 2.12 Å to 8.46 Å

For this case, we have to find the energy levels corresponding to the given radii. The formula to find the radius, r, of the nth orbit in hydrogen is: \(r_n = 0.529n^2 \ Å\) Solving for n for the two given radii, we get: \(n_1^2 = \cfrac{2.12}{0.529} \Rightarrow n_1 = 2\) \(n_2^2 = \cfrac{8.46}{0.529} \Rightarrow n_2 = 4\) Here, an electron transitions from an orbit with a radius of 2.12 Å (n=2) to an orbit with a radius of 8.46 Å (n=4). Since the electron transitions from a lower energy level (n=2) to a higher energy level (n=4), energy is absorbed.

(c) An electron adds to the H+ ion and ends up in the n=3 shell

For this situation, we have an electron joining an H+ ion, which is a hydrogen atom without an electron. When the electron joins and ends up in the n=3 shell, it goes from free space, where its energy is approximately 0 eV, to the n=3 orbit in the hydrogen atom. We already know from the analysis that the energy level for n=3 is: \(E_3 = -\cfrac{13.6 \text{ eV}}{3^2} = -1.51 \text{ eV}\) Since the electron transitions from a higher energy level (0 eV) to a lower energy level (n=3), energy is emitted. In summary: (a) Energy is emitted during the transition from n=4 to n=2. (b) Energy is absorbed during the transition from 2.12 Å to 8.46 Å. (c) Energy is emitted when an electron adds to the H+ ion and ends up in the n=3 shell.

Key concepts

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