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Chapter 6: Problem 35

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 \(2.12 \AA\) to one of radius \(8.46 \AA\) (c) an electron adds to the \(\mathrm{H}^{+}\) ion and ends up in the \(n=3\) shell?

Short Answer

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

Step by step solution

01

Calculate the energy levels for transition (a)

For finding if energy is emitted or absorbed for the transition from n=4 to n=2 in hydrogen, calculate the energy levels for both orbits first. E4 = -13.6 / (4^2) = -0.85 eV E2 = -13.6 / (2^2) = -3.40 eV
02

Determine if energy is emitted or absorbed for transition (a)

To determine if energy is emitted or absorbed, subtract the initial energy level from the final energy level: ΔE = E2 - E4 = (-3.40) - (-0.85) = -2.55 eV Because ΔE is negative, energy is emitted in this transition.
03

Calculate the principal quantum numbers for transition (b)

In part (b), the transition occurs from r1 to r2 (with r1 = 2.12 Å and r2 = 8.46 Å). To find the energy associated with these radii, we first need to determine the principal quantum numbers (n) for both orbits using the formula \(r_n = n^2 a_0\). n1² = r1 / a₀ = (2.12 / 0.529) n2² = r2 / a₀ = (8.46 / 0.529) Solve for n1 and n2: n1 ≈ 2 n2 ≈ 4
04

Calculate the energy levels for transition (b)

Using the energy levels formula, calculate the energy levels for n1 and n2. E2 = -13.6 / (2^2) = -3.40 eV E4 = -13.6 / (4^2) = -0.85 eV
05

Determine if energy is emitted or absorbed for transition (b)

Now, subtract the initial energy level (E2) from the final energy level (E4) to find ΔE: ΔE = E4 - E2 = -0.85 - (-3.40) = 2.55 eV Because ΔE is positive, energy is absorbed in this transition.
06

Determine if energy is emitted or absorbed for transition (c)

In part (c), an electron is added to the hydrogen ion (H⁺), and it ends up in the n=3 shell. The hydrogen ion (H⁺) has no electrons, so its energy level is 0 eV. Calculate the energy level for n=3: E3 = -13.6 / (3^2) = -1.51 eV Now, subtract the initial energy level (0 eV) from the final energy level (E3) to find ΔE: ΔE = E3 - 0 = -1.51 eV Because ΔE is negative, energy is emitted in this transition. #Summary# To summarize the results: (a) Energy is emitted during the electronic transition from n=4 to n=2. (b) Energy is absorbed during the electronic transition from a radius of 2.12 Å to 8.46 Å. (c) Energy is emitted when an electron is added to the H⁺ ion and ends up in the n=3 shell.

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Most popular questions from this chapter

(a) What is the relationship between the wavelength and the frequency of radiant energy? (b) Ozone in the upper atmosphere absorbs energy in the \(210-230-\mathrm{nm}\) range of the spectrum. In what region of the electromagnetic spectrum does this radiation occur?

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