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Chapter 2: Problem 62

Assume that a hydrogen atom's electron has been excited to the \(n=5\) level. How many different wavelengths of light can be emitted as this excited atom loses energy?

Short Answer

Expert verified
The hydrogen atom's electron is excited to the \(n=5\) level and can undergo a series of possible transitions as it loses energy and falls to lower energy levels. By identifying all possible transitions (5 → 4, 5 → 3, 5 → 2, 5 → 1, 4 → 3, 4 → 2, 4 → 1, 3 → 2, 3 → 1, 2 → 1), we find that the excited atom can emit a total of 10 unique wavelengths of light as it loses energy.

Step by step solution

01

Identifying the energy levels in a hydrogen atom

In a hydrogen atom, the energy levels are quantized and are given by the principal quantum number n. The electron in a hydrogen atom can only exist in specific energy levels (n = 1, 2, 3, and so on). In this exercise, the electron is excited to the n=5 level. We will now identify the possible transitions as the electron loses energy and falls to lower energy levels.
02

Identifying the possible transitions

When the electron moves to a lower energy level, it gives off energy in the form of light. The wavelengths of light emitted during these transitions can be found using the formula for the energy difference between two energy levels in a hydrogen atom. The possible transitions for an electron in the n = 5 level are: 1. 5 → 4 2. 5 → 3 3. 5 → 2 4. 5 → 1 However, we should also consider the subsequent transitions when the electron is at the other energy levels: 5. 4 → 3 6. 4 → 2 7. 4 → 1 8. 3 → 2 9. 3 → 1 10. 2 → 1
03

Counting the unique wavelengths

Now that we have identified all possible transitions for the electron, we can count up the unique wavelengths emitted during these transitions: 1. 5 → 4 2. 5 → 3 3. 5 → 2 4. 5 → 1 5. 4 → 3 6. 4 → 2 7. 4 → 1 8. 3 → 2 9. 3 → 1 10. 2 → 1 There are a total of 10 unique wavelengths of light that can be emitted as the excited hydrogen atom loses energy.

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The Heisenberg uncertainty principle can be expressed in the form $$ \Delta E \cdot \Delta t \geqq \frac{h}{4 \pi} $$ where \(E\) represents energy and \(t\) represents time. Show that the units for this form are the same as the units for the form used in this chapter: $$ \Delta x \cdot \Delta(m v) \geq \frac{h}{4 \pi} $$

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