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Chapter 2: Problem 36

In the hydrogen atom, what is the physical significance of the state for which \(n=\infty\) and \(E=0 ?\)

Short Answer

Expert verified
The physical significance of the state for which \(n=\infty\) and \(E=0\) in a hydrogen atom represents the electron being completely removed from the atom and infinitely far away from the nucleus. This state is known as the ionization limit, as reaching this state would cause the electron to become ionized and separate from the atom.

Step by step solution

01

Recall the Bohr model of the hydrogen atom

The Bohr model is a simplified representation of a hydrogen atom with one proton in its nucleus and one electron orbiting around it. In this model, electrons can only occupy specific discrete energy levels, represented by the principal quantum number n. As n increases, the electron orbits at a larger distance from the nucleus and has higher energy.
02

Examine the relationship between energy and quantum number

The energy levels of a hydrogen atom in the Bohr model can be represented by the equation: \[E_n = -\frac{13.6 eV}{n^2}\] Where \(E_n\) is the energy of a specific energy level, and n is the principal quantum number. As n increases, the energy levels become closer together and approach zero.
03

Consider the limit as n approaches infinity

As n approaches infinity (\(n \to \infty\)), the energy of the electron (\(E_n\)) approaches zero: \[\lim_{n \to \infty} \left(-\frac{13.6 eV}{n^2}\right) = 0\] Simultaneously, as n approaches infinity, the radius of the electron's orbit also increases. In the Bohr model, the radius of an electron orbit can be represented by: \[r_n = n^2 \cdot a_0\] Where \(r_n\) is the radius of the orbit, n is the principal quantum number, and \(a_0\) is the Bohr radius (approximately \(0.529 \times 10^{-10} m\)). As \(n \to \infty\), the radius of the electron's orbit also approaches infinity: \[\lim_{n \to \infty} \left(n^2 \cdot a_0\right) = \infty\]
04

Physical significance of the state \(n=\infty\) and \(E=0\)

When the principal quantum number n approaches infinity and the energy of the electron approaches zero, this essentially represents the electron being completely removed from the hydrogen atom (forming an ion) and being infinitely far away from the nucleus. This state is known as the ionization limit, as adding enough energy to the electron to reach this state would cause it to become ionized and separate from the atom.

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Most popular questions from this chapter

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