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Chapter 7: 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
In the hydrogen atom, the physical significance of the state for which \(n=\infty\) and \(E=0\) is that the electron is becoming infinitely far away from the nucleus and is no longer bound to it, turning into a free electron. This state represents the ionization limit, where the hydrogen atom loses its electron and becomes ionized.

Step by step solution

01

Understand the meaning of principal quantum number n

The principal quantum number, n, is a positive integer that describes the electron shell of an electron within an atom. Higher values of n correspond to shells with higher energy and larger radii. In other words, as n increases, the electron is further away from the nucleus and has a higher energy.
02

Analyze the implications of n approaching infinity

When the principal quantum number n approaches infinity, it means that the electron is moving very far away from the nucleus. In fact, the electron is becoming so far from the nucleus that the attractive force between them becomes weaker. Eventually, the electron is hardly bound to the nucleus at all.
03

Evaluate the significance of the energy state E being 0

The energy of an electron in a hydrogen atom is given by the formula: \[E = \frac{-13.6 \, eV}{n^2}\] As n approaches infinity, the energy E effectively becomes 0. A 0 energy state means that the electron is no longer bound to the nucleus. It has gone from a bound state within an atom to a free state, where it is considered a free electron.
04

Conclude the physical significance of the state for which n approaches infinity and E=0

In conclusion, the physical significance of the hydrogen atom state for which n approaches infinity and E=0 is that the electron is becoming infinitely far away from the nucleus and is no longer bound to it, turning into a free electron. This is essentially the ionization limit, where the hydrogen atom loses its electron and becomes ionized.

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Most popular questions from this chapter

A particle has a velocity that is \(90 . \%\) of the speed of light. If the wavelength of the particle is \(1.5 \times 10^{-15} \mathrm{~m}\), calculate the mass of the particle.

Which of the following sets of quantum numbers are not allowed in the hydrogen atom? For the sets of quantum numbers that are incorrect, state what is wrong in each set. a. \(n=3, \ell=2, m_{\ell}=2\) b. \(n=4, \ell=3, m_{\ell}=4\) c. \(n=0, \ell=0, m_{\ell}=0\) d. \(n=2, \ell=-1, m_{\ell}=1\)

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