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Chapter 7: Problem 42

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 hydrogen atom state when the principal quantum number \(n\) approaches infinity and the energy \(E\) is zero is that the electron is on the verge of being detached from the hydrogen atom. This state corresponds to the transition of the hydrogen atom from an atomic existence to its ionized state.

Step by step solution

01

Recall the hydrogen atom energy levels formula

The energy levels of a hydrogen atom can be calculated using the following formula: \[ E = -\frac{13.6 eV}{n^2} \] where \(E\) is the energy of the electron in the hydrogen atom, \(n\) is the principal quantum number, and 13.6 eV is the ionization energy of the hydrogen atom.
02

Calculate the energy for n approaching infinity

As the principal quantum number, \(n\), approaches infinity, we calculate the energy as follows: \[ E_{n \to \infty} = -\frac{13.6 eV}{n^2} \] Since \(n\) approaches infinity, the expression \(n^2\) will also approach infinity. Therefore: \[ E_{n \to \infty} = -\frac{13.6 eV}{\infty} = 0 \]
03

Consider the electron's behavior for E=0

When the energy of an electron in a hydrogen atom approaches zero, it means that the electron is moving further away from the nucleus of the hydrogen atom. At a very high value of the principal quantum number, the atom's electron moves away from the nucleus and the distance between the electron and the nucleus increases, eventually reaching a point where the electron is almost detached from the hydrogen atom.
04

Physical Significance of n approaching infinity and E=0

The physical significance of the hydrogen atom state when the principal quantum number approaches infinity and the energy is zero indicates that the electron is on the verge of being detached from the hydrogen atom. The hydrogen atom becomes ionized as the electron reaches the ionization energy, which is equal to 13.6 eV. Essentially, this state corresponds to the transition of the hydrogen atom from anatomic existence to its ionized state.

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Most popular questions from this chapter

One bit of evidence that the quantum mechanical model is 鈥渃orrect鈥� lies in the magnetic properties of matter. Atoms with unpaired electrons are attracted by magnetic fields and thus are said to exhibit paramagnetism. The degree to which this effect is observed is directly related to the number of unpaired electrons present in the atom. Consider the ground-state electron configurations for Li, N, Ni, Te, Ba, and Hg. Which of these atoms would be expected to be paramagnetic, and how many unpaired electrons are present in each paramagnetic atom?

Element 106 has been named seaborgium, Sg, in honor of Glenn Seaborg, discoverer of the first transuranium element. a. Write the expected electron configuration for element 106. b. What other element would be most like element 106 in its properties? c. Predict the formula for a possible oxide and a possible oxyanion of element 106.

The four most abundant elements by mass in the human body are oxygen, carbon, hydrogen, and nitrogen. These four elements make up about 96% of the human body. The next four most abundant elements are calcium, phosphorus, magnesium, and potassium. Write the expected ground-state electron configurations for these eight most abundant elements in the human body.

Calculate the maximum wavelength of light capable of removing an electron for a hydrogen atom from the energy state characterized by \(n=1,\) by \(n=2\)

Although no currently known elements contain electrons in g orbitals in the ground state, it is possible that these elements will be found or that electrons in excited states of known elements could be in \(g\) orbitals. For \(g\) orbitals, the value of \(\ell\) is \(4 .\) What is the lowest value of \(n\) for which \(g\) orbitals could exist? What are the possible values of \(m_{\ell} ?\) How many electrons could a set of \(g\) orbitals hold?
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