Question

Atkins' Physical Chemistry

Compute the following for a 2s election in the hydrogen atom: The most probable distance of the electron from the nucleus The average distance of the electron from the nucleus The distance from the nucleus of the maximum probability density.

Answer 1

The most abundant isotope, hydrogen-1, protium, or light hydrogen, contains no neutrons and is just a proton and an electron. Protium isstable and makes up 99.9885% of naturally occurring hydrogen by absolute number (not mass).

Deuterium contains one neutron and one proton. Deuterium is stable and makes up 0.0115% of naturally occurring hydrogen and is used in industrial processes like nuclear reactors and Nuclear Magnetic Resonance.

Tritium contains two neutrons and one proton and is not stable, decaying with a half-life of 12.32 years. Because of the short half life, Tritium does not exist in nature except in trace amounts.

Higher isotopes of hydrogen are only created in artificial accelerators and reactors and have half lives around the order of 10⁻²² seconds.

The formulas below are valid for all three isotopes of hydrogen, but slightly different values of the Rydberg constant (correction formula given below) must be used for each hydrogen isotope.

Hydrogen ion[edit]

Main article: Hydron (chemistry)

Hydrogen is not found without its electron in ordinary chemistry (room temperatures and pressures), as ionized hydrogen is highly chemically reactive. When ionized hydrogen is written as "H⁺" as in the solvation of classical acids such as hydrochloric acid, the hydronium ion, H₃O⁺, is meant, not a literal ionized single hydrogen atom. In that case, the acid transfers the proton to H₂O to form H₃O⁺.

Ionized hydrogen without its electron, or free protons, are common in the interstellar medium, and solar wind.

Theoretical analysis[edit]

The hydrogen atom has special significance in quantum mechanics and quantum field theory as a simple two-body problem physical system which has yielded many simpleanalytical solutions in closed-form.

Failed classical description[edit]

Experiments by Rutherford in 1909 showed the structure of the atom be a dense, positive nucleus with a light, negative charge orbiting around it. This immediately caused problems on how such a system could be stable. Classical electromagnetism had shown that any accelerating charge radiates energy described through the Larmor formula. If the electron is assumed to orbit in a perfect circle and radiates energy continuously, the electron would spiral into the nucleus with a fall time of:^[2]

{\displaystyle t_{\text{fall}}\approx {\frac {a_{0}^{3}}{4r_{0}^{2}c}}\approx 1.6\cdot 10^{-11}{\text{s}}} ${\displaystyle t_{\text{fall}}\approx {\frac {a_{0}^{3}}{4r_{0}^{2}c}}\approx 1.6\cdot 10^{-11}{\text{s}}}$

Where {\displaystyle a_{0}} $a_{0}$ is the Bohr radius and {\displaystyle r_{0}} $r_{0}$ is the classical electron radius. If this were true, all atoms would instantly collapse, however atoms seem to be stable. Furthermore, the spiral inward would release a smear of electromagnetic frequencies as the orbit got smaller. Instead, atoms were observed to only emit discrete frequencies of light. The resolution would lie in the development of quantum mechanics.

Bohr-Sommerfeld Model[edit]

Main article: Bohr model

In 1913, Niels Bohr obtained the energy levels and spectral frequencies of the hydrogen atom after making a number of simple assumptions in order to correct the failed classical model. The assumptions included:

1. Electrons can only be in certain, discrete circular orbits or stationary states, thereby having a discrete set of possible radii and energies.
2. Electrons do not emit radiation while in one of these stationary states.
3. An electron can gain or lose energy by jumping from one discrete orbital to another.

Bohr supposed that the electron's angular momentum is quantized with possible values:

{\displaystyle L=n\hbar } ${\displaystyle L=n\hbar }$ where {\displaystyle n=1,2,3,...} ${\displaystyle n=1,2,3,...}$

and {\displaystyle \hbar } $\hbar$ is Planck constant over {\displaystyle 2\pi } $2\pi$ . He also supposed that the centripetal force which keeps the electron in its orbit is provided by the Coulomb force, and that energy is conserved. Bohr derived the energy of each orbit of the hydrogen atom to be:^[3]

{\displaystyle E_{n}=-{\frac {m_{e}e^{4}}{2(4\pi \epsilon _{0})^{2}\hbar ^{2}}}{\frac {1}{n^{2}}}} ${\displaystyle E_{n}=-{\frac {m_{e}e^{4}}{2(4\pi \epsilon _{0})^{2}\hbar ^{2}}}{\frac {1}{n^{2}}}}$ ,

where {\displaystyle m_{e}} ${\displaystyle m_{e}}$ is the electron mass, {\displaystyle e} ${\displaystyle e}$ is the electron charge, {\displaystyle \epsilon _{0}} ${\displaystyle \epsilon _{0}}$ is the electric permeability, and {\displaystyle n} ${\displaystyle n}$ is the quantum number (now known as the principal quantum number). Bohr's predictions matched experiments measuring the hydrogen spectral series to the first order, giving more confidence to a theory that used quantized values.

For {\displaystyle n=1} $n=1$ , the value

{\displaystyle {\frac {m_{e}e^{4}}{2(4\pi \epsilon _{0})^{2}\hbar ^{2}}}={\frac {m_{\text{e}}e^{4}}{8h^{2}\varepsilon _{0}^{2}}}=1Ry=13.605\;692\;53(30)\,{\text{eV}}} ${\displaystyle {\frac {m_{e}e^{4}}{2(4\pi \epsilon _{0})^{2}\hbar ^{2}}}={\frac {m_{\text{e}}e^{4}}{8h^{2}\varepsilon _{0}^{2}}}=1Ry=13.605\;692\;53(30)\,{\text{eV}}}$ ^[4]

is called the Rydberg unit of energy. It is related to the Rydberg constant {\displaystyle R_{\infty }} ${\displaystyle R_{\infty }}$ of atomic physics by {\displaystyle 1\,{\text{Ry}}\equiv hcR_{\infty }.} ${\displaystyle 1\,{\text{Ry}}\equiv hcR_{\infty }.}$

The exact value of the Rydberg constant assumes that the nucleus is infinitely massive with respect to the electron. For hydrogen-1, hydrogen-2 (deuterium), and hydrogen-3 (tritium) the constant must be slightly modified to use the reduced mass of the system, rather than simply the mass of the electron. However, since the nucleus is much heavier than the electron, the values are nearly the same. The Rydberg constant R_M for a hydrogen atom (one electron), R is given by

{\displaystyle R_{M}={\frac {R_{\infty }}{1+m_{\text{e}}/M}},} $R_M = \frac{R_\infty}{1+m_{\text{e}}/M},$

where {\displaystyle M} is the mass of the atomic nucleus. For hydrogen-1, the quantity {\displaystyle m_{\text{e}}/M,} $m_{\text{e}}/M,$ is about 1/1836 (i.e. the electron-to-proton mass ratio). For deuterium and tritium, the ratios are about 1/3670 and 1/5497 respectively. These figures, when added to 1 in the denominator, represent very small corrections in the value of R, and thus only small corrections to all energy levels in corresponding hydrogen isotopes.

There were still problems with Bohr's model:

1. it failed to predict other spectral details such as fine structure and hyperfine structure
2. it could only predict energy levels with any accuracy for single–electron atoms (hydrogen–like atoms)
3. the predicted values were only correct to {\displaystyle \alpha ^{2}\approx 10^{-5}} ${\displaystyle \alpha ^{2}\approx 10^{-5}}$ , where {\displaystyle \alpha } $\alpha$ is the fine-structure constant.

Most of these shortcomings were repaired by Sommerfeld's modification of the Bohr model. Sommerfeld introduced two additional degrees of freedom allowing an electron to move on an elliptical orbit, characterized by its eccentricity and declination with respect to a chosen axis. This introduces two additional quantum numbers, which correspond to the orbital angular momentum and its projection on the chosen axis. Thus the correct multiplicity of states (except for the factor 2 accounting for the yet unknown electron spin) was found. Further applying special relativity theory to the elliptic orbits, Sommerfeld succeeded in deriving the correct expression for the fine structure of hydrogen spectra (which happens to be exactly the same as in the most elaborate Dirac theory). However some observed phenomena such as the anomalous Zeeman effect remain unexplained. These issues were resolved with the full development of quantum mechanics and the Dirac equation. It is often alleged, that the Schrödinger equation is superior to the Bohr-Sommerfeld theory in describing hydrogen atom. This is however not the case, as the most results of both approaches coincide or are very close (a remarkable exception is the problem of hydrogen atom in crossed electric and magnetic fields, which cannot be solved in the framework of the Bohr-Sommerfeld theory self-consistently), and their main shortcomings result from the absence of the electron spin in both theories. It was the complete failure of the Bohr-Sommerfeld theory to explain many-electron systems (such as helium atom or hydrogen molecule) which demonstrated its inadequacy in describing quantum phenomena.