1. Helium Atom Diagram
2. Helium Atomic Number And Mass
3. Helium Atomic Weight
4. Helium Atom

Helium-8 atom is the radioactive isotope of helium with relative atomic mass 8.03392 and half-life of 119.0 ms.

The second element in the periodic table provides our first example of a quantum-mechanical problem which cannot be solved exactly. Nevertheless, as we will show, approximation methods applied to helium can give accurate solutions in perfect agreement with experimental results. In this sense, it can be concluded that quantum mechanics is correct for atoms more complicated than hydrogen. By contrast, the Bohr theory failed miserably in attempts to apply it beyond the hydrogen atom.

1. Since the one electron of the positive helium ion behaves like the one electron of the ordinary hydrogen, then in the same way the two electrons of the helium atom behave like the two electrons of the negative hydrogen ion leading to the quantum mechanics of the two-electron orbitals.
2. Of the helium atom. Thus, just as we thought of the electrons in the ground state as being “paired”, we say the electrons in Case 2 are paired. The spatial parts of three of the states above (cases 1,3 and 4) are the same. Case 2 has a different spatial part. Because the Hamiltonian only depends on.

The helium atom has two electrons bound to a nucleus with charge Z = 2. The successive removal of the two electrons can be diagrammed as

[ce{He}xrightarrow{textit{I}_1}ce{He}^++e^-xrightarrow{textit{I}_2}ce{He}^{++}+2e^-label{1}]

Thefirst ionization energy I₁, the minimum energy required to remove the first electron from helium, is experimentally 24.59 eV. The second ionization energy, I₂, is 54.42 eV. The last result can be calculated exactly since He⁺ is a hydrogen-like ion. We have

[textit{I}_2=-textit{E}_{ 1textit{s}}(ce{He}^+)=dfrac{Z^2}{2n^2}=2 mbox{ hartrees}=54.42mbox{ eV}label{2}]

The energy of the three separated particles on the right side of Equation (ref{1}) is, by definition, zero. Therefore the ground-state energy of helium atom is given by (E_0=-(textit{I}_1+textit{I}_2)=-79.02mbox{ eV}=-2.90372mbox{ hartrees}). We will attempt to reproduce this value, as close as possible, by theoretical analysis.

Schrödinger Equation and Variational Calculations

The Schrödinger equation for He atom, again using atomic units and assuming infinite nuclear mass, can be written

[bigg{-dfrac{1}{2}nabla^2_1-dfrac{1}{2}nabla^2_2-dfrac{Z}{r_1}-dfrac{Z}{r_2}+dfrac{1}{r_{12}}bigg}psi(text{r}_1,text{r}_2)=Epsi(text{r}_1,text{r}_2)label{3}]

The five terms in the Hamiltonian represent, respectively, the kinetic energies of electrons 1 and 2, the nuclear attractions of electrons 1 and 2, and the repulsive interaction between the two electrons. It is this last contribution which prevents an exact solution of the Schrödinger equation and which accounts for much of the complication in the theory. In seeking an approximation to the ground state, we might first work out the solution in the absence of the 1/r₁₂-term. In the Schrödinger equation thus simplified, we can separate the variables r₁ and r₂ to reduce the equation to two independent hydrogen-like problems. The ground state wavefunction (not normalized) for this hypothetical helium atom would be

[psi(text{r}_1,text{r}_2)=psi_{1s}(r_1)psi_{1s}(r_2)=e^{-Z(r_1+r_2)}label{4}]

and the energy would equal (2times(-Z^2/2)=-4) hartrees, compared to the experimental value of (-2.90) hartrees. Neglect of electron repulsion evidently introduces a very large error.

A significantly improved result can be obtained with the functional form ( Equation (ref{4})), but with Z replaced by a adjustable parameter (alpha), thus:

[tilde{psi}(r_1,r_2)=e^{-alpha(r_1+r_2)}label{5}]

Using this function in the variational principle [cf. Eq (4.53)], we have

[tilde{E}=dfrac{intpsi(r_1,r_2)hat{H}psi(r_1,r_2)dtau_1tau_2}{intpsi(r_1,r_2)psi(r_1,r_2)dtau_1dtau_2}label{6}]

where (hat{H}) is the full Hamiltonian as in Equation (ref{3}), including the (1/r_{12})-term. The expectation values of the five parts of the Hamiltonian work out to

[leftlangle-dfrac{1}{2}nabla^2_1rightrangle=leftlangle-dfrac{1}{2}nabla^2_2rightrangle=dfrac{alpha^2}{2}]

[leftlangle-dfrac{Z}{r_1}rightrangle=leftlangle-dfrac{Z}{r_2}rightrangle=-Zalpha, leftlangledfrac{1}{r_{12}}rightrangle=dfrac{5}{8}alphalabel{7}]

The sum of the integrals in Equation (ref{7}) gives the variational energy

[tilde{E}(alpha)=alpha^2-2Zalpha+dfrac{5}{8}alphalabel{8}]

This will be always be an upper bound for the true ground-state energy. We can optimize our result by finding the value of (alpha) which minimizes the energy (Equation (ref{8})). We find

[dfrac{dtilde{E}}{dalpha}=2alpha-2Z+dfrac{5}{8}=0label{9}]

giving the optimal value

[alpha=Z-dfrac{5}{16}label{10}]

This can be given a physical interpretation, noting that the parameter (alpha) in the wavefunction (Equation (ref{5})) represents an effective nuclear charge. Each electron partially shields the other electron from the positively-charged nucleus by an amount equivalent to (5/8) of an electron charge. Substituting Equation (ref{10}) into Equation (ref{8}), we obtain the optimized approximation to the energy

[tilde{E}=-left(Z-dfrac{5}{16}right)^2label{11}]

For helium ((Z = 2)), this gives (-2.84765) hartrees, an error of about (2%) ((E_0 = -2.90372)). Note that the inequality (tilde{E} > E_0) applies in an algebraic sense.

In the late 1920's, it was considered important to determine whether the helium computation could be improved, as a test of the validity of quantum mechanics for many electron systems. The table below gives the results for a selection of variational computations on helium.

The third entry refers to the self-consistent field method, developed by Hartree. Even for the best possible choice of one-electron functions (psi(r)), there remains a considerable error. This is due to failure to include the variable (r_{12}) in the wavefunction. The effect is known as electron correlation.

The fourth entry, containing a simple correction for correlation, gives a considerable improvement. Hylleraas (1929) extended this approach with a variational function of the form

[psi(r_1, r_2, r_{12})=e^{-alpha(r_1+r_2)} times textrm{polynomial in} r_1, r_2, r_{12}]

and obtained the nearly exact result with 10 optimized parameters. More recently, using modern computers, results in essentially perfect agreement with experiment have been obtained.

Spinorbitals and the Exclusion Principle

The simpler wavefunctions for helium atom in Equation (ref{5}), can be interpreted as representing two electrons in hydrogen-like 1s orbitals, designated as a 1s² configuration. According to Pauli's exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers, the two 1s electrons must have different spins, one spin-up or (alpha), the other spin-down or (beta). A product of an orbital with a spin function is called a spinorbital. For example, electron 1 might occupy a spinorbital which we designate

[phi(1)=psi_{1s}(1)alpha(1) textrm{or} psi_{1s}(1)beta(1)label{12}]

Spinorbitals can be designated by a single subscript, for example, (phi_a) or (phi_b), where the subscript stands for a set of four quantum numbers. In a two electron system the occupied spinorbitals (phi_a) and (phi_b) must be different, meaning that at least one of their four quantum numbers must be unequal. A two-electron spinorbital function of the form

[Psi (1, 2) = dfrac{1}{2} bigg( phi_a(1)phi_b(2) - phi_b(1)phi_a(2)bigg)label{13}]

automatically fulflls the Pauli principle since it vanishes if (a=b). Moreover, this function associates each electron equally with each orbital, which is consistent with the indistinguishability of identical particles in quantum mechanics. The factor (1/sqrt{2}) normalizes the two-particle wavefunction, assuming that (phi_a) and (phi_b) are normalized and mutually orthogonal. The function (Equation (ref{13})) is antisymmetric with respect to interchange of electron labels, meaning that

Helium Atom Diagram

[Psi (2,1) = -Psi (1, 2)label{14}]

This antisymmetry property is an elegant way of expressing the Pauli principle.

We note, for future reference, that the function in Equation (ref{13}) can be expressed as a ( 2 times 2) determinant:

[Psi (1, 2) = dfrac{1}{sqrt{2}}begin{vmatrix}phi_a(1) & phi_b(1)phi_a(2) & phi_b(2)end{vmatrix}label{15}]

For the 1s² configuration of helium, the two orbital functions are the same and Equation (ref{13}) can be written

[Psi (1, 2) = psi_{1s}(1)psi_{1s}(2) times dfrac{1}{sqrt{2}}bigg(alpha(1)beta(2) - beta(1)alpha(2)bigg)label{16}]

For two-electron systems (but not for three or more electrons), the wavefunction can be factored into an orbital function times a spin function. The two-electron spin function

[sigma_{0,0}(1, 2) = dfrac{1}{sqrt{2}}bigg(alpha(1)beta(2) - beta(1)alpha(2)bigg)label{17}]

represents the two electron spins in opposing directions (antiparallel) with a total spin angular momentum of zero. The two subscripts are the quantum numbers S and M_S for the total electron spin. Eqution (ref{16}) is called the singlet spin state since there is only a single orientation for a total spin quantum number of zero. It is also possible to have both spins in the same state, provided the orbitals are different. There are three possible states for two parallel spins:

[sigma_{1,1}(1, 2) = alpha(1)alpha(2)]

[sigma_{1,0}(1, 2) = dfrac{1}{sqrt{2}}bigg(alpha(1)beta(2) + beta(2)alpha(2)bigg)]

[sigma_{1,-1}(1, 2) = beta(1)beta(2)label{18}]

These make up the triplet spin states, which have the three possible orientations of a total angular momentum of 1.

Excited States of Helium

The lowest excitated state of helium is represented by the electron configuration 1s2s. The 1s 2p configuration has higher energy, even though the 2s and 2p orbitals in hydrogen are degenerate, because the 2s penetrates closer to the nucleus, where the potential energy is more negative. When electrons are in different orbitals, their spins can be either parallel or antiparallel. In order that the wavefunction satisfy the antisymmetry requirement (Equation (ref{14})), the two-electron orbital and spin functions must have opposite behavior under exchange of electron labels. There are four possible states from the 1s 2s configuration: a singlet state

[Psi^+ (1, 2) = dfrac{1}{sqrt{2}}bigg(psi_{1s}(1)psi_{2s}(2) + psi_{2s}(1)psi_{1s}(2)bigg) sigma_{0, 0}(1, 2)label{19}]

and three triplet states

[Psi^-(1, 2) = dfrac{1}{sqrt{2}}bigg(psi_{1s}(1)psi_{2s}(2) - psi_{2s}(1)psi_{1s}(2)bigg)begin{cases} sigma_{1,1}(1, 2) sigma_{1,0}(1, 2) sigma_{1,-1}(1, 2)end{cases}label{20}]

Using the Hamiltonian in Equation (ref{3}), we can compute the approximate energies

[E^{pm}=iintPsi^{pm}(1,2) hat{H} Psi^{pm}(1,2)dtau_1dtau_2label{21}]

After evaluating some fierce-looking integrals, this reduces to the form

[E^{pm}=I(1s)+I(2s)+J(1s, 2s) pm K(1s, 2s)label{22}]

in terms of the one electron integrals

[I(a)=int psi_a(textrm{r})left{-dfrac{1}{2}nabla^2-dfrac{Z}{r}right} psi_a(textrm{r})dtaulabel{23}]

the Coulomb integrals

[J(a, b)=iintpsi_a(textrm{r}_1)^2dfrac{1}{r_{12}}psi_b(textrm{r}_2)^2dtau_1dtau_2label{24}]

and the exchange integrals

[K(a, b)=iintpsi_a(textrm{r}_1)psi_b(textrm{r}_1)dfrac{1}{r_{12}}psi_a(textrm{r}_2)psi_b(textrm{r}_2)dtau_1dtau_2label{25}]

The Coulomb integral represents the repulsive potential energy for two interacting charge distributions (psi_a(textbf{r}_1)^2) and (psi_b(textbf{r}_2)^2). The exchange integral, which has no classical analog, arises because of the exchange symmetry (or antisymmetry) requirement of the wavefunction. Both J and K can be shown to be positive quantities. Therefore the lower sign in (22) represents the state of lower energy, making the triplet state of the configuration 1s 2s lower in energy than the singlet state. This is an almost universal generalization and contributes to Hund's rule, to be discussed in the next Chapter.

Contributors and Attributions

• Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor)

• Gerardo Ochoa (Hope College, Holland, MI)

Helium is a colorless, odorless, tasteless chemical element, one of the noble gases of the periodic table of elements. Its boiling and melting points are the lowest among the elements; except in extreme conditions, it exists only as a gas. The second most abundant element in the universe, significant amounts are found on Earth only in natural gas. It is used in cryogenics, in deep-sea breathing systems, for inflating balloons, and as a protective gas for many purposes. Helium is not toxic and has no biological effect.

History

Helium was first detected in 1868 as a bright yellow line in the spectrum of the chromosphere of the Sun, by French astronomer Pierre Janssen during a solar eclipse in India. The same year, English astronomer Norman Lockyer also observed a previously unknown yellow line in the solar spectrum and concluded that it was caused by an element unknown on Earth. He and English chemist Edward Frankland named the element with the Greek word for the Sun, helios. In 1895, British chemist William Ramsay isolated helium on Earth by treating cleveite with mineral acids. These samples were identified as helium by Lockyer and British physicist William Crookes. It was independently isolated from cleveite the same year by Swedish chemists Per Teodor Cleve and Nils Langlet.

In 1905, American chemists Hamilton Cady and David McFarland discovered that helium could be extracted from natural gas. In 1907, Ernest Rutherford and Thomas Royds demonstrated that an alpha particle is a helium nucleus. Helium was first liquefied by Dutch physicist Heike Kamerlingh Onnes in 1908 by cooling the gas to less than one kelvin. It was first solidified in 1926 by his student Willem Hendrik Keesom. In 1938, Russian physicist Pyotr Leonidovich Kapitsa discovered that helium-4 has almost no viscosity at temperatures near absolute zero, a phenomenon now called superfluidity. In 1972, the same phenomenon was observed in helium-3 by American physicists Douglas D. Osheroff, David M. Lee, and Robert C. Richardson.

Under standard temperature and pressure, helium exists only as a monatomic gas. It solidifies only under great pressure, the variation of which can drastically change the volume of the solid. Below its boiling point of 4.21 kelvin and above the lambda point 2.1768 kelvin, the isotope helium-4 exists in a normal liquid state, called helium I. Below the lambda point, it begins toexhibit unusual characteristics, in a state called helium II. Less is known about such properties in the isotope helium-3.

Helium II

Helium II exhibits characteristics of two distinct fluids, one a normal, viscous liquid and the other a superfluid apparently without internal friction. It has a mobile, rapid flow through even the smallest of capillaries and, in the fountain effect, can rise over the rim of a containment vessel in a thin film that appears unaffected by gravity. In addition, its thermal conductivity is greater than that of any other known substance. When introduced, heat will rapidly propagate through the substance in waves, a phenomenon called second sound.

Reactions

Helium is chemically unreactive under all normal conditions. With electric glow discharge or electron bombardment, however, helium can form compounds with tungsten, iodine, fluorine, sulfur and phosphorus.

Isotopes

Although there are six known isotopes of helium, only helium-3 and helium-4 are stable. The others, radioactive, rapidly decay into other substances. The most common isotope, helium-4, is produced by alpha decay from heavier radioactive elements; its nucleus is an alpha particle. It is an unusually stable nucleus because its nucleons are arranged into complete shells. There is only a trace amount of helium-3 on Earth, produced from the beta decay of tritium.

Abundance

Helium is the second most abundant element in the known universe after hydrogen and constitutes nearly a quarter of the mass of the universe. It is concentrated in the stars, where it is formed from hydrogen by the nuclear fusion of the proton-proton chain reaction and CNO cycle. According to the Big Bang model of the early development of the universe, the vast majority of helium was formed in the first three minutes after the Big Bang.

Helium Atomic Number And Mass

However, the concentration of helium in the Earth's atmosphere is only 1 part in 200,000, largely because most helium in the Earth's atmosphere escapes into space due to its inertness and low mass. All considerable helium on Earth is a result of radioactive decay. The decay product is found in minerals of uranium and thorium, including cleveites, pitchblende, carnotite, monazite and beryl. There are also small amounts in mineral springs, volcanic gas, meteoric iron. The greatest concentrations on the planet are in natural gas, from which most commercial helium is derived. The principal source in the world is the natural gas wells of the American states of Texas, Oklahoma, and Kansas.

Synthesis

Helium can be synthesized by bombardment of lithium or boron with high-velocity protons.

Uses

Pressurized helium is commercially available, extracted from natural gas.

Helium is used for many purposes:

Helium Atomic Weight

• Because it is lighter than air, airships and balloons are inflated with helium for lift. Helium is advantageous in airships because it is not flammable and has 92.64% of the lifting power of the alternative hydrogen.
• The voice of a person who has inhaled helium becomes temporarily high-pitched, because the speed of sound in helium is nearly thrice that in air, with a corresponding increase in the resonant frequencies of the larynx. Although this is a novel amusement, it can be lethally dangerous with concentrated helium.
• Trimix, an atmosphere of helium, oxygen, and nitrogen, is used in deep-sea breathing systems to reduce the risk of nitrogen narcosis and oxygen toxicity at high pressures.
• Because of its extremely low melting and boiling points, helium is used as a coolant in magnetic resonance imaging, nuclear reactors, superconducting magnets, and cryogenics.
• Because it is inert, helium is used as a protective gas in growing silicon and germanium crystals, in titanium and zirconium production, in arc welding, and in gas chromatography.
• Helium used to pressurize liquid fuel rockets.
• Helium is used in supersonic wind tunnels.
• The gain medium of the helium-neon laser is a mixture of helium and neon.

External links

Helium Atom

• WebElements: Helium(http://www.webelements.com/webelements/elements/text/He/key.html)
• EnvironmentalChemistry.com – Helium(http://environmentalchemistry.com/yogi/periodic/He.html)
• It's Elemental – Helium(http://education.jlab.org/itselemental/ele002.html)
• Hoax: Helium Causes Death (CIAC Hoaxbusters)(http://hoaxbusters.ciac.org/HBUrbanMyths.shtml#helium)