Chemistry - Quantum mechanical model of atom

Quantum mechanical model of atom :

A new branch of science, called quantum machanics, was developed in 1926 by Werner Heisenberg and Erwin Schrodinger based on uncertainty principle and wave motion, respectively. Quantum mechanics based on the ideas of wave motion will be discussed here. Schrodinger developed the fundamental equation of quantum mechanics which incorporates wave particle duality of matter. The Schordinger equation or wave equationis written as :

Here H∧ is a mathematical operator called Hamiltonian, ψ (psi) is the wave function and E the total energy of the system. Solving Schrodinger equation is beyond the scope of this book. It may, however, be noted that solution of Schrodinger equation gives E and ψ.

Schrodinger equation : 

When Schrodinger equation is solved for hydrogen atom, the possible values of energy (E) that the electron may have along with the corresponding wave function (ψ) are obtained As a natural consequence of solving this equation, a set of three quantum numbers characteristic of the quantized energy levels and the corresponding wave functions are obtained. These are : 

Principal quantum number (n)
azimuthal quantum number (l)
magnetic quantum number (ml).

The solution of Schrodinger wave equation led to three quantum numbers and successfully predicted features of hydrogen atom emission spectrum. Splitting of spectral lines in multi-electron atomic emission spectra could not be explained through such model. These were explained by George Uhlenbeckand Samuel Goudsmit (1925) who proposed the presence ofthe fourth quantum number called electron spin quantum number, ms. Wave function, ψ, as such does not have any physical meaning. The probability of finding an electron at a point within an atom is proportional to ψ 2 in the neighbourhood of that point (within a tinyvolume element) around it. 

Atomic orbitals and quantum numbers :

Many wave function are possible for an electron, and therefore, many atomic orbitals in an atom. Thus the wave functions or atomic orbitals form the basis of the quantum mechanical electronic structure of an atom. Various orbitals in an atom differ in size, shape and orientation with respect to the nucleus depending upon the value of ψ 2. Each orbital is designated by three quantum numbers labelled as n, l and ml, and each electron being assigned with four quantum numbers, viz, n, l, ml and ms.

The principal quantum number ‘n’ is a positive integer with values of n being 1, 2, 3, 4, .............. It identifies the shell. Atomic orbitals, having the same value ‘n’ belong to the same shell. With increase of ‘n’, the number of allowed orbitals in that shell increases and given by ‘n2’. A set of orbitals with given value of ‘n’constitutes a single shell. Shells are represented by symbols K, L, M, N,......... so on. Table. shows ‘n’ and electronic shell symbol.

Allowed orbitals in the first four shells

With an increase of ‘n’, the distance from the nucleus and size of the shell increases and also the energy increases (becoming lesser and lesser negative). In hydrogen-like species the energy of orbital depends only on the value of ‘n’. In the case of multi-electron atoms the energy of orbital depends on two quantum numbers ‘n’ and ‘l’ as well.The azimuthal quantum number, l, is also called subsidiary quantum mumber. Atomic orbitals with the same value of ‘n’ but different values of ‘l’ constitute a subshell belonging to the shell for the given ‘n’. The number of subshells in a shell is equal to ‘n’. Thus, the third shell contains three subshells (with three different values of ‘l’), the second shell contains two subshells and the first shell contains only one subshell. The values of ‘l’ range from 0 to (n - 1). Thus, the K shell (with n = 1) contains only one subshell having l = 0. The subshells or sub-levels have ‘l’ to be 0, 1, 2, 3, ..... which are represented by the symbols s, p, d, f,............., respectively.

The magnetic orbital quantum number, ml, gives information about the relative spatial orientation of the orbitals in a given subshell. For any subshell (defined by ‘l’ value) (2l + 1) values of ml are possible which range through:

ml = - l, - ( l - 1 ),- ( l - 2 )..........,0,.....( l - 2 ), ( l - 1 ), l.

Thus for the subshell ‘s’ with l = 0, the only allowed value of ml = 0. In other words, ‘s’ subshell has only one orbital in it. For the subshell ‘p’ having l = 1, the allowed values of ml are -1, 0, +1. Thus ‘p’ subshell contain three orbitals having distinct orientations, and so on. The sum of orbitals in a constituent subshell gives the total number of orbitals in a concerned shell and is given by n2 .

Distribution of orbitals in shells and subshells

Electron spin quantum number, ms, specifies the spin state of the electron in an orbital. An electron spins around its axis. This imparts spin angular momentum, to the electron.The two orientations which the spin angular momentum of an electron can take up give rise to the spin states which can be distinguished from each other by the spin quantum number, ms, which can be either + 1/2 or -1/2. The two spin states are represented by two arrows, ↑ (pointing up) and ↓ (pointing down) and thus have opposite spins. “An orbital can accomodate maximum two electrons and they must have opposite spins.” This is known as Pauli exclusion principle.

Shapes of atomic orbitals : 

The probability of finding an electron at a given point in an atom is proportional to square of the wave function ψ2 at that point. According to Max Born ψ2 at a point in an atom is the probability density of electron at that point.

(a) Shapes of 1s and 2s orbitals

Figure (a) shows the probability density diagrams of 1s and 2s atomic orbitals. These diagrams appear like a cloud. The electron cloud of 2s orbital shows one node, which is a region with nearly zero probability density and displays the change of sign for its corresponding wavefunction.

(b) Shapes of 1s and 2s orbitals

Figure (b) shows the boundary surface diagram of atomic orbitals 1s and 2s, which are spherical in shape. Here, a boundary surface is drawn in space for an orbital such that the value of probability density ψ 2 is constant and encloses a region where the probability of finding electron is typically more than 90%. Such a boundary surface diagram is a good representation of shape of an orbital. 

The s orbitals are spherical in shape. Their size increases with increase of n. It means that the electron is located farther away from the nucleus as the principal quantum number n increases.

Shapes of 2p Orbitals

Figure displays the boundary surface diagram with the nucleus being at the origin, for the three 2p orbitals (n = 2 and l =1). It can be seen that each p orbital has two lobes on the two sides of a nodal plane passing through the nucleus. (A nodal plane has ψ2 very close to zero) The size and energy of the dumbell shaped three 2p orbitals are the same. Their orientations in space are, however, different. The lobes of the three 2p orbitals are along thex, y and z axes. Accordingly the corresponding orbitals are designated as 2px, 2py and 2pz. The size and energy of the orbitals in p subshell increase with the increase of principal quantum number.

Shapes of 3d Orbitals

There are five orbitals associated with d subshell. Designated by dxy, dyz, dxz, dx2-y2. and dz2. The shapes of the five 3d orbitals are shown in Figure. In spite of difference in 

their shapes, the five d orbitals are equivalent 

in energy. The shapes of 4d, 5d, 6d...... orbitals 

are similar to those of 3d orbitals, but their 

respective size and energies are large or they 

are said to be more diffused.