Quantum Numbers
To understand the concept of Quantum Numbers, we must known the meaning of some terms clearly so as to avoid any confusion.
First energy level (K or) has one sub-shell designated as 1s, the second energy level (L and 2) has two sub-shell as 2s & 2p, the third energy level (M or 3) has three sub shell as 3s, 3p and 3d, and the fourth energy level (N or 4) has four sub-shells as 4s, 4p, 4d and 4f. The energy of sub-shell increases roughly in the order: s < p < d < f.
To describe or to characterize the electrons around the nucleus in an atom, a set of four numbers is used, called as Quantum Numbers. These are specified such that the states available to the electrons should follow the laws of quantum mechanism or wave mechanics.
Higher is the value of n, greater is its distance from the nucleus, greater is its size and also greater is its energy. It also gives the total electrons that may be accommodated in each shell, the capacity of each shell is given by the formula 2n^{2}, where n: principal quantum number.
It can assume all integral values from 0 to n –1. The possible values of l are:
0, 1, 2, 3, ……., n – 1
Each value of l describes a particular sub-shell in the main energy level and determines the shape of the electron cloud.
When n = 1, l = 0, i.e., its energy level contains one sub-shell which is called as a s-sub-shell. So for l = 0, the corresponding sub-shell is a s-sub-shell. Similarly when l = 1, 2, 3, the sub-shell are called p, d, f sub-shell respectively.
As you known for n = 1, l = 0, there is only one sub-shell. It is represented by 1s. Now for n = 2, l can take two values (the total number of values taken by l is equal to the value of n in a particular level). The possible values of l are 0, 1. The two sub-shell representing the II^{nd} energy level are 2s, 2p. In the same manner, for n = 3, three sub-shell are designated as 3s, 3p, 3d corresponding to l = 0, 1, 2, and for n = 4, four sub-shells are designated as 4s, 4p, 4d, 4f corresponding to l = 0, 1, 2, 3, .
The orbital Angular momentum of electron =
m can be have any integral values between –l to + l including 0, i.e., m = –l, 0+, l, ……0, 1, 2, 3, 4,……….l, –1, +l.
We can say that a total of (2l + 1) values of m are there for a given value of l–2, –1, 0, 1, 2, 3.
In s sub-shell there is only one orbital [l = 0, → m = (2l + 1) = 1].
In p sub-shell there are three orbitals corresponding to three values of m: –1, 0 + 1. [l = 1 → m = (2l + 1) = 3]
These three orbitals are represented as p_{x}, p_{y}, p_{z} along X, Y, Z axes perpendicular to each other.
In d sub-shell, there are five orbitals corresponding to –2, –1, +2, [l = 2, → m = (2 × 2 + 1) = 5]. These five orbitals are represented as d_{xy}, d_{zx}, .
In f sub-shell there are seven orbitals corresponding to –3, –2, –1, 0, +1, +2, +3, [l = 3 → m = (2×3 + 1) = 7]
Aufbau Principle:
Aufbau is a German word meaning ‘building up’. This gives us a sequence in which various sub-shells are filled up depending on the relative order of the energy of the subs-hells. The sub-shell with minimum energy is filled up first and when this obtains maximum quota of electrons, then the next sub-shell of higher energy starts filling.
Exceptions of Aufbau Principle: In some cases it is seen that actual electronic arrangement is slightly different from arrangement given by Aufbau principle. A simple reason behind this is that half-filled and full-filled sub-shell have got extra stability.
Cr(24) → 1s^{2}, 2s^{2}2p^{6}, 3s^{2}3p^{6}3d^{4}, 4s^{2} (Wrong)
→ 1s^{2}, 2s^{2}2p^{6}, 3s^{2}3p^{6}3d^{5}, 4s^{1} (Right)
Cu(29) → 1s^{2}, 2s^{2}2p^{6}, 3s^{2}3p^{6}3d^{9}, 4s^{2} (Wrong)
→ 1s^{2}, 2s^{2}2p^{6}, 3s^{2}3p^{6}3d^{10}, 4s^{1} (Right)
Similarly the following elements have slightly different configurations than expected.
Nb → [Kr] 4d^{4}5s^{1}
No → [Kr] 4d^{4}5s^{1}
Ru → [Kr] 4d^{7}5s^{1}
Ph → [Kr] 4d^{8}5s^{1}
Pd → [Kr] 4d^{10}5s^{0}
Ag → [Kr] 4d^{10}5s^{1}
Pt → [Xe] 4d^{14}5s^{9}6s^{1}
Au → [Xe] 4d^{14}5d^{10}6s^{1}
Shapes of Atomic Orbitals
The orbital with the lowest energy is the 1s orbital. It is a sphere with its center of the nucleus of the atom. The s-orbital is said to spherically symmetrical about the nucleus, so that the electronic charge is not concentrated in any particular direction. 2s orbital is also spherically symmetrical about the nucleus, but it is larger than (i.e., away from) the 1s orbit.
Dual Character
In case of light some phenomenon like diffraction and interference can be explained on the basis of its wave character. However, the certain other phenomenon such as black body radiation and photoelectric effect can be explained only on the basis of its particles nature. Thus, light is said to have a dual character. Such studies on light were made by Einstein in 1905.
Louis de-Broglie, in 1942 extended the ideal of photons to material particles such as electron and he proposed that matter also has a dual character-as wave and as particle.
Derivation of de-Broglie equation: The wavelength of the wave associated with any material particle was calculated by analogy with photon, if it is assumed to have wave character, its energy is given by
E = hν ……….(i) (according to the Planck’s quantum theory)
Where ν is the frequency of the wave and ‘h’ is Plack’s constant
If the photon is supposed to have particle character, its energy is given by
E = mc^{2}…….(ii) according to Einstein’s equation)
Where ‘m’ is the mass of photon, ‘c’ is the velocity of light.
By equating (i) and (ii)
hν = mc^{2}
But ν = c/λ
Or λ = h/mc
The above equation is applicable to material particle if the mass and velocity of photon is replaced by the mass and velocity of material particle. Thus for any particle like electron.
λ = h/mv (or)
Where mv = p is the momentum of the particle.
Heisenberg’s Uncertainty Principle:
All moving objects that we see around us e.g., a car, a ball thrown in the air etc, move along definite paths. Hence their position and velocity can be measured accurately at any instant of time. Is it possible for subatomic particle also:
As a consequence of dual nature of matter. Heisenberg, in 1927 gave a principle about the uncertainties in simultaneous measurement of position and momentum (mass × velocity) of small particles. This principle states.
It is impossible to measure simultaneously the position and momentum of a small microscopic moving particle with absolute accuracy or certainty i.e., if an attempt is made to measure any one of these two quantities with higher accuracy, the other becomes less accurate.
The product of the uncertainty in position (Δx) and the uncertainty in the momentum (Δp = m.Δv where m is the mass of the particle and Δv is the uncertainty in velocity) is equal to or greater than h/4p where h is the Planck’s constant.
Thus, the mathematical expression for the Heisenberg’s uncertainty principle is simply written as
Explanation of Heisenberg’s uncertainty Principle: Suppose we attempt to measure both the position and momentum of an electron, to pin point the position of the electron we have to use light so that the photon of light strikes the electron and the reflected photon is seen in the microscope. As a result of the hitting. The position as well as the velocity of the electron is distributed. The accuracy with which the position of the particle can be measured depends upon the wavelength of the light used. The uncertainty in position is +l. The shorter the wavelength, the greater is the accuracy. But shorter wavelength means higher frequency and hence higher energy. This high energy photon on striking the electron changes its speed as well as direction. But this is not true for macroscopic moving particle. Hence Heisenberg’s uncertainty principle is not applicable to macroscopic particles.
Pauli’s Exclusion Principle: According to this principle
No two electrons in an atom can have the same set of all the quantum numbers or one can say that no two electrons can have the same quantized states.
Consider an electronic arrangement in 1^{st} energy level (n = 1). For n = 1, l = 0, and m = 0. Now s can have to values corresponding to each value of m i.e., s = +1.2, –1/2 (n, 1, possible designation of an electron in a state with n = 1 is 1, 0, 0, +1/2 and 1, 0, 0, –1/2 (n, l, m, s) i.e., two quantized states. This implies that an orbital can accommodate (for n = 1, m = 0, → one orbital) maximum of two electrons having opposite spins.
The maximum number of electrons in the different sub-shells = 2 (2l + 1)
s-sub-shell = 2, p-sub-shell = 6, d-sub-shell = 10 and f-sub-shell = 14
Hund’s Rule of Maximum Multiplicity
This means an electron always occupies a vacant orbital in the same sub-shell (degenerate orbital) and pairing starts only when all of the degenerate orbitals are filled up. This means that the pairing starts with 2^{nd} electron in a sub-shell, 4^{th} electron is p-sub-shell, 6^{th} electron in d-sub-shell and 8^{th} electron in f-sub-shell.
By doing this, the electrons stay as far away from each other as possible. This is highly reasonable if we consider the electron-electron repulsion. Hence electron obey Hund’s rule as it results in lower energy state and hence more stability.
Node and Nodal Plane
Node is defined as a region where the probability of finding an electron is zero.
The planes passing through the angular nodal points are called nodal planes.
Nodes
No. of radial or spherical nodes = n – l – 1
No. of angular nodes,= l, Total number of nodes = n – 1