The atomic orbital is a complex mathematical function called a wave function that determines the energy, angular momentum, and location of an electron. A better way to define the atomic orbital is the space around the nucleus, which has a high probability of finding the electron.
The simplest orbital of all is the 1s orbital, which is spherical in shape (see image below). The nucleus is at the center of the orbital and the electron revolves around the nucleus.
The region around the core is the probabilistic region, not the discrete one. The probability of finding the electron in this area is high (about 90%) and the electron can exist beyond that. Therefore, an orbital is an electronic probabilistic cloud around the nucleus.
In quantum mechanics, a wave function is a complex mathematical description of a quantum state. It gives the probability distribution of an electron and is denoted by Ψ. Each orbital has its own Ψ. |Ψ|2represents the probability density function. The probability of an electron in the positionxbetween a and b is given by the following equation.
We can normalize the equation.
Thus, when the probability of finding an electron is 100%, the size of the orbital is equal to the size of the universe, that is, −∞ to ∞. In practice, we define the orbital based on 90% probability, not 100%.
The hydrogen atom has only one electron and is the only atom for which the Schrödinger equation can be solved exactly. The wave function for the hydrogen atom in the spherical coordinate system is as follows:
R(R) is the radial function that determines the distance from the nucleusYM
eu(EU,ϕ)is a spherical harmonic that determines the direction of the orbital. Here,N,eu, EMare the principal, azimuthal, and magnetic quantum numbers.R(R) is expressed below.
Here,A0is the Bohr radius,, Eeu2eu+1
N−eu−1is a generalized Laguerre polynomial.
As you can see above, the quantum numbersN,eu, EMDecide on the equation for the Ψ wave function. For each set of these three quantum numbers, we get a new wave function. From this we can conclude that these three quantum numbers represent an orbital.
The main quantum numberNis a set of positive integers;N=1, 2, 3… The azimuthal quantum numbereuis a set of non-negative integers in the range 0 to (N−1);eu=0, 1, 2 … (N−2), (N−1). Finally, the magnetic quantum numberMis a set of integers from −euForeu;M=−eu, −(eu−1) … −1, 0, 1 … (eu−1),eu.
An atom can have a large number of orbitals. The three quantum numbers mentioned in the above section are used to identify an orbital in an atom.
In an atom, electronic space is divided into layers. Each layer is represented by a principal quantum number value.N. For example,N=1 is the lowest energy shell, called the K shell,N=2 is the second shell, L shell and so on.
Within each shell are subshells. The number of subshells in a shell is determined by the azimuthal quantum numbereu. For each value ofN, the possible values ofeuare 0, 1, 2...N−2,N−1. Thus, for layer K (N=1),eu=0, para L-shell (N=2),eu=0, 1; para M Shell (N=3),eu=0, 1, 2 and so on. any value ofeucorresponds to a sublevel.eu=0 is a sublevel s,eu=1 is a p-subshell,eu=2 is a sublevel d,eu=3 is a subshell f, etc. The following table summarizes the same.
|principal quantum numberN||manga||Azimuthal Quantum Numbereu||lower shell||number of subshells|
|2||eu||0, 1||s, s||2|
|3||M||0, 1, 2||s, p, d||3|
|4||N||0, 1, 2, 3||s, p, d, f||4|
|5||Ö||0, 1, 2, 3, 4||s, p, d, f, g||5|
|6||P||0, 1, 2, 3, 4, 5||s, p, d, f, g, h||6|
|7||Q||0, 1, 2, 3, 4, 5, 6||s, p, d, f, g, h, i||7|
|8||R||0, 1, 2, 3, 4, 5, 6, 7||s, p, d, f, g, h, i, k||8|
|…||…||0, 1, 2, 3, 4, 5, 6, 7…||s, p, d, f, g, h, i, k…||…|
|Note: When naming subshells, the letter j is ignored.|
From the table above, theN-th shell temNsublayers of. Each of these subshells consists of a number of orbitals determined by the magnetic quantum numberM. For eacheu, the possibleMare 0, ±1, ±2 … ±(eu−1), ±eu. any value ofMcorresponds to an orbital.
For a sublevel s (eu=0),M=0. Therefore, it has only one orbital called an s orbital. For a p-subshell (eu=1),M=0, ±1. Therefore, a p subshell has three orbitals: pz(M=0) e pxe Pj(M=±1).
In a subshell d (eu=2), there are five orbitals: dz2(M=0), dxze de(M=±1) e dxye dx2−j2(M=±2).
In an f-subshell (eu=3), there are seven orbitals: fz3(M=0), fxz2e fe2(M=±1), fxyze fz(x2−j2)(M=±2) e fx(x2−3j2)e fj(3x2−j2)(M=±3).
|lower shells||Azimuthal Quantum Numbereu||magnetic quantum numberM||Orbital|
Other higher orbitals, viz. g, h, i etc. are very complicated and rarely encountered.
As can be seen from the previous table, the number of orbitals strangely increases with the azimuthal quantum number. The number of orbitals of a dieeuis 2eu+1.
For a given value of the principal quantum numberN, the number of orbitals is given byN2.
The possible values ofeuper diceNare 0, 1, 2...N−2,N−1 and for alleuthere are 2eu+1 orbitals. The number of orbitals for a given valueN:
An orbital can hold the top two electrons, so given the number of electrons for aNis 2N2.
Usually, the orbital is named with a combination of numbers and letters. consider 4dz2orbital; the first number represents the principal (or shell) quantum number. Here is 4 (n shell). The number is followed by an alphabet, where d represents a subshell. The third is the alphabet index. Indicates the orientation of an orbital. then 4dz2Orbital is oriented along the d-sublevel of the 4th levelz-Axle.
Orbitals can also be simply represented by the three quantum numbers, for example 4dz2is 420 (N=4,eu=2,M=0).
Let's go to another example:N=5,eu=3,M=±1 corresponds to 5fxz2and 5fe2orbitals. Both are at sublevel f of 5th level. 5fxz2rests onxzlevel for 5fe2rests oneLevel.
Each orbital has a unique shape, and the shape becomes more complex and difficult to follow as we move to higher orbitals.
The s orbital is spherical; The nucleus is at the center of the sphere. It is not oriented in any direction. In other words, it is undirected.
There are three dumbbell-shaped p orbitals. Each orbital has two lobes aligned on one of three axes. the pzorbital is aligned with thez- Axis, Sxasx-axis and finally pjasj-Axle.
o Dz2Orbital has two lobes on thez-Axis and a donut on thexyLevel. The remaining four d orbitals each have two dumbbells. Dxze delie on thexzEeplanes, while the other two, Ixye dx2−j2, They are inxyLevel. Dxze deare identical to each other. They have the same radial component of the wave function and differ only by an angle of 90°. In the same way dxye dx2−j2are identical and differ by an angle of 45°.
o fz2Orbital has two lobes along thezshaft and two donuts in the middle. fxz2e fe2are similar to dxze deexcept they have two additional bean-shaped lobes lined up on thex- that's me fxz2E noj- that's me fe2. Fxyze fz(x2−j2)it has eight lobes; four of them are below thatxyAirplane and the remaining four above. fx(x2−3j2)e fj(3x2−j2)each has six lobes; All lobes are placed in thexyLevel. All three pairs above are identical and have the same radial component but different angular components.
With the exception of the s orbital, all orbitals are directional – they point in a specific direction.
For s orbitals applyeuEMare always zero, the only onesNvaries. ifN=1, it is 1s orbital. The wave function of the 1s orbital for the hydrogen atom can be obtained by substitutionN,eu, EMas 1, 0, 0 in the generalized wave function mentioned above.
PS2is the probability density function. The probability density function is the probability of finding an electron per unit volume. it's at the maximumR=0 and approaches zero as the value increasesR(see first chart below). We get the radial probability by multiplying the area of the sphere by 4πR2, for the probability density. The third graph shows the radial probability versus the radius. At theR=0 eR=∞, the probability is zero. From the graph, the probability peaks atR=52.9 pm, which is also the Bohr radius (A0).
SeN=2 eN=3 we get the 2s and 3s orbitals. Its wave functions are as follows:
The following graph is the radial probability (Ψ2R2) versus radius for 1s, 2s, and 3s orbitals.
As can be seen from the diagram above, the 1s orbital has only one peak, while the 2s and 3s have two and three peaks, respectively. As a result, there is a minimum probability point between every two peaks. With the exception of extremes, the probability at these points becomes zero. Those points where the probability becomes zero are called nodes. The 1s orbital has zero nodes, the 2s orbital has one node, and the 3s orbital has two nodes; in general we can sayNorbital hat s (N−1) it.
Note: The different colors in the graphs indicate different phases of the orbital, as well as peaks and troughs in the wave.
Nodes in orbitals are similar to standing wave nodes, where they are defined as points with zero amplitudes. You can think of nodes as the space between spheres that are placed inside each other.
From the above diagram it can also be seen that the size of the orbital increases with the principal quantum number – that is, 3s>2s>1s.
The p orbital corresponds toeu=1. The p orbital, unlike the s orbital, is not spherical; it is dumbbell shaped. As already mentioned, there are three p orbitals: pz(eu=1;M=0) e pxe Pj(eu=1;M=±1).
ForN=2 the wave functions are as follows:
PS210is the wave function for 2pzorbital e Ψ21±1is for 2pxand 2pj. OEUEϕA part of Ψ decides the orientation of the orbital. In the graphs below, only the radial part of Ψ is considered, i.e. H.EUEϕis ignored.
Unlike the s orbital, the probability density of the p orbital is not maximal atR=0. We can see in the figure above that it is at zeroR=0. In the middle, the value reached its peak. The radial probability plot also shows similar behavior. Since the probability density in the nucleus is zero, the electron spends more time outside the nucleus in the p orbital than in the s orbital, where the probability density in the nucleus is maximum.
Also, the probability is zero in thexyplane for 2pzorbital. Likewise, the corresponding aircraft for 2pxand 2pjThey areeExz. These levels, at which the probability becomes zero, are called node levels. There are three levels of nodes in p orbitals, as shown below.
These levels of nodes are called angle nodes. Thus, p orbitals have three angular nodes. The s orbital has no angular nodes; they have only radial nodes. The p orbitals have radial and angular nodes.
The 3p wave functionsz, 3pxe 3pjThey are as follows:
The diagram below shows the radial node in 3pzorbital. 2pzit has no radial nodes.
me 4 pzOrbital, not shown in the figure above, has two radial nodes. In short, theNp-orbital hat (N−2) radial nodes.
There are five d orbitals: dz2,Dxz,De,Dxy, e dx2−j2. Of these orbitals, dz2it's unique; has two lobes on thez-axis and a donut-shaped lobe on thexyLevel. The d orbitals emanate from the 3rd shell and their wave functions are mentioned below.
In all the above wavefunctions, the angular parts are different, while the radial parts are multiples of each other.
Like p orbitals, d orbitals have angular nodes. Each d orbital has 2 angular nodes.
The radial nodes for theNd are orbitals (N−3) us. So 3d has no radial nodes. Radial nodes in the d orbital start at 4d.
The f orbital is much more complex than the d orbital and is observed in heavy elements. There are seven f orbitals: fz3, Fxz2, Fe2, Fxyz, Fz(x2−j2),x(x2−3j2), e fj(3x2−j2).
Each of these orbitals has 3 angular nodes. Radial nodes start at 4f;Nf orbital hat (N−4) radial nodes.
As discussed above, nodes are the region with zero probability of finding electrons. They can be divided into two types: angular and radial. Radial nodes are obtained from the radial component of the wave function, while angular nodes are obtained from the angular component, ie H. spherical harmonics.
|orbital||N||eu||radial nodes||angular nodes||us in general|
From the table above, we generalize the formula for ourselves. The angular nodes depend only on the value ofeuand it's the sameeu. On the other hand, radial nodes depend on bothNEeuand is given byN−eu−1. Therefore, the total number of nodes is the sum of both and isN−1.
Orbit x Orbit
Orbits and orbitals are often confused with each other, especially by beginners. They sound the same, but they are completely different concepts and should not be used interchangeably. The table below explains the difference between the two.
|Orbits are fundamental to the Bohr model, a predecessor of the quantum mechanical model; describes the orbit as a circular path followed by an electron. Orbits are concentric circular paths of electrons.||An orbital is a space around the nucleus of an atom that has a high probability of finding an electron.|
|The orbits come from the Bohr-Rutherford atomic model.||The notion of orbitals was derived from the quantum model of an atom.|
|Orbits are always concentric circles. In the modified Bohr model, the orbits have an elliptical shape.||Orbitals come in multiple forms and their shapes become mind-boggling as we move to higher orbitals. The simplest of these is the s orbital, which is spherical in shape.|
|The orbit is a two-dimensional figure.||Orbitals are three-dimensional.|
|They are direction independent.||They are directional except for the s orbitals.|
|Since orbits have a specific path, we can predict their position and momentum. This contradicts Heisenberg's uncertainty principle.||We cannot estimate the exact position of an electron in an orbital. We can only find the probability of one electron; uncertainty always remains.|
|Orbits are not real; they do not exist and electrons do not revolve around the nucleus in a specific orbit.||Orbitals are realistic.|
- principal quantum number
- Azimuthal Quantum Number
- Bohr's Atomic Model
- Rutherfords Atommodell