The Quantum Mechanical model of an Atom

Developments leading to quantum mechanical model of Atom:

As, the drawbacks of Bohrsinglequotes model became obvious, a new model of atom was required and efforts were made to develop it.

There were two fundamental theories behind the development of the new model:

i. de Broglie priniciple of dual nature of matter suggested by the french scientist Louis de Broglie.

ii. Heisenberg uncertainty principle.

Lets look at these in detail:

de Broglie priniciple and dual nature of matter and radiation:

Einstein in 1905 had already suggested that light had dual nature both that of a particle and a wave.In 1924, french physicist Louis de Broglie suggested that like photons of light, all particles such as an electron or a cricket ball (everything macroscopic and microscopic) had dual nature , and the wave associated with a particle was called matter wave or de Broglie wave.Hence , according to him matter was wave-like and any particle moving with a linear momentum p should have a wavelength given by the de Broglie relation:

                                                           λ =  h/p

i.e., particles moving with high linear momentum has short wavelength and vice versa.So, macroscopic bodies  have such a high mometa (becuase their mass is so high in magnitude ) that their wave -like can not be observed.

The wavelength of a particle was calculated the analogy with photon as:

In case of photon , since it is assumed to have wave caharacter, (by the dual nature light):

                                E = h f ............................  (i)

where f is the frequency and h is the plancksinglequotes constant.

Using the energy equation for a particle:

                             E = mc²  ........................................ (ii)

From, (i) and (ii) we have,

                          hf = mc²

Also,                   f = c/λ 

?                         λ = h/mc

So, the valocity c of light is replaced by velocity v of the particle and mass of the photon is replaced by the mass of the particle. So, the above equation becomes:

                            λ = h/mv  = h/p , since mv = p is the momentum of the particle.

In the equation above, λ ( called de Broglie wavelength ) is inversely proportional to the wavelength.De Brogliesinglequotes equation relates a particle character with the  wave character of matter.

There were a of experiments carried out which proved the dual character of matter.Some of them are:

i) Davisson and Germersinglequotes experiment:American physicist Clinton Davisson and Lester Germer observed diffraction of electrons by a crystal.Diffraction is an interference caused by an object in the path of waves.Depending on whether the interference is constructive or destructive a pattern or increased or reduced intensity was obtrained.This experiment was later also repeated with other particles like α particles and molecular hydrogen and it was established that particles have wave like properties.

ii)Thomsonsinglequotes experiment: G P Thompson, working in scotland,conducted a experiment with a beam of electrons and a gold foil.He showed that the beam was diffracted when passed through a thin gold foil.This formed the basis of electron diffraction technology used in microscopy.This also established that particles had wave-like property.

Similarly particle character of electrons was also established through experiments , like milikan oil drop experiment, black body radiation , photo-electric effect and so on.

De brogliesinglequotes wavelength of the electron when accelerated through an potential difference V:

If an electron is accelerated through a potential difference , it acquires kinetic energy.Energy gained by an electron when accelerated through a potential difference of 1V is one electron-volt. (1 eV = 1.602 X 10^-19 J).The energy acuired is the product of charge on the electron and the potential difference.This is the kinetic energy of the electron.

Also, kinetic energy of a particle is 1/2mv²

So, we get,

1/2 mv² = eV .

?v = ?(2eV/m)

λ = h/mv = h/m X ?(2eV/m)

                    = h/?(2emV)

Putting  value of the constants we get,

 λ = 1.226 X 10-9 / ?V metres . 

 Hence, wavelength of electron can be calculated  is the potential is known .The wavelengths of electrons found are in the order of the sizes of the molecules.So, electron diffraction is used in the study of determaination of molecular structure.

Heisenbergsinglequotes Uncertainty Priniciple:

Heisenbersinglequoteg Uncertainty priniciple states that , it is impossible to specify simultaneously , with absolute precesion,both  the momentum and the position of a  sunbatomic particle.The product of uncertainty in position Δx and uncertainty in momentum Δp is contant.Therefore, if we increase the certainty in the position then the uncertainty in momentum increases and vice versa,which means that we if measure position with position perfectly precisely then momentum becomes completely unpredictable.

The quantitative version of the priniciple is given by : 

                    ΔpΔx ? h/2π

or,                    mΔvΔx ? h/2π  ( since , p = mv )

Also , it should be noted that the position and momentum (and theiruncertainty) are measured along the same axis.

Heisenberg uncertainty principle only holds significance for microscopic particles.

So, the Bohrsinglequotes model of definite postion (fixed circular orbits) and definite velocity was replaced by probability of the electron having a given position and momentum.

[SolvedExample]

Example6: Show that the electron can not exist in a nucleus.

Solution: The diameter of nucleus in an atom is 10^-15 m.So, the maximum uncertainty in position is 10^-15 m. singlequote

The uncertainty principle is m Δv  ? h/2π,

  The minimum uncertainty in the velocity would be,  ( mass of the electron = 9.1 X 10-31 Kg) 

                                                               Δv = h/ 2π X m X Δx

                                                    = 6.626 X 10^-34 / 2 X 3.14 X 9.31 X 10^-31 X 10^-15

                                                    = 5.77 X 10¹⁰ m sec^-1

So, the uncertainty in velocity is more than the speed of ligt which is not possible.

Hence, an electron cansinglequotet exist inside a nucleus.

 [/SolvedExample]

After the discovery of uncertainty principle the Bohrsinglequotes model of fixed orbit with fixed energy and velocity became unacceptable so there was a need for new model of an electron, and it is when quantum mechanical model of an atom came into picture. The branch of science is called Quantum mechanics and is different from Classical mechanics of Isaac Newton, which successfully explained the motion of macroscopics particles.

Quantum mechanical model of atom - atomic orbital and Schrodinger wave equation.

The Schrodinger wave equation developed by Ervin Schrodinger in 1926 , and it describes the motion of the electron in three dimensional space surrounding the nucleus.It forms the basis of quantum mechanics.

       
The wave function ψ doesnsinglequotet  hold any physical significance as such but the square of the wave function represents the intensity of the electron wave which according to the Schrodinger wave equation represents the probability of the electron at that point i.e., electron density at that point.

In quantum mechanical model the postion of electron is not described with certainty but with a probability at a particular position around the nucleus at any given instant of time.So, if the position of electron around the nucleus at any time is represented by a dot then the postion of electron over a certain period of time can be seen as (called electron cloud):

Electron cloud for an electron sorounding the nucleus

Image Source: universe-review.ca

The probability of finding an electron is not zero evn at large distances from the nucleus.Though, ity goes on diminishing as the distance from the nucleus increases.Thats why, electron clouds do not have sharp boundaries but the boundary slowly fades away.But, for sake of representation, the points of equal probability of the electron from the centre is connected to enclose a closed volume within which the probability of finding the electron is maximum , say upto 90%.This region is called atomic orbital.

Donsinglequotet confuse between orbit (which we discussed during Bohrsinglequotes model of an atom) and orbital.The differences between them are:

 Important features of Quantum mechanical model:

 The quantum mechanical model of an atom is essentially based on Schrodinger wave equation.Its main features are:

i. The energy of the electron is quantized and can be obtained through Schrodinger wave equation.

ii.The wave function ψ is also found by solving, the Schrodinger wave equation.Though the funciton ψ doesnsinglequotet have any physical significance as such but the square of the function (ψ²) represents the probability density of the electron at that point.

iii.By estimating the value of the wave funciton ψ at different points around the nucleus, a region can be defined where the probability of presence of electron is maximum.These region is called atomic orbital.So, the wave function is also called orbital wave function or simply atomic orbital.

iv.An electron can have multiple values of wave function which means an atom can have multiple atomic orbitals.

Application of the Schrodinger wave function

For hydrogen atom, the solution of the Schrodinger wave equation gives the value of energy (E) which the electron can have.The values of energy are called eigen values and the corresponding values of wave function (ψ) are called eigen functions.

 The motion of electron, its energy etc, is characterized by a set of four numbers called quantum numbers.Wesinglequotell see that in the next part of the chapter.