# Quantum Chemistry

At the end of 19th century, Newton’s mechanics had been completely developed and many physicists felt that no more laws were needed to explain the natural phenomena. The Newton’s laws could explain the motion of everyday objects and planets and even that of atoms and subatomic particles. However, with the development of technology more facilities were available to conduct experiments under conditions impossible till then. As a result experimental evidence accumulated which showed that the Newtonian mechanics, now called classical mechanics was not applicable to small particles such as atoms (microscopic particles). A new mechanics had to be developed which could explain the behaviour of microscopic particles.
According to classical physics, the trajectory or the path of a particle can be accurately found and its position at any instance can be calculated if that at present is known. There were several experimental evidences which showed that classical mechanics failed when applied to microscopic particles.
Failures of classical physics
Let us review some experimental evidences which showed that several concepts of classical physics are untenable.
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**1. Black body radiation**

On heating an iron rod it becomes red followed by white and then blue. The wavelength of the emitted radiation moves from higher to lower values while the frequency moves from lower to higher values. The exact frequency spectrum varies from one body to another. However, we have an ideal body called the black which absorbs and emits radiation of all frequencies. A pin hole in a heated container is the nearest example of a black body.

A study of the radiation emitted by a black body as a function of temperature showed that the plot of Intensity versus the wavelength or frequency had a guassian shape with a maximum. The maximum of the peak shifts to longer frequency or shorter wavelength as the temperature is raised. Wien formulated an empirical law relating the λmax and the temperature.
Joseph Stefan considered the total energy density **[math]\varepsilon[/math]**, the total electromagnetic energy in a region divided by the volume of the region

- [math]\varepsilon=\frac{E}{V}\,[/math]

and related it to the temperature. Using these two laws we could calculate the λmax if the temperature is known and the energy density if the temperature is known.

Theoretical study of the black body radiation was carried out by Rayleigh. His calculations were based on the assumption that the electromagnetic field is a collection of oscillators of all possible frequencies. He formulated the Rayleigh-Jeans law. The raditiaon curves obtained using this law overlapped with those obtained experimentally at longer wavelengths but at shorter wavelengths, there was no correlation. According to the Rayleigh –Jeans law event the oscillators of very short wavelength (corresponding to uv, X rays, gamma rays etc) are strongly excited at room temperature. Therefore as per the theoretical equation which is actually based on the classical mechanics, large amount of energy is radiated in the high frequency region of the electromagnetic spectrum. This absurd result is called ultraviolet catastrophe. We expect that even cool objects should glow in darkness, there should in fact be no darkness!!!

## 2.Planck distribution

The German physicist Max Planck offered a successful explanation of the black body radiation. He assumed that the radiation emitted by the black body was caused by the oscillations of electrons in the constituent particles of the body. These electrons were pictured as oscillating in an atom much like electrons oscillate in an antenna to give off radio waves. In these “atomic antennae”, however, the oscilllations occur at much higher frequency i.e., uv, vis, IR rather than radio wave region. Rayleigh and Jeans assumed that the energies of the electronic oscillators responsible for the emission of the radiation could have any value; one of the basic assumptions of classical physics. In classical physics, the variables that represent observables (such as position, momentum and energy) can take on a continuum of values.

Since this assumption could not explain the radiation curves of a black body, Planck made a revolutionary assumption that the energies of the oscillators were discrete (only certain values could be taken) and had to be proportional to an integral multiple of the frequency or in equation form

- [math]E = nh\nu\,[/math]

where **[math]E[/math]** is the energy of an oscillator, **[math]n[/math]** is an integer, **[math]h[/math]** is proportionality constant called Planck’s constant;

- [math]h=6.626\text{ x }10^{-34}\rm Js\,\![/math]

A ray of light of a certain frequency can be thought of consisting of a stream of particles called photons, each one having an energy **[math]h\nu[/math]**. The oscillators of the electromagnetic field could be excited only when they are supplied with an energy hν. When this value is large, which is the case with high frequency oscillators or low wavelength oscillators, such oscillators are not excited because the particles of the body are not able to supply that much energy. By restricting the energy to discrete values the contribution from the high frequency oscillators is restricted. The Planck’s distribution relates the energy density to the wavelength. The mathematical forma of the Planck’s equation is such that the energy density approaches zero at high frequencies, in agreement with the experimental observation. At longer wavelengths the Planck distribution reduces to the Rayleigh-Jeans law and of course agrees well with the observations.

In essence, Planck found that oscillators cannot possess arbitrary amounts of energy. The discovery of quantisation (Latin quantum – amount) led to quantum mechanics. The whole structure of physics had to be revised.

## 3. Heat Capacities

If you need to account for blackbody radiation then you must also examine how energy is taken up by the electromagnetic field. And accounting for heat capacities involves examining how energy is taken up by the vibration of particles. A study of heat capacity also supported quantization.
As per the Dulong and Petit law the molar heat capacity at constant volume is C_{v,m} = 3R and is independent of temperature. With the advancement of technology, scientists could make measurements at lower temperatures also. Einstein tested Dulong and Petit law at low temperatures and found significant deviations. All metals were found to have a value of 3R for molar heat capacities at higher temperatures while at lower temperatures these were lower than 3R at low temperatures. The values approached zero as temperature approached zero.

Heat capacity

- 3R at higher temperatures
- lower than 3R at lower temperatures.
- approaching zero as temp. approached zero.

(to be contd...)