Wein’s Laws

John Strutt, 3rd Baron Rayleigh, Senior Wrangl...

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Various workers tried to explain the problem of energy distribution in black body radiation and finally the problem was successfully solved by German Physicist Max Planck. Before him, German Physicist Wilhelm Wein and British Physicist Lord Rayleigh & James Jean have tackled this problem and have given important laws. In fact, the work of their scientists paved the way for Planck to give his famous theory of radiation.

In this series of articles, I shall be discussing the various laws, special concentration on Planck’s law, concerning the black body in the brief.
This article is very first of the series & in I shall discuss briefly about Wein’s Laws. Other two useful topics will be discussed later.

Wein’s Formula & Wein’s Laws

The problem of black body radiation was first theoretically tackled by Wein in 1893. Besides giving a general formula for the energy distribution in the blackbody radiation, he gave following important and useful laws.

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Largest Prime Numbers

What is a Prime Number?

An integer, say $ p $ , [ $ \ne {0} $ & $ \ne { \pm{1}} $ ] is said to be a prime integer iff its only factors (or divisors) are $ \pm{1} $ & $ \pm{p} $ .

As?

Few easy examples are:
$ \pm{2}, \pm{3}, \pm{5}, \pm{7}, \pm{11}, \pm{13} $ …….etc. This list goes up to infinity & mathematicians are trying to find the larger one than the largest, because primes numbers has no distinct pattern (as any one cannot guess the next prime after one.) As of now the biggest prime number found is  $ M-47 $ , called as Mersenne’s 47. This has an enormous value of $ 2^{43112609} -1 $ . It is very hard to write it on paper because it consists of $ 12978189 $ digits.
»M47 was Invented in 2008.

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Albert Einstein and His introduction to the Concept of Relativity

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Albert Einstein

 

This name need not be explained. Albert Einstein is considered to be one of the best physicists in the human history.

The twentieth century has undoubtedly been the most significant for the advance of science, in general, and Physics, in particular. And Einstein is the most luminated star of the 20th century. He literally created cm upheaval by the publication, in quick succession, in the year 1905, two epoch-making papers, on the concept of the photon and on the Electrodynamics of moving bodies respectively, with yet another on the Mathematical analysis of Brownian Motion thrown in, in between.

A Curvature

The Electrodynamics of moving bodies was the biggest sensation and it demolished at one stroke some of the most cherished and supposedly infallible laws and concepts and gave the breathtaking new idea of the relativity of space and time.

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Derivative of x squared is 2x or x ? Where is the fallacy?

As we know that the derivative of $ x^2$ , with respect to $ x$ , is $ 2x$.

i.e., $ \dfrac{d}{dx} x^2 = 2x$

However, suppose we write $ x^2$ as the sum of $ x$ ‘s written up $ x$ times..

i.e.,

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Solving Ramanujan’s Puzzling Problem

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Consider a sequence of functions as follows:-

$ f_1 (x) = \sqrt {1+\sqrt {x} } $
$ f_2 (x) = \sqrt{1+ \sqrt {1+2 \sqrt {x} } } $

$ f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } } $

……and so on to

$ f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt {1+3 \sqrt {\ldots \sqrt {1+n \sqrt {x} } } } } } $

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Its a Mystery! – Simulacrum in Eagle Nebula

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Of all the exotic & strangest photos ever taken from outer space, the most curious ones are of Eagle Nebula. According to the images received through various space cameras, Eagle Nebulae appear to made of gaseous clouds and they give birth to new stars. Eagle Nebulae are named after their bird alike shape.

First taken through Hubble, when this photo was first shown on CNN, hundreds of responses came in from people reporting that they could see a face in the cloud. When the color of the photo was adjusted, a large human form seemed to appear within the cloud.  Couldn’t you see a human (or a wolf-man) in above image? As Eagle Nebulae are more productive and quite younger than other Nebula types, they were named ‘Star Queen Nebula‘ later.

 

Researchers argued such illusions to be incidental. As clouds can take any shape, Eagle Nebulae do the same too. The Nebula clouds work as same as water clouds do. The one major difference is only that one yields stars and latter yields  water. NASA occasionally updates eagle nebula images and each time, a simulacrum appears. This is still a mystery.

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Raman Effect- Raman Spectroscopy- Raman Scattering

In contrast to other conventional branches of spectroscopy, Raman spectroscopy deals with the scattering of light & not with its absorption.

Raman Effect

Raman effect

Raman Effect: An Overview

Chandrasekhar Venkat Raman discovered in 1928 that if light of a definite frequency is passed through any substance in gaseous, liquid or solid state, the light scattered at right angles contains radiations not only of the original frequency (Rayleigh Scattering)  but also of some other frequencies which are generally lower but occasionally higher than the frequency of the incident light.

The phenomenon of scattering of light by a substance when the frequencies of radiations scattered at right angles are different (generally lower and only occasionally higher) from the frequency of the incident light, is known as Raman Scattering or Raman effect.
The lines of lower frequencies as known as Stokes lines while those of higher frequencies are called anti-stokes lines.

If $f$  is the frequency of the incident light &  $f’$  that of a particular line in the scattered spectrum, then the difference   $f-f’$  is known as the Raman Frequency. This frequency is independent of the frequency of the incident light. It is constant and is characteristic of the substance exposed to the incident light.

A striking feature of Raman Scattering is that Raman Frequencies are identical, within the limits of experimental error, with those obtained from rotation-vibration (infrared) spectra of the substance.
Here is a home-made video explaining the Raman Scattering of Yellow light:

And here is another video guide for Raman Scattering:

Advantage of Raman Effects

  •  Raman Spectroscopy can be used not only for gases but also for liquids & solids for which the infrared spectra are so diffuse as to be of little quantitative value.
  • Raman Effect is exhibited not only by polar molecules but also by non-polar molecules such as $O_2$, $N_2$, $Cl_2$ etc.
  • The rotation-vibration changes in non-polar molecules can be observed only by Raman Spectroscopy.
  • The most important advantage of Raman Spectra is that it involves measurement of frequencies of scattered radiations, which are only slightly different from the frequencies of incident radiations. Thus, by appropriate choice of the incident radiations, the scattered spectral lines are brought into a convenient region of the spectrum, generally in the visible region where they are easily observed. The measurement of the corresponding infrared spectra is much more difficult.
  • It uses visible or ultraviolet radiation rather than infrared radiation.

Uses

  •  Investigation of biological systems such as the polypeptides and the proteins in aqueous solution.
  •  Determination of structures of molecules.

C. V. Raman was awarded the 1930 Physics Nobel Prize for this.

Classical Theory of Raman Effect

The classical theory of Raman effect, also called the polarizability theory, was developed by G. Placzek in 1934. I shall discuss it briefly here. It is known from electrostatics that the electric field $ E $ associated with the electromagnetic radiation induces a dipole moment $ \mu $ in the molecule, given by
$ \mu = \alpha E $ …….(1)
where $ \alpha $ is the polarizability of the molecule. The electric field vector $ E $ itself is given by
$ E = E_0 \sin \omega t = E_0 \sin 2\pi \nu t $ ……(2)
where $ E_0 $ is the amplitude of the vibrating electric field vector and $ \nu $ is the frequency of the incident light radiation.

Thus, from equations (1) & (2),
$ \mu= \alpha E_0 \sin 2\pi \nu t $ …..(3)
Such an oscillating dipole emits radiation of its own oscillation with a frequency $ \nu $ , giving the Rayleigh scattered beam. If, however, the polarizability varies slightly with molecular vibration, we can write
$ \alpha =\alpha_0 + \frac {d \alpha} {dq} q $ …..(4)
where the coordinate q describes the molecular vibration. We can also write q as:
$ q=q_0 \sin 2\pi \nu_m t $ …..(5)
Where $ q_0$ is the amplitude of the molecular vibration and $ \nu_m $ is its (molecular) frequency. From equations. 4 & 5, we have
$ \alpha =\alpha_0 + \frac {d\alpha} {dq} q_0 \sin 2\pi \nu_m t $ …..(6)
Substituting for $ alpha $   in (3), we have
$ \mu= \alpha_0 E_0 \sin 2\pi \nu t + \frac {d\alpha}{dq} q_0 E_0 \sin 2\pi \nu t \sin 2\pi \nu_m t $ …….(7)
Making use of the trigonometric relation $ \sin x \sin y = \frac{1}{2} [\cos (x-y) -\cos (x+y) ] $ this equation reduces to:
$ \mu= \alpha_0 E_0 \sin 2\pi \nu t + \frac {1}{2} \frac {d\alpha}{dq} q_0 E_0 [\cos 2\pi (\nu – \nu_m) t – \cos 2\pi (\nu+\nu_m) t] $ ……(8)
Thus, we find that the oscillating dipole has three distinct frequency components:

1• The exciting frequency $ \nu $ with amplitude $ \alpha_0 E_0 $
2• $ \nu – \nu_m $
3• $ \nu + \nu_m $ (2 & 3 with very small amplitudes of $ \frac {1}{2} \frac {d\alpha}{dq} q_0 E_0 $ . Hence, the Raman spectrum of a vibrating molecule consists of a relatively intense band at the incident frequency and two very weak bands at frequencies slightly above and below that of the intense band.

If, however, the molecular vibration does not change the polarizability of the molecule then $ (d\alpha / dq )=0$ so that the dipole oscillates only at the frequency of the incident (exciting) radiation. The same is true for the molecular rotation. We conclude that for a molecular vibration or rotation to be active in the Raman Spectrum, it must cause a change in the molecular polarizability, i.e., $ d\alpha/dq \ne 0$ …….(9)

Homo-nuclear diatomic molecules such as $ \mathbf {H_2 , N_2 , O_2} $ which do not show IR Spectra since they don’t possess a permanent dipole moment, do show Raman spectra since their vibration is accompanied by a change in polarizability of the molecule. As a consequence of the change in polarizability, there occurs a change in the induced dipole moment at the vibrational frequency.

REFERENCES:-

  • Principles in Physical Chemistry, Puri, Sharma & Pathania
  • Physics Chemistry, Atkins
  • Spectroscopy, Raj Kumar
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