Index of refraction - gas density formula

[Goodver]

Could anyone please write the derivation (or forward me to the source of derivation) of the formula related these two properties.

http://www.mathpages.com/home/kmath187/kmath187.htm

the first fomula in this article.

thanks

 

[dauto]

I think this is an empirical relation. Note the giveaway word "typical" in the first paragraph.

 

[dlgoff]

The index of refraction also depends on the wavelength of the light.

Chromatic dispersion is the change of index of refraction with wavelength.

 

[UltrafastPED]

Let the relative permeability of the gas be ε_rel = 1 + χ, where χ is the susceptibility;
χ=α*N/V=αρ, where α is the molecular polarizability, N is the number of molecules, V the volume, and ρ=N/V is the density.


Then the index of refraction for a homogeneous, isotropic, non-magnetic dielectric such as a gas is:
n = √ε_rel = √(1 + χ)≈1 + χ/2=1+αρ/2.

Then let k = α/2ρ_0, where ρ_0 is the reference density of the gas - hence the given formula:
n ≈ 1 + k(ρ/ρ_0).

 

[sardar]

The index of refraction of a gas is defined as the ratio of the speed of light in vacuum to the speed of light in the gas. It is a measure of how much the speed of light is reduced when passing through the gas. The formula for the index of refraction is n = c/v, where c is the speed of light in vacuum and v is the speed of light in the gas.

The gas density, on the other hand, is a measure of the mass of the gas per unit volume. It is typically denoted by the symbol ρ and is expressed in units of kg/m^3. The formula for gas density is ρ = m/V, where m is the mass of the gas and V is the volume it occupies.

To derive the formula for the relationship between the index of refraction and gas density, we can start by considering the behavior of light as it passes through a gas. When light enters a gas, it interacts with the particles of the gas, causing them to vibrate. These vibrations can be thought of as waves that propagate through the gas at a certain speed. This speed is related to the index of refraction of the gas.

Now, let's consider a beam of light passing through a thin layer of gas with a thickness of Δx. As the light travels through this layer, it will experience a change in speed due to the interactions with the gas particles. This change in speed can be expressed as Δv = v - c, where v is the speed of light in the gas and c is the speed of light in vacuum.

We can also express the change in speed as Δv = v - c = v - (c/n), where n is the index of refraction of the gas. This is because the speed of light in the gas is equal to the speed of light in vacuum divided by the index of refraction.

Now, let's consider the number of gas particles in this thin layer of gas. This can be expressed as N = ρΔx, where ρ is the gas density and Δx is the thickness of the layer. The number of gas particles will affect the strength of the interactions with the light, which in turn affects the change in speed of the light.

Combining these equations, we get Δv = (v - c)/n = ρΔx. Rearranging this equation, we get nρΔx = v - c. Now