Atoms emitted from a tube (Atomic Oven)

In summary, the scenario presented involves an evacuated metal tube with a metal source emitting gaseous metal atoms in all directions. The question asks for the fraction of emitted atoms that will exit the free end of the tube. Using trigonometric calculations, it is determined that this fraction is equal to 2/pi times the ratio of the length of the tube to the diameter of the tube. This fraction approaches 0 as the length of the tube approaches infinity and approaches 1 as the diameter of the tube approaches infinity.
  • #1
Sachin Vaidya
2
0

Homework Statement


Consider a evacuated long metal tube of length 'l' and diameter 'd' containing a metal source at one end. The metal source is connected to an oven and emits gaseous metal atoms in all possible directions. If an atom hits the walls of the tube, it will get stuck and will not bounce back. What fraction of the emitted atoms will be emitted out of the free end of the tube? (Neglect inter atomic collisions, consider a 2D situation)https://fbcdn-sphotos-d-a.akamaihd.net/hphotos-ak-xaf1/v/t1.0-9/18884_10152998608889958_7600213725768080208_n.jpg?oh=8038f95eb741b39ad8ac3bb448afcb2d&oe=55BFDFAC&__gda__=1442863826_7f94b06130ac37c9fff88440e5fc45b7

Homework Equations

The Attempt at a Solution


I consider an element dx at a distance x from the top (as shown in the figure). Atoms are emitted in all directions spanning angles from 0 to π. The atoms that will make it out must lie between the triangle shown in the figure (lines joining the element dx and the end points of the tube). So the ratio of atoms that will make it out will be the ratio of the angle subtended within the triangle and π (all possible angles).
Fraction of "useful" atoms =
[tex]fr_x=\frac{1}{\pi}\left[\tan^{-1}\left({\frac{x}{l}}\right)+\tan^{-1}\left({\frac{d-x}{l}}\right)\right][/tex]

Now how do I proceed? How do I integrate this over the whole length d? I have tried something but I'm not sure if this is correct.
[tex]F(l,d)=\frac{\frac{1}{\pi}\int_0^d \left[\tan^{-1}\left({\frac{x}{l}}\right)+\tan^{-1}\left({\frac{d-x}{l}}\right)\right]dx}{\int_0^d dx}[/tex]

Which on solving becomes:
[tex]F(l,d)=\frac{2}{\pi}\left[\tan^{-1}\left(\frac{d}{l}\right)-\frac{l}{2d}\log\left(\frac{d^2}{l^2}+1\right)\right][/tex]

In the limit [itex]d<<l[/itex], this becomes almost linear:
[tex]F(l,d)=\frac{2}{\pi}\frac{d}{l}[/tex]
and when [itex]l\rightarrow \infty[/itex] this goes to 0.

On the other hand, in the limit [itex]d>>l[/itex], this becomes like an orifice and all the atoms are emitted out. This is reflected when the limit is taken of [itex]F(d,l)[/itex] and that gives 1.
 
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  • #2
For d<<L you can argue that the vertical location of the source does not change the angle much and as the angle in radians is approximately d/L shouldn't the d<<L limit be d/πL. I don't get the factor of 2?

Anyway, it looks like you know what you are doing.
 

Related to Atoms emitted from a tube (Atomic Oven)

1. What is an Atomic Oven?

An Atomic Oven, also known as an Atomic Emission Spectrometer, is a scientific instrument used to analyze the chemical composition of a sample by measuring the intensity of light emitted by the atoms in the sample.

2. How do Atoms get emitted from a tube in an Atomic Oven?

The Atomic Oven uses high temperatures to vaporize a sample into a plasma, which is a hot, ionized gas. This plasma produces a light emission that is characteristic of the elements present in the sample. The light is then passed through a spectrometer, which separates the different wavelengths of light and allows for the identification and quantification of the elements in the sample.

3. What types of samples can be analyzed using an Atomic Oven?

An Atomic Oven can analyze a wide range of samples, including liquids, solids, and gases. It is commonly used in environmental, pharmaceutical, and material testing industries to determine the elemental composition of substances.

4. How accurate are the results from an Atomic Oven?

The accuracy of the results from an Atomic Oven depends on various factors, such as the quality of the instrument, the sample preparation technique, and the experience of the operator. However, in general, an Atomic Oven can provide highly accurate and precise results, with a margin of error of less than 1%.

5. Is an Atomic Oven safe to use?

Yes, an Atomic Oven is safe to use as long as proper safety precautions are followed. The high temperatures used in the instrument can pose a potential hazard, but the instrument is designed with safety features such as interlocks and shielding. Proper training and knowledge of safety protocols are necessary for safe operation of an Atomic Oven.

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