Temperature for small systems

From the Maxwell-Boltzmann distribution we know that :

\[v_{\text{rms}}^{2}=\frac{3RT}{M}\]

Labelling all particles of the gas with an index we know that :

$v_{\text{rms}}^{2}=\dfrac{\displaystyle\sum_{i=1}^{n} v_i^{2}}{n}$

where n is the number of particles in the gas

So making $T$ the subject of the equation we get :

$T= \dfrac{\displaystyle M \sum_{i=1}^{n} v_i^{2}}{3Rn}$

This is assuming the case that the system containing our gas is not in motion, if in motion we should be taking the velocities deviating from the velocity of the centre of mass of the gas

Keeping the average velocity as $\overline v$

Our new equation to combat moving systems should be

$T= \dfrac{\displaystyle M \sum_{i=1}^{n} (v_i-\overline v)^{2}}{3Rn}$

$\text{Both } v_i \text{ and } \overline v \text{ are to be treated as vectors}$

This can be applied to smaller systems easily, so I was wondering whether this is a good argument or not, as I have heard you can’t define a temperature for smaller systems

If true, will such systems obey the law of thermodynamics, as in my sight there are just statistical laws which are more true as the size of the system increases

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Comments

@Siddharth Chakravarty Have any idea about this? Whether it could be right/wrong, or know someone who will know ?

Jason Gomez - 3 months, 2 weeks ago

Im still yet to complete Thermodynamics completely, I cant mention the people I know ugh.

Siddharth Chakravarty - 3 months, 2 weeks ago

Why?

Is it because you know no one who would be interested in this, or is it because the people you know(who would know about this) won’t be interested in this

No, no actually when we try to mention a person we can see the name of the person pop-up, and thus clicking on it, the person gets mentioned but when I am trying to mention those people I can't see their name pop-up.

@Siddharth Chakravarty – Yeah you need to find them somewhere they commented and use Frisk’s technique, that is why I made a new note for keeping mention ids

@Jason Gomez – now that is gonna be a long process, I will give you the names rather Josh Silverman, Aryan Sanghi, Alakh Bhattacharya, Agnishom Chattopadhaya, etc.

@Siddharth Chakravarty – Ok I think I know where to find Aryan Sanghi and Alakh Bhattacharya

@Siddharth Chakravarty – Couldn’t find the others

@Siddharth Chakravarty – Who do you want to mention btw?

@Aryan Sanghi

The laws of thermodynamics are universal, they could be applied to every system, big or small.

Considering the last equation you wrote for $T$ , it is based on Kinetic Theory of Gases which is applicable only for large systems, so the last equation might differ a good amount from experimental value if applied to small systems. Also, temperature is defined for all systems, big or small, but the only problem is that Kinetic theory of gases is not valid for small systems, so you can't define temperature for small systems using KTG. Hope it helps. :)

Aryan Sanghi - 3 months, 2 weeks ago

Thank you, this helps a lot

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$