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Question

You have but one molecule of Hydrogen (H2), meandering about in the vast emptiness of a 1cm3 closed box, at a speed of 500 m/s, bouncing off from one wall to the next. What are the rms speed and the temperature inside the box?


A
512 m/s; 20k
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B
250000 m/s; 20 k
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C
500 m/s; 20 k
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D
500 m/s; temperature cannot be determined
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Solution

The correct option is D 500 m/s; temperature cannot be determined

Well, finding the vrmsfor this one is easy! Since we have only one molecule its speed will be its own "rms” speed -

vrms=(v2)=(5002)ms=500 m/s.

Now, what can we say about the temperature, T ?

Kinetic theory says, for a simple, ideal gas at a fixed volume and pressure, T and vrms are related as -

vrms=3RTM=13MRv2rms.

But can we even use this definition here? If we have just one or a few gas molecules in a fixed volume, the assumption of kinetic theory that - at any given instant the velocity distribution of the molecules is homogenous (uniform density throughout) and isotropic (no special average direction) - will be immediately violated if we have just one molecule in the system, or if it's a small number. We cannot apply our kinetic theory of gases here to determine the temperature from the vrms.

In such cases, the definition of temperature comes from a much more generalized statistical physics point of view - as the slope of a heat gained versus entropy function, at any given value of entropy. We will have another look at this in the section for the second law of thermodynamics.


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