Saturday, July 24, 2010

Kinetic theory of gas paradox

Consider a long narrow tube with circular face with perfect insulator and one end (A) maintained at 100 degree and other end (B) maintained at 0 degree. Tube is filled with ideal gas. Temperature at A (100) and B(0) are maintained at respective temperatures by a infinite heat source at A and infinite heat sink at B.
1) Will heat flow from A to B?
2) Will the Temperature at A higher than that at B?
3) Will the pressure at both end same?
4) Will the density at both end same?
If we consider the kinetic theory of gas, the pressure is proportional to the density of molecules (more molecules more hits to the boundary), velocity molecules (more velocity more number of hits to the boundary) and momentum of molecules (more momentum more force). Momentum = velocity * mass. Hence pressure is proportional to the mass, density and square of velocity.
However energy transfer is proportional to density, velocity and kinetic energy of each molecules. Kinetic energy = m*square of velocity. Heat transfer is proportional to mass, density and cube of velocity (This is evident from the fact that thermal conductivity of ideal gas is proportional to temperature).
Hence in equilibrium, density at A will be lower. Average velocity of molecules at A will be higher such that density * square of velocity at both side is equal. However heat transfer is proportional to cube of velocity heat will flow from A to B. Mass transfer is proportional to square of velocity there will not be any mass transfer (which is a necessary constraint).
However now heat flows from A to B where as there is no temperature difference.
What are alternatives?
We need to re-define temperature to such that temperate is proportional to cube of velocity. We may need to redefine the measurements as well.
Will further post on the impact of this on other equation of temperature.
How to verify?
Verify this by conducting an experiment with above set up and observing the temperature, pressure and density at both end.
Some other corollaries:
1) T = pū Where ū is the average velocity of the molecules.
2) Ideal gas equation instead of pV = mRT, it should be:
p3V = CT2 Where C is a constant.
Yet another explanation:
There will be a pressure gradient between A and B.
This is equivalent (or similar) to pressure gradient because of velocity gradient (bernoulli's principle).
That means when there is a temperature between two points in gas, there will be a density gradient and pressure gradient between these points.
Consider a cross section across the pipe, at any arbitrary point. velocity of particles at side A will be greater than velocity pf particles at the other side (side of B). Density of particles at Side A shall be less than that of side B.
At any point the number of particles crossing the cross section shall be equal in number. Consider the circular surface adjacent to the cross section, at both sides the number of particles hitting the surface shall be same. However the velocity of the particles is higher at side A. Hence the pressure at side A shall be higher.
That means there shall be pressure gradient across the tube.

Will post the corresponding equations later.


Unknown said...

AMMA didnt understand much of it but 10 years from now I will be proud of my Einstein Aacha.
My aacha THE BEST


Unknown said...

This is more like a hypothetical situation and more clarity is required. If external actions are not in place to maintain the temperature delta soon the gas will reach equilibrium inside the pipe with density, temperature and velocities becoming same.
If we assume you have given the starting condition then
1. Heat will flow from A to B, the high velocity molecules will impinge the ones nearby transferring heat unil a uniform temperature is achieved.
2. Yes, Until temperature is uniform the difference will stay.
3. Yes, the total pressure at both end will be same...the density will vary across the tube until temperature is uniform.
4. No, the density will not be uniform until all molecules are at the same temperature. It will be less at hot end and more at cold end as you have rightly stated.

The equilibrium with temp difference cannot be achieved without external action and hence cannot comment on the heat transfer part.

Rejeev Divakaran said...

To clarify Rajesh's assumption;
Temperature at A is maintained at 100 and B at 0 degrees. At equilibrium, heat should flow from A to B isn't it?

Rejeev Divakaran said...

If my hypothesis true then:
Delta of temperature = velocity of molecules * delta of pressure.