Question

Thermodynamics and Statistical Mechanics problem:

n regard to the Maxwell speed distribution, you might wonder why all the molecules in a gas in thermal equilibrium don’t have exactly the same speed. After all, when two molecules collide, doesn’t the faster one always lose energy and the slower one always gain energy? And if so, wouldn’t repeated collisions eventually bring all the molecules to some common speed? Describe an example of an elastic billiard ball collision in which this is not the case: the faster ball gains energy and the slower ball loses energy. Include numbers and be sure that your collision conserves both energy and momentum.

Answer 1

For the first question, assuming that a molecule 'A' collides with another molecule 'B' with a lesser energy as compared to 'A', then molecule 'B' gains the energy of molecule 'A' and molecule 'A' gains the energy of 'B' i. e. Overall there is exchange in momentum among the molecules A and B, thus it doesn't matter how many collision takes place among the molecules, the momentum only keeps on exchanging. There is no way of achieving an average momentum and thereby average speed. Thus in a gas at thermal equilibrium, the molecules do not have exactly the same speed.

Now coming to the second question :

Now considering a billiard board consisting of say 10 balls such that these balls can collide with each other randomly according to the way described in the question i. e. The collision will is conserve both energy and momentum and when a collision will take place between two balls the one with greater speed gains the eneiof other ball in collision.

No let these balls be numbered from 1 to 10. Let all these balls have random unequal energy with which they are moving. Now if ball 'x' has the highest speed among all the remaining 9 balls. It will absorb all the energy by colliding with them one by one or in any fashion and will alone have motion and all the balls will come to a halt as there energies have been absorbed.

Thus concluding we can say for such a given system, ultimately a single billiard ball will gain energy of all other balls. Thus the energy of whole system will be contained in a single ball.