Conservation of momentum and Energy

[MotoPayton]

In an isolated system, momentum is conserved. I understand this. However, I also learned that that the motion and energy of the particles inside of an isolated system is defined by the average kinetic energy (temperature).

Do we have to assume that every collision is elastic and that kinetic energy is always conserved?

If there are inelastic collisions inside of the the system what happens to the energy?
Does the temperature of the system remain constant? How could it if energy is lost?

Thanks for the help.

 

[Calrid]

Yes we have to assume energy is never lost it is conserved, it merely changes form. The terms are elastic the energy is not.

If energy is lost in that system then the temperature falls. Momentum is conserved in the same way energy is, it is transferred as energy that is lost from the system. It might for example heat the air around it. Overall if we take the whole system into account nothing is lost.

 

[Rap]

In an isolated system, momentum is conserved. I understand this. However, I also learned that that the motion and energy of the particles inside of an isolated system is defined by the average kinetic energy (temperature).

Do we have to assume that every collision is elastic and that kinetic energy is always conserved?

If there are inelastic collisions inside of the the system what happens to the energy?
Does the temperature of the system remain constant? How could it if energy is lost?

Thanks for the help.

There are ways for a gas (or any material) to hold energy other than kinetic energy. A hydrogen molecule (composed of two atoms) can spin like a top, for example. A single atom can have one of its electrons kicked up to a higher energy level as the result of a collision. If a collision occurs between two non-rotating molecules and makes one molecule rotate, the total kinetic energy of the two colliding particles will be less than the total kinetic energy before, but the total kinetic energy plus the energy of rotation will be equal to the kinetic energy before. On the other hand, a molecule colliding with a rotating molecule might pick up energy from the rotating molecule, with the rotating molecule losing rotational energy. Then the total kinetic energy of the colliding particles will be greater after the collision than before. But again, the total energy will be the same.

Saying that the temperature is proportional to the average kinetic energy is mostly true, but not the whole picture. These other ways for an atom or a molecule to store energy are called "degrees of freedom". If you have just point atoms that cannot have any energy except kinetic energy, then it has 3 degrees of freedom - x, y, and z - the number of coordinates you need to specify its position. If it can rotate, like a diatomic molecule, you need 2 more to specify its orientation, so it has 5 degrees of freedom.

The best way to think of the relationship between energy and temperature is that each degree of freedom has energy kT/2. So a gas of point atoms has energy 3kT/2 per atom. A diatomic gas has energy 5kT/2 per molecule. So if you have inelastic collisions, the energy doesn't "go" anywhere, because its already there. Every collision can give stored energy to kinetic energy, or kinetic energy to stored energy. When everything is at equilibrium the rates are the same, and there is no change in the total stored energy or kinetic energy.

The simple picture above is for "classical" atoms and molecules. In reality, the rotation of a diatomic gas is a quantum phenomenon, which means the energy stored is in discrete amounts, not continuous. If the temperature gets low enough, this can interfere with the idea that there is energy kT/2 per degree of freedom. But you can worry about that later.