As per my understanding we know that molecules of an ideal gas are identical in all aspects (size, shape, mass). Since collisions are elastic in nature, they don't lose their kinetic energy. That means that kinetic energy of each molecule doesn't change over time. Then how do the molecules move with different velocity regardless of possessing same mass and kinetic energy ?
[Physics] Why gas molecules move with different speed at a given tempreture
kinetic-theorystatistical mechanicsthermodynamics
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As the comments to the question have stated, in real gasses ( contrasted to ideal gasses which just bounce around elastically) there exist both elastic and inelastic scatterings controlled by quantum mechanical interactions.
Photons are generated leading to what we call Black Body radiation and an isolated gas volume will lose energy according to the Stephan Boltzmann law.
the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant exitance or emissive power), is directly proportional to the fourth power of the black body's thermodynamic temperature T:
Thus the gas does lose energy if the temperature of matter surrounding it is lower.
In answer to
I think another way to phrase this is, how do elastic collisions not lose any energy in the exchange
Elastic means an interaction of two particles where before and after , kinetic energy is conserved. If one assumes that only kinetic energies exist for this scatter ( as in the ideal gas) then energy is conserved because what one particle loses the other gains . If there are other forms of energy that can contribute to the two particle interaction then it is the total energy that is conserved. With billiard balls classically friction has to be taken into account with the energy balance, the same with the bouncing ball, and the kinetic energies stop being the total energy of the system. For particles in a gas it is the quantum mechanical framework, described above.
Gravity makes molecules gradually accelerate downwards. Neglecting collisions, the molecules closer to the earth would thus be (on average) moving faster.
You cannot neglect collisions, at least not in the part of the atmosphere where the atmosphere acts like a gas. Collisions remain important until you get to the exobase. Above the exobase, the atmosphere is so rare that collisions can be ignored. Modeling the exosphere is messy because now you have to worry about solar flares, the Earth's magnetic field, and a varying distribution of components (e.g., the exosphere is dominated by hydrogen).
Collisions are extremely important in the thick part of the atmosphere (up to 120 km or so). At sea level, the mean free path (average distance between collisions) in the atmosphere is less than 1/10 of a micrometer. At 20 kilometers altitude, the mean free path is about one micrometer. By the time you get to 120 kilometers, the mean free path grows to about a meter. Collisions remain important until you get to the exobase.
Temperature in the atmosphere follows a complex profile. The highest temperatures are in the two highest layers of the atmosphere, the thermosphere and exosphere.
Your second argument suffers many of the same problems as your first. You cannot ignore collisions. Collisions are an essential part of what make a gas a gas. Things are a bit murky in the one part of the atmosphere, the exosphere, where collisions can be ignored. There is one easy way to model the exosphere: It's essentially a vacuum. Getting past that easy model is non-trivial. Even the thermosphere (the next layer down, where the space station orbits) is problematic to model. The atmosphere is still thin enough in the thermosphere that it doesn't act quite like a gas.
It's the mesosphere on down where the atmosphere acts like a gas. That the components are constant colliding with one another is what makes pressure in the lower atmosphere equal to the weight of all the stuff above. This isn't necessarily true in the upper atmosphere.
Best Answer
Here is the misunderstanding:
Only in the center of mass of two colliding particles the collisions have equal and opposite energy , not in the laboratory frame of the containing box. When one puts all the "identical molecules of an ideal gas" means the "molecules" not the energy momentum vector of each molecule in the laboratory frame of the box.When introduced in the box they will have an average kinetic energy according to the temperature, but there will be a distribution of possible energies and momenta. The elastic center of mass collisions of individual pairs will transform back to the lab with different energies due to the angles of scattering.
It gets worse, because of the spill over electric fields of molecules , the collisions quantum mechanically will allow for radiation, black body radiation, which will eventually lower the temperature to an equilibrium with the outside the box temperature.