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My school teacher had told me that one of the assumptions of Kinetic Theory of Gases was that the molecules of a given gas were all identical and to be considered as very small elastic spheres. However, I noticed later on during the derivation for degrees of freedom for a diatomic molecules, she took the atoms to be spherical and assumed a spring attaching them. This clearly doesn't represent spherical molecule. I asked her about this and she simply disregarded my question. Also, if the diatomic gas were to be attached by a spring, would it potentially lead to a loss in energy? For example if I took two diatomic molecules and say only one atom of each were to collide, Then due to the action of the spring, eventually both atoms would have to move together.

Situation I'm trying to describe

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3 Answers 3

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You are comparing apples and oranges. The atoms in an ideal gas have no internal degrees of freedom, and the ideal gas law, which can be derived from the kinetic theory of gases, only applies approximately to non-monatomic gases. Deriving accurate equations of state from first principles for a real gas (which does not meet the requirements of an ideal gas) is considerably more difficult. I suspect you are reading too much into an over-simplified explanation.

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  • $\begingroup$ How do you claim that an ideal gas has no degrees of freedom. If that is so then poissons ratio would not be defined for it. But there are clearly different distinct values of poissons ratio for mono,dia and polyatomic ideal gases $\endgroup$
    – Rexquiem
    Dec 18, 2023 at 1:43
  • $\begingroup$ @Rexquiem The atoms in an ideal gas have no internal degrees of freedom because its atoms (which are regarded as point particles) have no internal structure. Each atom in an ideal gas has $3$ degrees of translational freedom, corresponding to its $3$ spatial co-ordinates. $\endgroup$
    – gandalf61
    Dec 18, 2023 at 7:49
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Almost any area of science involves a series of approximations. One begins with the simplest model that does a reasonably accurate job of matching the phenomena, and then one refines the model. In the present example, the model based on spherical particles in a gas is a good way to begin. It furnishes good insights into way the pressure comes about and how the pressure relates to the velocity distribution and how the velocity distribution relates to the temperature.

Now turning to diatomic molecules, one has two jobs to do. First one wants to calculate the new degrees of motion, which are rotation and vibration of individual molecules. Next, one wants to check how far the previous model (with spherical particles) can still be trusted. It turns out that the simpler model is still mostly ok for pressure and things like that, because of the rapid rotations of the molecules. In each individual collision the diatomic molecules are not like spheres, not even approximately, but because there are many collisions and they happen at all possible angles, the average effect is just as if you had spherical particles. Hence the formulae relating pressure to temperature remain very accurate. They are not perfectly accurate, but nothing in science is perfectly accurate. Every model has simplified to a larger or smaller degree from the complexities of the actual physical world.

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It's all about simplifying approximations really. You can imagine that the spring is really small and really firm so in collisions the whole thing acts more or less like a solid ball. Doesn't really matter, it's just a simplistic picture of molecules. The point is simply to add some internal vibrational degrees of freedom to the simple model of an ideal gas.

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