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Say that during an adiabatic process, the pressure changes from $p_0$ to $a*p_0$, and let the heat capacities ratio be denoted by b ; using the basic adiabatic relations for an ideal gas, we can find that the temperature changes from $T_0$ to $a^{1-1/b}*T_0$. As the degrees of freedom of a molecule in the gas tend to infinity, b will tend to 1, which implies that for high degrees of freedom there is little to no temperature change during an adiabatic process i.e. it becomes isothermal. Why is this?

My intuition tells me that high D.O.F. implies more places to store energy, requiring more energy for a given V,P,& T; hence an adiabatic process w/ high D.O.F. will have so much stored energy that the temperature change caused by moving from $p_0$ to $a*p_0$ will be negligible. Is this the right idea?

On a different note, how valid is the ideal gas law when it comes to gases with high D.O.F ? What gases have the highest D.O.F ?

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In my judgment, your assessment is basically correct. In an adiabatic process, for a specified amount of work done, the change in internal energy is likewise specified. And, for a particular internal energy change, the higher the heat capacity of the gas, the lower the temperature change. So increasing the heat capacity makes the process closer to isothermal.

With regard to your question about the ideal gas law, all gases approach ideal gas behavior in the limit of low densities (irrespective of the DOF). More precisely, in the limit of low reduced pressures.

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Well, i think, there is a highest limit how many degrees of freedom a molecule can actually have. The bigger the molecule, the more degrees of freedom you'll get. This will constitute not only the translational and rotational degrees, but also degree of freedom due to the movement of atoms relative to each other(vibrational). The formula for vibrational degree of freedom is 3N-6, where N is the total number of atoms in the gas molecule.

Mostly, gases are mono, di or tri atomic. So the total number of DOF don't become very large. But if you consider a hypothetical large molecule with many atoms, in my opinion, the intermolecular attraction would be too much to keep following ideal gas law(in fact it may lead the gas to condense, thus, the gas won't be even a gas)

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