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 ?