Thermodynamic definition of an adiabatic process
I am posting about this because it seems to be a big issue and misconception in the thermodynamic literature. My issue is about adiabatic processes. As I see it there are two intrinsically different definitions of adiabatic processes:
1. Processes for which $\delta Q_\mathrm{irr}=Td_iS+Td_eS=0$ ($Td_iS$ is the irreversible heat produced and $Td_eS$ the heat due to heat transfer). This means that in these processes there is no heat generation whatsoever. 749
This also means that any adiabatic process is isentropic. Actually, I think this definition is wrong, because every irreversible process will produce entropy $Td_iS$ which cannot be compensated, because the system is thermally isolated ($Td_eS=0$), so that $\delta Q_\mathrm{irr}>0$.
2. (I think the right definition) Processes for which $\delta Q_\mathrm{rev}=Td_eS=0$. This means that no heat transfer is allowed into the system, but still irreversible processes can generate heat.
The second one should be in principle correct, as an adiabatic, irreversible expansion of a gas can heat it up due to entropy production. To name an example for definition 2, I could name the expansion of the universe which is adiabatic in the sense of no heat transfer (no environment). Still, the entropy is increasing, since $Td_eS\neq 0$ (while it is assumed to be 0 in definition 1).
However, large parts of the literature work with the first definition also.
One example for the use of the first definition is https://chemistry.stackexchange.com/questions/16260/derivation-of-the-relation-between-temperature-and-pressure-for-an-irreversible or https://chemistry.stackexchange.com/questions/38127/reversible-and-irreversible-adiabatic-expansion Here, the authors claim that they derived the expression for the volume change of an irreversible, adiabatic process and starts with the equation (I have seen exactly the same equation in Lecture notes of my Thermodynamics class and other books):
$\mathrm{d}U=-p_\mathrm{ex}\mathrm{d}V$
However, according to the first law:
$\mathrm{d}U=\delta Q_\mathrm{irr}+\delta W_\mathrm{irr}=\delta Q_\mathrm{irr}-p_\mathrm{ex}\mathrm{d}V$
So to reproduce the equation above $\delta Q_\mathrm{irr}=0$ which means that the use of the above equation assumes definition 1, which makes no sense in my opinion.
In my opinion, for any process we should have:
$\mathrm{d}U=\delta Q_\mathrm{rev}+\delta W_\mathrm{rev}=0-pdV=-pdV$
which means that no matter the reversibility, a specific volume change will always induce the same change in the internal energy.
I would appreciate any opinion on this issue. 749