Example of an irreversible process using this formal definition

Question

Some time ago, I asked for a definition of thermodynamic reversibility without mentioning entropy, and a user came up with the following formal definition of a reversible process:

Definition:

Reversible process means that given the outside controllable mechanical, electrical, magnetic, chemical, etc., macroscopic parameters $\hat x_1,\hat x_1,\hat x_2,...,\hat x_n$ of the surroundings and its temperature $\hat T$ at which heat exchange can also take place any and all internal thermodynamic properties (parameters), say $z$, of the system at any time instant during the process can be written as a function of said external parameters: $z(t) = f(\hat T(t), \hat x_1(t),\hat x_2(t),...,\hat x_n(t))$. Notice the function depends only on the instantaneous values and not on the time rates of the external parameters. The $t$ in the function is just a process index by which the various consecutive stages of the thermodynamic process is marked, i.e., time."

Using this definition, can you come up with an example of irreversible process and explain why it is irreversible, again, using the provided definition or explain why this definition is wrong?

Answer 1

The most obvious example is magnetic hysteresis. The current value of magnetization $M=M(H_{ext})$ does not just depend on the current bias field but it also depends on the history of how the sample has been biased. What makes this example so interesting is that actual bias history can be quasi-static, the time rate of bias can be arbitrarily low, and it is still irreversible. This is very much unlike the usual example, say a quick piston movement against a gas in a cylinder, that creates all kinds of internal disequilibrium, and the time-dependent relaxation is irreversible.

Answer 2

I think there are some cases that are not covered by this definition of reversibility and are, therefore, irreversible. The way I interpret this definition is that the state is a unique function of the control parameters. One way to violate it is by finding a state that is not uniquely determined by its parameters (one example is already given by @hyportnex), but the other way is by finding a situation where the external parameters cannot be controlled (or different at the beginning and the end of the process.)

For example:

• Gas expansion into vacuum If we were able to put the gas back into the container that it came from, so that its volume is the same as it initially was, we conceivably could reverse the process - that is, it falls under the definition... but we cannot reverse it.
• Irreversible chemical reactions - e.g., one can have a mixture of oxygen and hydrogen in a container, which exists for a very long time as is. It may spontaneously explode and transform into water (provided that the container is strong enough to withstand the explosion - a situation often discussed in the context of combustion.) The external parameters remain the same, but the situation has drastically changed.

Related:
Difference between Reversible and Irreversible processes in Physics vs. Chemistry
Can we call rusting of iron a combustion reaction?

Answer 3

If I'm reading the definition right, I'm not sure there is any "reversible process" according to it. The problem is this part of the definition:

"the function depends only on the instantaneous values and not on the time rates of the external parameters"

In practice, physical systems always have some amount of inertia, such that sufficiently rapid changes in their external environment are only reflected in their internal state after some delay. Indeed, if we had a parameter to which a system reacted truly instantaneously, with no delay at all, it's hard to see how such a parameter could be described as "external" to the system.

---

But you were asking for an example of an "irreversible process" according to the definition you quoted. As I argue above, I think almost any process should qualify as irreversible according to it, but the obvious example that comes to mind reading it would be heat exchange.

Specifically, let our system consist of a 3-dimensional block of solid material in a heat bath at temperature $\hat T(t)$, and let the one of the internal thermodynamic properties $z(t)$ be the temperature at the center of the block, which I'll call $T_c(t)$.

At thermal equilibrium, of course, $T_c(t) = \hat T(t)$, at least assuming that no heat is generated within the block. But if we change $\hat T(t)$ sufficiently fast (or, technically, at all!), then $T_c(t)$ will lag behind $\hat T(t)$, since it will take time for heat to diffuse in or out of the block and for a new thermal equilibrium to be reached.