# Can a constant pressure expansion occur reversibly?

I'm currently reading section 19-7 in McQuarrie's physical chemistry textbook, but I'm getting a bit hung up on the possibility of reversible processes at constant pressure, since up to this point the only reversible processes treated were isothermal.

I think I understand the whole point of postulating a reversible process. At every point in a reversible process, the system and surroundings are essentially in equilibrium. This implies that the state variables will change smoothly, and so the techniques of integral calculus can be used to calculate the changes in the state functions which result from the reversible process.

Crucially, since state functions are path-independent, the reversible process can act as a bridge between states, and the changes it evokes in a state function will hold, despite reversible processes being non-physical.

In the case of a constant volume process (assuming that the system and surroundings are electrically neutral) there is no work performed, and so the change in internal energy associated with a process will simply be the energy transferred as heat.

Since most processes (e.g. chemical reaction) take place at constant pressure, we also find that the energy transferred as heat will be equal to the change in internal energy minus the work (which reduces to the external pressure times the change in volume for the case of constant pressure).

I'm trying to imagine a small amount of gas expanding and while (quite obviously) it won't affect the pressure of the rest of the universe, it seems possible to imagine that the system's pressure is still infinitesimally greater at every step, expanding the volume of the system while negligibly affecting the external pressure. Now that I've typed this all out, it indeed does seems necessary that this is the case, or else the expansion would never occur, but I'm going to post this in the hopes that any underlying flaws in my thought process will be pointed out.

Here are some reversible processes at constant pressure.

1. Ideal gas: pull a piston out (so volume increases) while simultaneously providing heat at the system temperature (so that the heat transfer is reversible), at each state along the path, maintaining $$T/V$$ constant.

2. Gently boil water at 100 celcius and 1 atmosphere of pressure.

3. Reversible chemical reactions in a vessel whose pressure is constant.

Not sure if this answers your question, but the main point is that it is no big deal: there are many reversible processes that can happen at constant pressure.

• In the case of chemical reactions, you cannot merely allow the reaction to proceed spontaneously. You need to start at an equilibrium state, and gradually add reactants (through semi-permeable membranes) while at the same time removing corresponding amounts of products (through semi-permeable membranes), such that the equilibrium is maintained. – Chet Miller Feb 15 at 15:45
• You don't need to pull the piston out. Adding heat gradually, while pushing to maintain the pressure constant, will automatically allow the volume to increase in proportion to the temperature.. – Chet Miller Feb 15 at 15:48
• @ChetMiller thanks; yes I agree in both cases; the comments speak for themselves. – Andrew Steane Feb 15 at 16:13

In order for a constant pressure process to be reversible, the difference between the external pressure and the pressure of the gas has to be infinitesimal at each point during the process, so that the gas and surroundings are essentially always in equilibrium. For this to happen, the process has to be carried out very slowly (quasi-statically) and without any mechanical friction.

So if you are expanding a small amount of gas (meaning the gas is doing work on the surroundings) and want the process to be reversible, the expansion has to be carried out very slowly so that the pressure of the gas is always infinitesimally greater than the external pressure at every stage of the expansion process.

Hope this helps.

• And note that "very slowly" means longer than the lifetime of a human being. – David White Feb 15 at 3:14
• @David White Yes, which is why an ideal reversible process is impractical – Bob D Feb 15 at 3:40
• @DavidWhite, I would not be so anthropomorphic. Human life is not always the correct time scale. It is enough that the process is slower than the slowest relaxation time of the system. For a gas at laboratory conditions, it may be very short. – GiorgioP Feb 15 at 11:48
• If, by gas pressure, you mean the force per unit area that the gas exerts on the inside piston face, and by external pressure you mean the force per unit area that the inside piston face exerts on the gas, then, by Newton's 3rd law, the two must always be equal, not just for reversible processes but for irreversible processes as well. However, in an irreversible process, the force exerted by the gas is not given by the ideal gas law and requires viscous stresses resulting from piston motion at finite velocity to match the external force exerted by the piston. – Chet Miller Feb 15 at 12:42
• @ChetMiller By gas pressure I meant the pressure of the gas within the volume, not the pressure at the interface. I understand the pressure at the interface is the same as the external pressure per N3. But for the irreversible process there is a pressure gradient going into the bulk of the gas. – Bob D Feb 15 at 12:46