# Ideally, how to achieve isothermal expansion of an ideal gas?

In an isothermal expansion of an ideal gas, there cannot be any change in the internal energy otherwise the temperature would change. So hence we know that all the heat goes into work. Knowing that $$PV=nRT$$ as the equation of state and $$T$$ being constant in this process, we see that $$P$$ is inversely proportional to $$V$$.

Does that mean in order to feasibly realize an isothermal expansion we would need to do the following:

1. Put the ideal gas in a cylinder with a piston and some amount of weight on it to reach some pressure.
2. Find a heat reservoir equal to the temperature of the gas.
3. Slowly lift weight off of the piston according to $$P \propto 1/V$$.

The last part is kind of the strange part. We have to lift some weight off otherwise the volume won't change. But why can't we quickly lift the weight off? So the external pressure changes from some large number to a very smaller number? Would that be approximately an adiabatic expansion?

Editted: Just want to confirm the necessity of the last part where we are slowly decreasing the force applied by the piston. Not sure how to do that practically, you would need some kind of electronically controlled device to slowly "lift the weights off".

• Just want to confirm the necessity of part 3. Jun 1, 2022 at 15:28
• I think my confusion was I was thinking too much about the pressure of the gas and the "weight" on top of the piston. I can simply consider a large heat reservoir and an ideal gas trapped in a cylinder with a top that I can move up and down. Not worrying about the exact pressure of the gas on the top/cap. Isothermal expansion of this ideal gas would mean the new state has the same temperature and hence by ideal gas law larger volume and P=1/V Jun 2, 2022 at 5:03

The isothermal expansion is a theoretical ideal. An isothermal process requires the system is in perfect equilibrium with its surroundings at all times so it would have to be done infinitely slowly. As you say in your question, any process done at a finite speed is necessarily out of equilibrium.

However in real life provided heat flow is fast enough processes can be so close to isothermal that we can treat them as perfectly isothermal. That is, the error involved in assuming they are isothermal is negligibly small.

• If you allow the expansion (or compression) to take place slowly and continuously, there's enough time at every point for the temperature to equilibrate: Since all the gas will be at same temperature as the reservoir. This is an iso-thermal process.
• If you do the expansion (or compression) rapidly, so fast there's no time for thermal energy to flow and the gas to equilibrate, then the pV work being done will change the temperature of the gas. That's not an iso-thermal process, because the temperature changes. If you do it fast enough, so no thermal energy is exchanged, then it's adiabatic.

So the same sequence of operations if done fast enough is adiabatic and done slow enough is isothermal. What's "enough"? Compare the energy that can flow during the ideal amount, and see how close it is.

If you lift the weight off quickly (so that the external force on the gas is much lower), the expansion will be irreversible. Under these circumstances, even though the cylinder is in contact with a constant temperature reservoir at its surface, only the surface of the gas will be at the specified temperature, and the interior of the gas will drop to a lower (non-spatially-uniform) temperature throughout the expansion (until final equilibrium is reached, and the gas is again at the reservoir temperature). In the interim, heat conduction will be occurring within the gas.

In addition, the force that the gas imposes on the piston will be a combination of the effect of viscous (tensile) stresses within the gas plus what we ordinarily would consider the gas pressure under static conditions. Initially, for the irreversible case, all the change in force will be supported by viscous stresses, which are proportional, not to the change in volume but, to the rate of change of volume. So, during the irreversible expansion, the gas ceases to behave like an ideal gas and ceases to satisfy the ideal gas law (which applies only at thermodynamic equilibrium) within the cylinder. As equilibrium is approached, the rate of change of volume decreases, and the viscous stresses make less of a contribution to the overall force of the gas on the piston. At final equilibrium, the viscous stresses are again zero, and the force on the piston is entirely described by the ideal gas law.

To get a better feel for all this, imagine that the gas behaves like a combination of a spring (preloaded in compression) in parallel with a viscous damper, with the combination joined mechanically to the rear end of the cylinder and to the piston. The spring mimics the ideal gas behavior of the gas and the viscous damper mimics the viscous behavior of the gas. If the piston moves very slowly, the force on the piston is dominated by the spring (ideal gas behavior). But, if the piston moves very rapidly, the change in the force on the piston is dominated by the viscous behavior. In line with this conceptualization, a crude approximation to the behavior of a gas experiencing either a slow- or a rapid deformation can be expressed as: $$\frac{F}{A}=\frac{nRT}{V}-\frac{k}{V}\frac{dV}{dt}$$where F is the force of the gas on the piston and k is a constant proportional to the gas viscosity. The first term captures the thermodynamic equilibrium behavior of the ideal gas, and the second term captures the viscous contribution.

To get a better understanding of the viscous Newtonian behavior of fluids (including gases), I highly recommend Transport Phenomena by Bird, et al.

Two ways: 1) Expand the ideal gas into a vacuum, and there is no change in the temperature. However, this is not the type of expansion I surmise you are asking about.

2) Expand the gas against a piston so that it does work. Now, since the work is accomplished through the individual collisions of gas atoms, each collision transfers a small amount of momentum and energy to the piston. So the gas cools. Or think about it as change in internal energy = heat + work. The temperature reflects the kinetic energy of the gas, which is also the total energy of the gas since it is ideal.

Since the gas is cooling during expansion as described so far, the only way to maintain the same temperature is to heat the gas in sync with the transfer of collisional energy to the piston. This heat could transfer from some other part of the container enclosing the gas.

It's fairly simple. Consider Joule's free expansion which is isothermal for Ideal gas. If using a piston cylinder arrangement what we can do is use a spring instead of weights and let the gas reach equilibrium with the spring force(we can use a spring with a desired spring constant). Now add infinitesimally small amount of heat and let the system regain equilibrium. The small amount of heat will result in a small increase of temperature and pressure but a small amount of expansion will counter that and bring back the isothermal condition and considering the very small changes we can assume it to be a constant temperature process.

• He asked about physical realizability and not how it can be achieved theoretically Oct 26, 2020 at 9:56
• All that is asked in the question can be explained by this experiment. Ideal gas in itself is a theoretical concept. Oct 26, 2020 at 10:05
• Yes it can't be, that is why the accepted answer discusses that it is impossible in real life. He was not asking how to theoretically pull it off but he was asking how to realize the process physically Oct 26, 2020 at 10:06
• @Buraian Still couldn't get why free expansion of Ideal gas is not realizable physically considering that Joule actually performed it and it wasn't some hypothetical one. Still added a piston cylinder process in the edit. Oct 26, 2020 at 10:27
• Joule had done the experiment but couldn't value the non zero value of the coefficient for a real gas.s It was an experimental error. A true ideal gas doesn't exist in reality Oct 26, 2020 at 16:55