# Can an isothermal process also be adiabatic?

I came across this question of "can an isothermal process also be adiabatic?", and at first thought that yes, why not? If the temperature is constant and there is no heat loss to the environment it seems perfectly possible.

Now, I don't have too much experience in this field but I came across alot of vague statements that this is not possible at all.

Could someone with a bit more expertise maybe give me a clear answer?

If a process is both isothermal and adiabatic, it is implied that the work done on the system is being stored somewhere other than the internal energy of the working fluid. (Or conversely, if the system is doing work, the energy is coming from somewhere other than internal energy.)

The classic example of such a process is free expansion of an ideal gas (also called Joule expansion), though in this case there is no work being done at all.

Another example I can think of is a well-insulated vertical piston-cylinder filled with a saturated mixture of gas and liquid. The piston is frictionless and is allowed to move freely up and down. If a stirring rod is immersed in the liquid and spun, it will do work on the liquid, thus raising its enthalpy. Normally, this would result in a temperature increase, however since the mixture is saturated and held at a constant pressure (via the piston), its temperature must also be constant as required by the Gibbs phase rule. The added energy from stirring is instead used to raise the piston. Basically, the shaft work passes through the fluid and into the piston in the form of increased potential energy; the only change in the working fluid is the proportions of gas and liquid.

A similar setup that is both adiabatic and isothermal might also be possible using a chemical reaction instead of a saturated mixture. Consider the same piston-cylinder as before, but this time it's filled with two gasses in chemical equilibrium with each other; i.e. a reaction like $$A{\rightleftharpoons}B$$ where the reaction is exothermic in one direction and endothermic in the other. If the piston is used to compress the gas, it will heat up, and the increased pressure will cause the reaction to seek a new equilibrium. If the new equilibrium is in the endothermic direction, it would reduce the temperature and possibly cancel out the increase from the compression. The work on the piston would go into the chemical potential of the mixture without changing its temperature. I don't know of any reaction like this, but I also can't think of any reason why it wouldn't be possible. Perhaps someone with more knowledge in chemistry can check me on this.

• Hello Carlton, I am very confused about your first example. You assumed that the energy transfer from the stirring is transferred into the piston, since the mixture is saturated then the constant pressure will be maintained and the piston will move up in an adiabatic and isothermal fashion. But why doesn't the stirring result in an increase of the quality of the mixture? (meaning more vapor less liquid, we are in a constant pressure situation, correct?) until the point of full saturation where the gas is further superheated (if the stirring continues),then the system is not isothermal anymore. – Algo Aug 18 '20 at 8:36
• The choice of the saturated mixture is very complicated in my opinion, and I think your explanation is quite logical but I couldn't push away the idea the the mixture might reach a superheated condition, instead of pushing the piston. – Algo Aug 18 '20 at 8:40
• @Algo You are absolutely correct; the quality will increase as more work (energy) is applied. This only continues until the vapor becomes superheated, though. At that point, it will continue expanding but not isothermally. So, my example is only valid as long as the mixture remains saturated. I chose a saturated mixture because it can absorb/emit energy without changing temperature, so it satisfies the isothermal part of the question. – Carlton Aug 18 '20 at 14:13

No

An isothermal process is, almost by definition, a process where the fluid beeing worked upon can keep its temperature constant by exchanging energy with an external reservor, while an adiabatic process is defined by that not happening.

All these processes are special cases of polytropic processes. An isothermal process as the poytropic coefficient $$n=1$$, an isentropic process - the reversible case of an adiabatic process - has $$n=\kappa$$ with $$\kappa=\frac{C_p}{C_v}$$. Looking at the relation between these heat capacties, this is impossible since they can't be equal.

Phase changes?

Carlton writes:

Another source of storing/releasing energy I can think of is a phase change, i.e. steam-water-ice. Two of those phases can exist simultaneously at a constant temperature but across a range of pressures. Thus there is some capacity to store/release energy without changing temperature.

This appears to be wrong: Take a steam/water mix at equlibrium in a pressure cylinder. The volume is reduced by external force, the pressure and temperature of the gas phase rise (adiabatic process). Due to the new pressure, a new equilibrium between the vapor and liquid phase will establish by condensation. The enthalpy of evaporation is then released as heat. For the phase change to store energy without rising temperature, the condensation would have to absorb heat, not release it.

Chemical processes?

Carlton also suggests:

Consider a chemically-reactive gas at equilibrium in an insulated piston-cylinder setup. As the piston is raised by the gas pressure, the temperature and pressure will drop and thus the chemical equilibrium is disturbed. The gas reacts (chemically), releasing energy in the process until a new equilibrium is established at the original temperature. Thus, the whole process would be both adiabatic and isothermal. I don't know of such a reaction, but it is certainly possible.

I'm not convinced such a reaction exists and I'd have to (re)learn a lot of chemical physics to begin to answer the question if it is theoretically possible. In effect we are looking for a reaction where a change into the gas phase is exothermal somehow, enough so to offset enthalpy of evaporation. Alternativly we could look for a reaction like this:

$$A_g+B_g {\rightleftharpoons}AB_g$$

with the synthesis reaction endothermal.

Now, I can't prove that this is impossible but I have a strong hunch it is: There would be a tremendous application in energy storage. A huge problem with compressed air energy storage (CAES) is the heat generated in adiabatic compression, that is also required for adiabatic expansion.

Traditionally, the expansion heat is supplied by burning natural gas with the expanded air (so only part of the energy supplied comes from the compressed air), a new proposal was the (canceld) ADELE plant that would have used huge packed bed thermal storage units.

If someone had found a chemical reaction that allows de-facto adiabatic-isothermal processes, the killer application for this already here and we would have likely seen it in action.

• Two nitpicks: isothermal means constant temperature, but not necessarily by exchanging heat with the surroundings - energy can be added/removed from a substance without changing its temperature. Also, not every process is polytropic; that is just a special case when $pV^n$ is constant. Your conclusions about isothermal and adiabatic polytropic processes are correct, but nothing in the question says the process has to be polytropic. I have (hopefully) improved my answer by the way. The original examples were a bit vague; I have tried to add more detail. – Carlton Aug 18 '20 at 5:30
• I'd say isothermal means heat exchange with the surrounding if we use a narrow definition, however using a wider definition (as you do) leads to more interesting answers. – mart Aug 18 '20 at 6:22

The temperature varies during adiabatic process and due to this reason this process is not at all iso-thermal

typical example of Adiabatic combined with iso-thermal process ; An axial flow air compressor with inter-coolers and after cooler :

The adiabatic compression process of a multi-stage axial flow air compressor with inter-coolers between stages from suction to final discharge is effectively very near to iso-thermal process as inter-coolers bring down the rise in temperature after each stage

As the air is compressed adiabatically the work done on shaft give rise to increase in pressure and temperature at exit of first stage This temperature in cooled down to initial temperature by inter-cooler

This actions repeat in subsequent stages and achieve final discharge pressure and final temperature is further reduced in after cooler to initial temperature

the whole process is adiabatic compression at stages with stage exit temperatures maintained almost constant as initial temperature incorporating inter-coolers facilities and an after cooler facility making the process from suction to discharge iso-thermal

• So, your argument is about the same as saying the compression ratio of an engine is 1:1 as the exhaust is at the same pressure as the inlet so there is no compression process. – Solar Mike Aug 17 '20 at 6:44
• in a multistage axial flow air compressor air pressure increases stage by stage to achieve desired high pressure the process is adiabatic - Inter- coolers reduces the exit air temperature at each stage making the air more denser before routing to next stage -Denser the air lesser the work needed to raise pressure - The inter-coolers make the whole process from suction to final higher discharge pressure to come closer to iso-thermal process – mukundakumar elattuvalappil Aug 17 '20 at 7:27
• So, the air is cooled between the compression processes NOT during the process... – Solar Mike Aug 17 '20 at 8:58
• yes A compression process is adiabatic and no heat transfer to surroundings But in iso-thermal process heat is removed to surroundings not allowing a rise in temperature -In our case the inter coolers remove heat and make the process very near iso-thermal and that is how this is an adiabatic compression process with facilities provided to tend the process towards iso-thermal The advantage :compressor work less to achieve desired pressure Disadvantage: loss of heat to cooling media – mukundakumar elattuvalappil Aug 17 '20 at 11:13
• If a compression process can happen without transferring heat to the surroundings, why do compressors get hot and need cooling? – Solar Mike Aug 17 '20 at 11:42