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Why can't a isothermal process be adiabatic?

"Why can't a isothemal process be adiabatic?"

This was the question my housemate, that studies geology and geography, asked me while studying for his thermodynamics test. As a chemical engineering student I was suppose to be able to answer him immediately, but it took me some pondering to give him a good answer.

Lets consider the isothermal expansion of gas, and lets use the convention used by most thermodynamic textbooks that work (W) done by the system on the environment is said to be negative.
Consider the diagram and the formula used for work done by isothermal expansion of an ideal gas and the heat formula.
Isothermal expansion


Work done by isothermal expansion of an ideal gas
Heat formula for expansion of an ideal gas in a closed system
And because, for reversible isothemal expansion the following is true:
For reversible isothermal expansion 
thus we can see that Q = -W, and W is not equal to 0 because the volume has changed, that means that Q is also not equal to 0.

Keeping the above mentioned in mind, we can now look at the conditions for reversible adiabatic expansion, that is Q=0, and because we know that Q is not equal to 0 in a isothermal expansion system,  this is the reason that a isothermal process cannot be adiabatic.

Now as I will always try to do is give an real life example that I think is fitting for the discussed theory. I don't know if you have ever noticed a can of deodorant becomes colder when you use it, or if you are a mountain biker like me you might have used one of those CO2 gas-bomb  tire inflation devices those things can literally freeze to your hand if your not carefull, well that is because of the expansion of gas, take a look at the regulator of a CO2 tank that froze up because of the gas expanding from the high pressure cylinder to atmospheric pressure.

CO2 regulator freezing up
So for this process to be isothermal on a closed piston cylinder assembly, heat needs to be transferred from the environment to the system to keep the temperature constant, thus Q is not equal to 0, making it isothermal but not adiabatic.
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Comments

  1. Unknown16 November 2015 at 20:32

    If it is a closed system and a free expansion it's possible.I would also be possible with a control volume, for instance take a look at the isenthalpic expansion of a gas.

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