Joules's Experiment & Ideal Gas Paradox
Reminder concerning entropy
Entropy remains the poorest understood parameter. In the 19th century, Rudolf Clausius realized that something when multiplied by temperature represented energy. Since then, entropy has taken on an array of different meanings. To many, its definition wrongly revolves around Boltzmannís consideration that entropy signifies a systemís disorder; essentially entropy represents the ďrandomness of matter in incessant motionĒ9. A more recent but equally poor definition is that entropy is ďthe dispersal of a systemís molecular energyĒ10 .
See blog on Entropy
Consider TS=E+PV 1)
Note eqn 1) can be obtained by integrating the isothermal isobaric relation;
Of course differentiating eqn 1) gives:
Seemingly traditional thermodynamics avoids the above by hiding the obvious. This is accomplished by writing enthalpy (H) as
Although enthalpy is really a chemical reaction thing, it also hides eqn 1). We should really write
The above is further hidden by the differential shuffle that starts with eqn 2)
Ideal Gas Law Paradox
Bearing in mind the above stated definitions of entropy, consider the ideal gas in Vessel A, as shown in Fig 1.1.2. A valve is opened and the gas is allowed to isothermally disperse into Vessel B, as is illustrated in Fig 1.1.3.
We expect that: PdV= -VdP. In other words, as the gasís volume doubles its pressure decreases by half.
As this ideal gasís volume increases, the moleculesí randomness, and/or the dispersal of energy, must increase. Therefore by definition, its entropy (S) should increase. If a systemís entropy is increasing, and there is no total energy change within the system, then shouldnít we expect that the ideal gasís temperature (T) will decrease, such that: TdS=-SdT? But that makes no sense because the process is isothermal (dT=0), allowing Boyleís law to remain valid, i.e.: PdV= -VdP. Certainly, if the internal energy is related to the potential associated with intermolecular bonding, then the ideal gasís internal energy does not change.
Seemingly, there is something wrong with our understanding of thermodynamics. What could it be? Perhaps it is because the ideal gas law is only an approximation! Although valid, such an argument lacks scientific gumption. Other possibilities:
1) Perhaps, we must reconsider TdS+SdT=dE+PdV+VdP 3)
If PdV=-VdP and dT=0, then eqn 3) implies:
If eqn 6) defines changes to our isothermally expanding ideal gas, then one cannot isothermally expand a gas and maintain constant internal energy within that system of gas. In which case the internal energy (E) of our isothermally expanding ideal gas has seemingly increased: . Does this mean that the internal energy changes, while the intermolecular bonding energy remains constant, as expected for an ideal gas? It all seems convoluted.
2) Perhaps we must reconsider what entropy is!
Since: PdV=-VdP and dT=0 and if dE=0, then dS=0. Consider our previously given two definitions of entropy. During the isothermal expansion of the ideal gas, both the randomness of molecules in incessant motion and/or the dispersal of the gas molecules energy, have increased, yet there is no predicted entropy change? Seemingly, the virtues of entropy should be queried!
Ultimately the ideal gas law has suffered a paradox. One might argue that our analysis is too simplified. But to do so alleges that the ideal gas is now complex, which it is not. Or that eqn 1) can not be obtained by the integration of eqn 2). And herein resides the issue. In order to circumnavigate such simple logic, like that given above, traditional thermodynamics has unwittingly complicated the simple.
The experiment illustrated by Fig 1.1.2 and 1.1.3 is known as Jouleís experiment for gases. James Prescott Joule concluded that his experiment shows that the gasís internal energy is a function of temperature but not volume. Obviously the isothermal expansion of an ideal gas implies that dE=0. Even if his result is accepted as correct, his experiment remains misguided; if energy were extracted from the surrounding heat bath (which it was not) then would it be measurable? Probably not!
Whether we consider Jouleís experiment, or simply the isothermal expansion of any ideal gas, the ideal gas law suffers a paradox. Traditionalists can circumnavigate this by shuffling the differential equations around, and then boldly subscribing to a convoluted logic. Understandably, those bestowed with such arguments can be awkward to reason with.