By Kent W. Mayhew

Blog:Joules Experiment & Ideal Gas Paradox

*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 Boltzmann’s consideration that entropy signifies
a system’s disorder; essentially entropy represents the “randomness of matter in incessant motion”^{9}. A more recent but equally poor
definition is that entropy is “the dispersal of a system’s molecular energy”^{10} .

See blog on Entropy

*Other Considerations*

Consider TS=E+PV 1)

Note eqn 1) can be obtained by integrating
the isothermal isobaric relation;

TdS=dE+PdV 2)

Of course differentiating eqn 1) gives:

TdS+SdT=dE+PdV+VdP 3)

Seemingly traditional thermodynamics avoids the above by hiding the obvious. This is accomplished by writing enthalpy (H)
as

H=E+PV 4)

Although enthalpy is really a chemical reaction thing, it also hides eqn 1). We should really write

TS=H=E+PV 5)

The above is further hidden by the differential shuffle that starts with eqn 2)

*
It must be stated that differentiating eqn 1 as stated by eqn 3) has issues concerning its logic when one realizes
that PdV is the work done onto thte atmophere *

*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 gas’s volume
doubles its pressure decreases by half.

As this ideal gas’s volume increases, the *molecules’ randomness*, and/or the *dispersal
of energy*, must increase. Therefore by definition, its entropy (S) should increase. If a system’s entropy is increasing, and there
is no total energy change within the system, then shouldn’t we expect that the ideal gas’s temperature (T) will decrease, such that:TdS=-SdT? But that makes no sense because the process is isothermal (dT=0), allowing Boyle’s 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 gas’s 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:

TdS=dE 6)

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.

Click: For an expanding piston-cylinder

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.

* Joule’s
Experiment*

The experiment illustrated by Fig 1.1.2 and 1.1.3 is known as Joule’s experiment for gases. James Prescott Joule concluded
that his experiment shows that the gas’s 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 Joule’s 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 convictions can be awkward to reason with.

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