Re: [Phys-l] T dS versus dQ

[John Mallinckrodt <ajm@csupomona.edu>]

Bob LaMontagne wrote:

> What characteristic of the final equilibrium state would tell me if  
> stirring occured or not?

Assuming that we are talking about a thermally-insulated gas that  
starts and ends in thermal equilibrium, stirring has occurred if and  
only if the entropy increases.  Period.

> While you reject the temperature-entropy relation I wrote, do you  
> also disagree with the final T value after the waves die out being  
> determined by the well defined energy input - in this particular  
> example? I'm trying to get through this one step at a time.

Well ... I fear that much of the confusion in this thread is the  
result of differing hidden assumptions about what is given so let's  
try to be very specific.  First, a few facts:

1. For a given gas at thermal equilibrium, both the energy and the  
entropy depend only on the temperature and the volume and, for a  
given volume, both the energy and the entropy increase monotonically  
with temperature.

2. All of the above is also true for an ideal gas with the slight  
simplification that the energy does *not* depend on the volume.


Now some cases.  In each, I assume a) that the gas is thermally  
insulated, b) that it starts and ends in thermal equilibrium, and c)  
that the initial state is given.

CASE 1: An irreversible compression to a given final volume, (i.e.  
Carl's case):

1. Stirring has occurred and the final entropy is greater than the  
initial entropy.  (Because we are explicitly told that the process is  
irreversible.)

2. The final energy is greater than it would have been had the  
process not been irreversible. (Because higher entropy => higher  
temperature => higher energy for a given volume.)

3. More work was done on the gas than would have been had the process  
not been irreversible. (Because the extra energy was entirely the  
result of extra work done.)


CASE 2: The energy input is specified:

1. We know the work done.  (Because the energy input was entirely the  
result of work.)

2. We don't know the final temperature.  (Because we don't know the  
final volume.)

3. We don't know the final entropy.  (Because we don't know the final  
temperature OR the final volume.)

4. We don't know if stirring occurred.  (Because we don't know if the  
entropy increased.)


CASE 3: The energy input is specified and the gas is ideal:

1. We know the work done.  (Because the energy input was entirely the  
result of work.)

2. We know the final temperature.  (Because we know the final energy.)

3. We don't know the final entropy.  (Because we don't know the final  
volume.)

4. We don't know if stirring occurred.  (Because we don't know if the  
entropy increased.)


CASE 4: The energy input and the final volume are specified.

1. We know the work done.  (Because the energy input was entirely the  
result of work.)

2. We know the final temperature.  (Because we know the final energy  
and the final volume.)

3. We know the final entropy.  (Because we know the final temperature  
and the final volume.)

4. We know if stirring has occurred.  (Because we know if the entropy  
increased.)

5. We know if the problem specification was unphysical.  (Because it  
is possible to specify an energy input and final volume that result  
in a final entropy that is less than the original entropy and *that*  
is physically impossible.)

John Mallinckrodt
Cal Poly Pomona