Temperature CONFUSION in derivation for Helmholtz Free Energy

[daniel_r35]

Dear all,

I have been spending 12 hours on this and cannot seem to come up with a solution.

We derive Helmholtz Free Energy (A) from the second law of thermodynamics.

dS(Total) = dS(system) + dS(surrounds)

we try to express dS(surrounds) with properties of the system.

Assuming constant volume (so q=W) and constant temperature (WHICH AS FAR AS I UNDERSTAND it means that the system and surrounds each remain at their original temperatures but CAN HAVE DEIFFERENT temperatures - which they must if not dS(Total) is zero and you can't derive A),

so we get dS(surrounds) = -dU(system)/T...Now NOBODY, not my books, not the internet, NOBODY!says what 'T' is! THEY SAY IT'S the ABSOLUTE TEMPERATURE! But is it that of the SYSTEM or SURROUNDS?!

If it WAS THE SYSTEM, then it WILL NOT MAKE SENSE, because:

dS(system) is ALREADY +dU(system)/Tsystem.

if you say dS(surrounds) is -dU(system)/Tsystem then dS(total) = ZERO and you can't prove A because A = -TdS!

IF it WAS THE SURROUNDS, IT STILL DOESN'T MAKE SENSE:

because we will get A = -TdS(Total) = dU(system) - T(surrounds).dS(system)

What on Earth will Tsurrounds.dSsystem get you?!

EVERY BOOK will say in Helmholtz free energy, we have expressed it in terms of the internal energy of the system and the TEMPERATURE OF THE SYSTEM! (As though they MEANT for the Temperature to be of the system and not the surrounds)!

but based on my knowledge of entropy shouldn't it be temperature of surrounds, because the entropy of the surrounds should be the heat exchanged AT THAT TEMPERATURE belonging to the surrounds?! we know the heat exchanged is -dUsystem. No doubt about that. But what about T?! Either case doesn't work!

PLEASE! HELP ME!

 

[DrClaude]

Let me quote from D.V. Schroeder, Thermal Physics (Addison Wesley Longman, 2000):

[The Helholtz free energy] is the total energy needed to create the system, minus the heat you can get for free from an environment at temperature $T$.