Entropy Qrev/T?

Hi aniketp,

One way of writing the First Law is

[tex]dU=q+w[/tex]

which means that the differential change in energy equals the heat transferred to a system plus the work done on a system. But another way is

[tex]dU=T\,dS-p\,dV[/tex]

where the work is expressed as a generalized force (pressure) times a generalized displacement (change in volume). The heat transfer term is expressed in an analogous way: a generalized force (temperature) times a generalized displacement (change in entropy).

If you haven't heard the terms generalized force and generalized displacement before, they're just ways to classify variables. Generalized forces are intensive, and they drive processes; a change in temperature drives heat flow, and a change in pressure drives mass flow. Generalized displacements, which are extensive, are the "stuff" that is transferred: entropy, volume.

[itex]S=Q_\mathrm{rev}/T[/itex] arises because at constant volume and temperature, and if the process is reversible (no excess entropy generated) we can integrate [itex]q=T\,dS[/itex] to give [itex]Q_\mathrm{rev}=T\Delta S[/itex].

Does this make sense?

 

No...

Entropy is defined as S = k*log(omega)
Temperature is defined as 1/T = dS/dU at constant V
Pressure is given by P = T * dS/dV at constant U

From the previous two relationships it follows that
dU = TdS - PdV, and S = Q/T is a special case.

For an intuitive discussion see, for example, An Introduction to Thermal Physics by Schroeder, sections 2&3.

 

I should mention that these expressions for T and P are derived (in Schroeder) by consideration of equilibrium conditions.

 

k is the boltzmann constant, and omega is the "multiplicity" of the system, or the number of possible microscopic configurations the system can have.

Simple example:
Say I have a system of 3 "oscillators" (that are quantized), with a total energy of "q" units. Say q = 0, then there is only one possible arrangement (Omega=1). Say q = 1, then there are three possible arrangements (Omega = 2). Say q =2, then there are 6 possible arrangements, ie (2,0,0) three times, and (1,1,0) 3 times. So Omega = 6. Say q = 3, then there are 10 possible arrangements, (3,0,0) 3 times, (2,1,0) 6 times, and (1,1,1) once. So Omega = 10.

 

See here for rigorous derivation of dE = T dS - P dV