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I found this concept of entropy production in wikipedia.

Mainly I am trying to figure out the formula the Clausius formula for entropy production. What exactly are the terms involved?

  1. For what process is $S-S_o$ written?

  2. For what process is $\int \frac{dQ}{T}$ written? What is the $T$ in this expression?

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The entropy change of a system is the sum of two parts:

  • Entropy transferred from the surroundings to the system (across the interface with the surroundings) as a result of heat flow, and given by $\int{\frac{dq}{T_B}}$, where dq is the differential heat flow across the boundary interface between the system and surroundings and $T_B$ is the temperature at the boundary through which this same heat flow takes place.
  • Entropy generation $\sigma$ within the system as a result of irreversibility driven by internal viscous friction, internal conductive heat transfer, and internal mass diffusion. In a reversible process, this contribution to the entropy change is zero, and, in an irreversible process, this contribution is always positive

So, $$\Delta S=\int{\frac{dq}{T_B}}+\sigma$$or, expressed as an inequality, $$\Delta S\geq \int{\frac{dq}{T_B}}$$Also, in a reversible process, the system and surroundings temperatures are equal, so that, at the boundary, $T_B=T$, where T is the system temperature.

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  • $\begingroup$ So, this whole entropy production expression for the system? I'm a bit confused how we can write an integral expression as the infinitesimal quantites in between the processes are undefined ( how do you define temperature of a substance not in equilibrium) $\endgroup$
    – Brian
    Commented Oct 3, 2020 at 15:25
  • $\begingroup$ I don't understand what you are asking. $\endgroup$
    – Chet Miller
    Commented Oct 3, 2020 at 15:30
  • $\begingroup$ " Entropy transferred from the surroundings to the system (" What do you mean by this? Secondly, I'm asking if this expression for entropy change is the one for the entropy change of system or surroundings or whole universe. And also, when I first introduced to entropy, the temperature in the expression was temperature of system and that was ok because temperature of system is constant through out. So, I thought we couldn't define the expression here because of that however you have used temperature of boundary. Now I ask how did we reach the conclusion that it is the temperature - $\endgroup$
    – Brian
    Commented Oct 3, 2020 at 15:38
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    $\begingroup$ Sorry.. My understanding is that, as with internal energy, heat flow across the boundary can change the internal energy of a system, in the case of entropy, transfer flux of $dq/T_B$ can change the entropy of a system. However, a difference between internal energy and entropy is that internal energy can not be generated within the system, and entropy can. $\endgroup$
    – Chet Miller
    Commented Oct 3, 2020 at 15:52
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    $\begingroup$ With regard to your second question, the expression I'm discussing is one for the entropy change of the system only. $\endgroup$
    – Chet Miller
    Commented Oct 3, 2020 at 15:57
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Suppose the entropy associated with the system and the surrounding at the start of thermodynamic process is $S_o$ and the entropy associated with it at the end is$S$. $$∆S=S-S_o$$

The entropy change through any reversible path connecting intial and final state can be given as- $$∆S_{rev}=\int\frac{dq}{T}$$ Here , $T$ is thermodynamic temperature.

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