[Physics] Is the total entropy change of any isothermal process always zero

entropyhomework-and-exercisesthermodynamics

I was asked to solve this following problem:

Two moles of a perfect diatomic gas at 300 K is expanded isothermally
from an initial pressure of 3.00 atm to a final pressure of 1.00 atm,
against a constant external pressure of 1.00 atm. Determine the values
of $\Delta U, \Delta H, \Delta S,$ $\Delta S_\mathrm{surroundings}$
and $\Delta S_\mathrm{total}$.

First, I'm assuming $\Delta S$ means $\Delta S_\mathrm{system}$. My plan to solve this problem was to find $q_\mathrm{rev}$ for this system, but realized that I'm not exactly sure how total entropy is calculated for a non-reversible process. My question is:

We define total entropy as: $$\Delta S_\mathrm{total}=S_\mathrm{system}+S_\mathrm{surroundings}$$

I know that for an isothermal AND reversible expansion: $S_\mathrm{total}= 0$

But what about an isothermal expansion at a constant pressure? Does $S_\mathrm{total}$ also equal 0? If not, then would the equation look something like this:
$$\Delta S_\mathrm{total}=\frac{q_\mathrm{rev}}{T}+\frac{q_\mathrm{surroundings}}{T}$$

where $q_\mathrm{surroundings}$ is the heat lost or gained by the surroundings by a process and $q_\mathrm{rev}$ is the heat lost or gained by the system IF that process was to become reversible.

Best Answer

Here are the steps:

  1. Use the first law of thermodynamics to establish the final thermodynamic equilibrium state of your system for the irreversible process process you are considering.

  2. Forget about the irreversible process path entirely.

  3. Devise a reversible process path between the same pair of initial- and final thermodynamic equilibrium states that were obtained with the irreversible process (i.e., devise a path consisting of a continuous sequence of thermodynamic equilibrium states for the system). This reversible process path does not have to bear any resemblance whatsoever to the actual reversible path. There are an infinite number of reversible paths that will get you between the two states, so choose one that is easy to work with for step 4.

  4. Calculate the integral of dQ/T for the reversible path you have focused on in step 3. This is the change in entropy for the system.

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