Mathematical Proof: Entropy Change of a Reversible Process

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Question
Possible mathematical definition of a process that has both pressure and temperature difference but is reversible
I have a question about relations for a process even though I am unsure about the scope of this. Still I wonder about a mathematical relation for the following:

I wonder about how one defines mathematically that a process is reversible for a process that has both pressure and temperature differences from definition of total entropy

Is there a mathematical way to show how much a factor the temperature is and how much a factor the pressure is for an ideal gas and then find a situation where the process is reversible for a state that has both pressure and temperature differences that is that dS_tot=0? That is finding a way to express dS_tot while only using p, T, V and C_v as variables while there are pressure and temperature differences and showing where the values of T and p are that creates dS_tot=0

Even if it would only be a strict mathematic definition it could be really helpful!

I need a computer written answer with mathematical signs. I hope it is ok

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Answer

Reversible Process
In thermodynamics, a reversible process, or reversible cycle if the process is cyclic, is a process that can be "reversed" by means of infinitesimal changes in some property of the system without entropy production (i.e. dissipation of energy). A reversible process changes the ...