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I know that reversible processes must be quasi-static, but are quasi-static processes necessarily reversible?

Can someone also please describe the circumstances under the relationships$$ \begin{alignat}{7} \mathrm{d} Q &~=~ &&T \, &&\mathrm{d} S \tag{a} \\[5px] \mathrm{d} W &~=~ − &&P \, &&\mathrm{d} V \tag{b} \end{alignat} $$hold?

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For heat to transfer reversibly, the temperature difference through which it flows must be infinitesimal. For work to be done reversibly by a piston there must be no friction between the piston and the cylinder in which it is moving (if we're considering the work as being done on something external to the cylinder and piston). These are necessary conditions, not sufficient conditions. I don't think that either of them is implied by the term 'quasi-static' which I'd class as another necessary condition – but I stand to be corrected.

I've always thought that a useful criterion for thermodynamic reversibility is that the process will run backwards if only infinitesimal changes are made to conditions (e.g. heat will go in the other direction if an infinitesimal change is made to a system's temperature or work would be done on, rather than by, the system if the system' pressure were lowered infinitesimally).

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    $\begingroup$ agree with philip; my own criterion for reversibility is the absence of temperature gradients anywhere within the system during the process. $\endgroup$ Dec 25, 2017 at 0:50
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    $\begingroup$ @niels nielsen What about velocity gradients and concentration gradients? $\endgroup$ Dec 25, 2017 at 3:02
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    $\begingroup$ regarding velocity gradients: in compressible media this implies pressure gradients, but as long as they are comfortably subsonic then the process should be reversible. I do not know about concentration gradients... $\endgroup$ Dec 25, 2017 at 3:25
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    $\begingroup$ @nielsnielsen velocity gradient in a fluid often implies friction => process can't be reversible. $\endgroup$ May 4, 2019 at 19:18
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    $\begingroup$ I specified compressible media, meaning gases. Sound waves in air do not leave the medium in a new thermodynamic state upon passage and would then constitute a reversible process. $\endgroup$ May 4, 2019 at 20:14

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