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In quasistatic process, intensive variables like temperature and pressure are constant throughout the system. Does it imply that there is no heat flowing in and out the system Q=0 since temperature remain constant? Are there any quasistatic process where heat flows?

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Process with no heat flow is generally referred to as an adiabatic process. Quasistatic processes might be adiabatic or not. In fact, neither isothermic, nor isobaric, nor isochoric processes are adiabatic.

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I think you are mixing up "constant" with "uniform" in the definition. A quasistatic process is a slow thermodynamic process, where the system remains in equilibrium at every moment in the transition from initial to final state. I think this is what you mean by intensive quantities remaining "constant": they are always considered to be uniform throughout the system, because the system is always in equilibrium. The values of temperature, pressure, and any other thermodynamic variables can change in a quasistatic transition, however, and any heat transfer is allowed.

Any thermodynamic process that is represented as a curve on the state space, e.g, on a $pV$-graph, is by definition a quasistatic process, because that curve represents a sequence of (equilibrium) states.

A process that is rapid (e.g., rapid compression of a gas); or, a process that is otherwise out of equilibrium except for the initial and final states (e.g., the free expansion of a gas into a vacuum, the free mixture of two gases, melting ice in warm water) cannot be represented by such a curve. The thermodynamic transition occurs "off of" the state space and the transition is represented just as the pair of initial and final points on the state space. That is a non-quasistatic process.

A quasistatic process is obviously an idealization: it is impossible to change a system such that is it always in equilibrium. It is the limit of making very small changes to the system, letting it come to equilibrium, and repeating.

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A quasi-static frictionless process is a reversible process where the system is constantly in thermal (temperature) and mechanical (pressure) equilibrium with its surroundings (an exception being an adiabatic process where thermal equilibrium is not relevant).

Heat is energy transfer between a system and its surroundings due solely to temperature difference. So you are right that that there can be no energy transfer in the form of heat in a truly reversible process. The same applies to work. This tells us that truly reversible process don't exist. All real processes are driven by disequilibrium (thermal, mechanical, chemical).

That said, reversible processes are idealizations used to determine limits to efficiencies of things like heat engine cycles. The temperature (and pressure) difference are infinitesimal (but not zero) making the rate of energy transfer by heat and work infinitesimal, and the time to carry out the process infinite.

Hope this helps.

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