Derivation of isobaric heat capacity

As I was thinking about isobaric processes, one thing came to mind: In a reversible isobaric process, the value of Cp (molar isobaric heat capacity) is bigger than Cv (molar isochoric heat capacity) by exactly one R (ideal cas constant). For example, for argon, Cp is 2.5R while for Cv it is 1.5R.


This is due to the derivation, where under the pV integral, pV can be substituted by RT for one mole of the gas. This is only possible though, if the gas is in equilibrium with the outside pressure the whole time:


<https://en.m.wikipedia.org/wiki/Isobaric_process>


So my question is, would the value of the molar isobaric heat capacity change, if we applied an irreversible isobaric change?


That is, the gas is heated much more quickly that it can expand, so that the pressure in the gas is higher than the outside pressure on the piston, so that the gas and outside pressure are not in equilibrium the whole time.




If the gas is initially in equilibrium with the surroundings and, in the final state, it is also in equilibrium with the surroundings, then $$p\_{gas,init}=p\_{gas, final}=P\_{ext}$$then$$W=p\_{ext}(V\_{final}-V\_{init})=p\_{gas,final}V\_{final}-p\_{gas,init}V\_{init}=\Delta (PV)\_{gas}$$


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