By Kent W. Mayhew

Isobaric vs Isometric Heating and Work by Kent W. Mayhew

Writing the first law, in terms of heat into the system (dQ) and changes to a system’s internal energy (dE) plus the irreversible work done onto the surrounding atmosphere by an expanding system i.e. lost work = PdV:

dQsys = dEsys + (PdV)atm = dEsys + PatmdVsys (1)

Equation (1) applies to Fig. 1, as is shown on the right. Note subscripts "sys" and "atm" respectively apply to system and atmosphere, thus giving the parameters clarity. See parameters.

As was discussed in the work vs energy blog:
Things become more obvious if we now rewrite eqn (1) in terms of isometric heat capacity (Cv) and temperature change (dT), i.e.:

dQsys = (CvdT)sys+ (PdV)atm (2)

We can equally rewrite eqn (2) is terms
of the isobaric heat capacity (Cp) as:

dQsys = (CpdT)sys (3)

If you have trouble with the above then see please blog on Specific Heats i.e. Cp = Cv + PdV

Also Blog on expanding piston cylinder

Also if you do not understand why it is wrong to write eqn (2) in terms of the expanding systems parameter

then see parameters

Now it should be stated that the above three equations assumes that the process
is absolutely isobaric which could only occur if the piston is massless and the piston-cylinder is frictionless.

If the piston were not both frictionless and massless then we would have to add to eqn (2) a term for work done in moving the piston.
I.e.:

dQsys = (CvdT)sys+ (PdV)atm+ Wpiston (4)

And if the horizontal moving
piston experiences friction plus does work onto something else (Welse) then eqn (5) becomes:

dQsys = (CvdT)sys + (PdV)atm + Wfriction +
Welse (6)

Of course all three work terms in eqn (6) are irreversible work! This assumes
that Welse does not include a potential energy increase, which can then be used to compress the piston-cylinder. Therefore any
process described by eqn (5) is deemed irreversible!

Next consider the isometric heating of the piston-cylinder
with the piston locked in position, as illustrated in Fig 2. In this case there is no work done and no work can be done onto anything
else. Herein we can simply write in terms of heat transfer (dQin) and isometric heat capacity (Cv).

dQsys = (CvdT)sys (7)

In eqn 7) CvdT represents an increase in the total energy within System 1 (A.K.A. internal energy increase).

** Increase in Potential to do Work**

For the case of System 1 being a gas as illustrated in Fig 2, as the gas's temperature increases,
its pressure increases hence its potential to work increases i.e.:

dWpotential = (VdP)sys (8)

The increase in System 1's ability to do work, as defined by eqn (8) is part of System 1's energy increase as defined by eqn (7). This is to say that at no point is the increase in potential to do work to be added to the its energy increase.

Moreover if the gas in Sytem 1 was ideal monatomic gas then then energy of that gas is 3PV/2 hence eqn (7) can be rewritten as:

dQsys = (3VdP/2)sys (9)

Remember all of a closed system's energy increase cannot be used to do work see Work vs Energy

** Gas doing Work**

And if the piston inside the cylinder is
suddenly unlocked, then the potential to do work can become work done onto the atmosphere as defined by Watm=(PdV)atm. Again
this work is not defined in terms of the expanding system's parameter i.e. see parameters!

And if the heat in (dQin) was
removed prior to the above unlocking, then as System 1 does work onto the surrounding atmosphere then the gas's temperature
within System 1) will decrease.

Isobaric vs Isometric Heating