Isobaric vs Isometric Heating and Work

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:
dQ=dE+PdV (1)
Equation
(1) applies to Fig. 1, as is shown on the right.
As 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.:
dQ=CvdT+ PdV (2)
We can equally rewrite eqn (2) is terms of the isobaric
heat capacity (Cp) as:
dQ=CpdT (3)
If you have trouble with the above then see please blog on Specific Heats
Also Blog on expanding piston cylinder
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 pistoncylinder 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.:
dQ=CvdT+ PdV+Wpiston (4)
And if the moving piston does work onto something else (Welse) then eqn (4) becomes:
dQ=CvdT+ PdV+Wpiston + Welse (5)
Of course all three work terms in eqn (5) are forms of irreversible work. Therefore any process described by
eqn (5) is deemed irreversible.
Next consider the isometric heating of the pistoncylinder 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 simple write in terms of heat transfer (dQin) and isometric heat capacity (Cv).
dQ=CvdT (6)
In eqn 6) CvdT represents the 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 (7)
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 W=PdV.
And if the heat in (dQin) was removed prior to the above unlocking,
then as System 1 does work onto the surrounding atmosphere the gas's temperature in System 1) will decrease.