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3. Reversible and Irreversible Processes and Cycle

3.1 Carnot Cycle
In order to have a specific example of a Carnot cycle for study, consider a system
comprised of a gas held in an insulated cylinder fitted with an insulating piston (Fig. 9).
The insulation of the cylinder head can be removed so that the cylinder can periodically
be placed in intimate contact with the energy reservoir at T 1 or the one at T 2.
Let the cycle begin with the gas in a state 1 as shown on the PV diagram in Fig. 9.
The temperature of the gas is T 1. Then the insulation is removed from the cylinder, and
the cylinder head is placed in contact with the energy reservoir at T 1. The gas expands
very slowly, doing work on the surroundings. The temperature of the gas tends to
decrease, but the flow of the heat from the energy reservoir maintains the gas
temperature constant at (T1 — dT). Notice that the transfer of heat is reversible only as
long as the temperature difference between the reservoir and the gas is infinitesimal. If
the temperature of the gas were to fall lower than (T 1 — dT), the process would be
externally irreversible. This reversible isothermal process continues until the piston
reaches a position 2. The piston is then stopped, holding the gas in state 2, while the
cylinder is removed from the reservoir at T 1, and the insulation is put back on the
cylinder head. The gas then pushes the piston farther outward as it expands reversibly
and adiabatically. Work is done by the gas, and there is no heat input; so the
temperature drops. The piston is allowed to move until the gas temperature becomes T 2.
The gas is then in state 3. Notice that any heat transfer between the gas and one of the
reservoirs while the gas was at a temperature between T 1 and T 2 would have made the
process externally irreversible.
Now, if the piston were to be pushed inward while the cylinder is still completely
insulated, the gas would be compressed adiabatically, and its temperature would again
rise. If the compression were reversible and adiabatic, the gas would retrace the path
between states 3 and 2. Obviously this will not help us obtain a net work output from
the cycle. Therefore, while the piston is in position 3, the cylinder-head insulation is
again removed, and the cylinder is placed in contact with the low-temperature reservoir.
Now, as the piston is pushed inward, the gas temperature tends to rise, but heat is
transferred from the gas to the cold reservoir at such a rate that the gas temperature
remains constant at (T 2 + dT). Since the heat is transferred across an infinitesimal
temperature difference, the process is externally reversible. This reversible isothermal
compression of the gas is continued until a state 4 is reached. State 4 is such that, if the
insulation is put back on the cylinder head and the gas is compressed reversibly and
adiabatically, its temperature and pressure will increase, and the gas will be returned to
state 1. Thus the cycle is completed.
A heat engine performs a cycle during which it receives some heat from a hot
reservoir, rejects a part of this heat to a cold reservoir, and converts the remaining
portion of this heat into mechanical work. The reservoir at higher temperature is called
“the boiler” or “the source”, and the reservoir at lower temperature is called “the
condenser” or “the exhaust”. Suppose a heat engine absorbs heat Q 1 at temperature T 1,
and reject heat Q 2 at temperature T 2. Let the work done by the engine during a cycle be
L. As the engine comes back to its initial state, the not change in its internal energy
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