Weíve been illustrating the concept of isothermal expansion and contraction using a one-dimensional,
single-atom gas model.
An obvious simplification is that we treat the fixed and moving walls, as well as the
heat reservoir, as ideal, infinitely-hard, solid structures.
In reality, they would be made of atoms, and we are neglecting the kinetic interactions
between the atoms of the gas and the atoms of these structures.
For example, we assume there is no transfer of heat between the gas and the moving wall.
The actual physics of a gas-solid interface is more like this conceptual simulation.
The blue balls represent atoms of a solid which, in addition to collisional forces are
also subject to elastic forces trying to keep them in their equilibrium positions in the
crystal lattice.
The red balls represent gas atoms subject only to collisional forces.
All kinetic energy is originally contained in the red atoms.
As time goes on atomic collisions transfer energy from the gas to the solid until their
temperatures reach equilibrium.
So, when we illustrate a piston system, and claim that the gray regions, the piston and
cylinder walls, are thermally insulated, that canít be rigorously true in any real system.
Weíre developing an idealized model to help us understand ultimate theoretical thermodynamic
limits.
A real system might approach these limits with a design that reduces thermal transfer
between the gas and the gray regions to small amounts.
Possibly, by using materials with small heat capacity.
But no real system will ever reach the theoretical limits.
With that in mind, letís consider the work done in isothermal expansion or compression.
The gas volume equals the piston cross-sectional area, ëAí, times the gas height ëxí.
The force on the piston is the gas pressure ëPí times the area ëAí.
From the ideal gas law the pressure is the number of gas atoms, ëNí, times Boltzmannís
constant, ëkí, times the temperature, ëTí, divided by the volume.
The force is then ëNí ëkí ëTí over ëxí.
When the piston moves a distance delta-x the work done is approximately ëFí times delta-x.
Approximately, because the force changes with ëxí.
If delta-x is very small this change is very small.
If the piston travels a large distance, from x-1 to x-2, we have the process illustrated
in this graph.
The orange curve shows the force as a function of position ëxí.
For a small displacement delta-x, the work done is approximately the area of one of the
blue rectangles, ëFí times delta-x.
We add all these areas to get the total work done in moving from x-1 to x-2.
As delta-x gets smaller the area of the blue rectangles more closely approximates the exact
work, which is the area under the orange curve.
The calculation of this area is called integration, and we write that the work done on the environment
equals ëNí ëkí ëTí times the integral from x-1 to x-2 of ëdxí over ëxí.
This equals ëNí ëkí ëTí times the natural logarithm of x-2 over x1.
Instead of expanding from x-1 to x2, we can push down on the piston and compress the gas
from x-2 back to x-1.
The same expression now represents the work done on the system.
Everything returns to its original state leaving no trace of the expansion-compression cycle.
We say that isothermal expansion and compression are reversible.
We now have enough theory to design a simple, but useful, heat engine, based on the so-called
Stirling cycle.
We start with the piston in position x-1.
The gas is maximally compressed and in contact with a hot heat reservoir at temperature T-h.
The gas undergoes isothermal expansion until the piston position is x-2.
During this the gas exerts a varying force ëFí on the piston.
The resulting work done on the environment is ëNí ëkí T-hot times natural logarithm
of x-2 over x-1.
The gas temperature doesnít change, so the gasís internal energy remains constant.
By the first law, therefore, an amount of energy in the form of heat, equal to this
work, must flow from the hot reservoir to the gas.
We now replace the hot reservoir with a cold reservoir at temperature T-c.
The gas cools to this temperature and its pressure decreases proportionally.
Exerting a force on the piston we isothermally compress the gas until the piston returns
to its original position x-1.
The work done on the system is ëNí ëkí T-cold times natural logarithm of x-2 over
x-1.
By the first law this energy in the form of work must be converted to energy in the form
of heat which is transferred to the cold reservoir.
The cycle is now complete.
We can replace the cold reservoir with the hot reservoir and repeat the cycle for as
many times as desired.
Because expansion occurs at high temperature and pressure, and compression occurs at low
temperature and pressure, the work done on the environment during expansion is larger
than the work done on the system during compression.
So, a net amount of work, ëNí ëkí T-hot minus T-cold times natural log of x-2 over
x-1, is produced by the engine.
We can plot the evolution of the state of the system on a P-V diagram.
The first leg of the cycle is isothermal expansion at temperature T-hot.
This takes us from state 1 to state 2 along an isothermal curve.
As volume increases, pressure decreases according to the Ideal Gas Law.
In the second leg of the cycle, the gas is cooled.
The pressure drops accordingly, and we move to a second isothermal curve at temperature
T-cold.
This takes the system from state 2 to state 3.
The third leg of the cycle is isothermal compression at temperature T-cold.
This takes us from state 3 to state 4.
As the volume decreases the pressure increases.
The final leg of the cycle is the reheating of the gas to temperature T-hot.
This increases the pressure and takes us from state 4 back to state 1.
During the transition from state 1 to state 2, heat energy Q-s is delivered to the system
and converted to work W-e delivered to the environment, equal to ëNí ëkí T-hot natural
log x-2 over x-1.
The cooling from state 2 to state 3 transfers heat from the gas to the environment, specifically
the cold reservoir.
This heat Q-e equals the heat capacity of the gas, 3-halves ëNí ëkí times the temperature
change, T-hot minus T-cold.
During the transition from state 3 to state 4, work W-s is done on the system and converted
to heat Q-e transferred to the environment, specifically the cold reservoir, equal to
ëNí ëkí T-cold natural log x-2 over x-1.
Finally, warming the gas to bring the system back to state 1 requires heat Q-s to be transferred
from the hot reservoir to the system, equal to 3-halves ëNí ëkí T-hot minus T-cold.
The area under the curve between states 1 and 2 is the work done on the environment.
The area under the curve between states 3 and 4 is the work done on the system.
The difference is the area between the curves, and this represents the net work produced
by the engine.
The theoretical Stirling cycle weíve just developed is an ideal that can only be approximated
by a practical design.
A popular implementation for demonstration purposes is the gamma-type Stirling engine.
Hereís a conceptual schematic.
The heat reservoirs are implemented by two metal plates.
A cold one on top and a hot one on bottom.
If the cold plate is isolated by a displacer, the enclosed air is primarily in contact with
the hot plate.
The resulting higher pressure pushes a piston upward, doing work on the environment.
Moving the displacer to isolate the hot plate leaves the air primarily in contact with the
cold plate.
The resulting lower pressure allows the piston to be pushed downward with less work than
was produced on the upward stroke.
The net work output by the engine can be used to turn a flywheel.
Placing one of these toy Stirling engines over a cup of hot water warms the bottom plate.
As the flywheel turns you can see the large, gray displacer moving between the hot and
cold plates, and the small white piston delivering a net positive amount of energy to the flywheel,
causing it to accelerate.