### Thermodynamic Cycles

We would like to be able to convert thermal energy to work in
order to operate machines of all sorts. Further, we have shown that
we can obtain the *maximum* work from a reversible expansion
(as opposed to an irreversible one).

- Wanted: $Q \rightarrow W$
- Known: $W_{rev}>W_{irrev}$
- Therefore: Use reversible expansion to build an "engine"

##### Note:

- $Q_{exp} = - W_{exp}$
- $Q_{comp} = - W_{comp}$
- $W_{exp} = - W_{comp}$!! (no net work)

However, we recall that our irreversible process did not cancel
out. Why not use that as an "engine"?

##### Note:

- $- Q_{exp} = W_{exp}$
- $- Q_{comp} = W_{comp}$
- $W_{exp} < - W_{comp}$

##### Definition:

A **Carnot cycle** is the most efficient engine
possible.

$W_{net} = W_{12} + W_{23} + W_{34} + W_{41}$

Since the work from $2 \to 3$ exactly cancels the work from $4
\to 1$, we get

$W_{net} = W_{12} + W_{34} = nRT_1\ln \left (\frac{P_2}{P_1}
\right ) + nRT_3\ln \left (\frac{P_4}{P_3} \right )$

$Q_{net} = Q_{12} + Q_{34} = Q_H - |Q_C|$

$\Delta U_{net} = 0$

##### Note:

Here we take advantage of the fact that $W_{12}$ and $W_{34}$
have differing signs (since one is work "out" and the other is
"in"). Similarly, the signs of $Q_{12}$ and $Q_{34}$ differ, so
that the net value is obtained by adding them. When we switch to
the $Q_{H}$ and $Q_{C}$ notation, we explicitly use the absolute
value so that we can emphasize that the net heat flow is the
difference between the amount absorbed from the hot reservoir and
that expelled to the cold reservoir.

##### Definition:

Running a Carnot cycle "forward" is called an engine and has an
efficiency of :

$\eta = \frac{W_{net}}{Q_{H}}$

Running a Carnot cycle "backward" is called a refrigeration
cycle and has an efficiency of:

$COP = \frac{Q_{C}}{W_{net}}$ [$COP \equiv$
"coefficient of performance"]

##### Outcome:

Identify, formulate, and solve simple engineering problems (such
as expansion/compression and power cycles)