Irreversible processes

The laws of mechanics are invariant under time reversal:

And yet...


The laws of mechanics are reversible. But here are examples of processes occurring in mechanical systems that go spontaneously in just one direction...

Irreversibility: the Second Law of Thermo

Clausius statement: It is impossible to construct a device that operates in a cycle and whose sole effect is to transfer heat from a cooler body to a hotter body.

...that is, with no net work input.

Kelvin-Planck statement: It is impossible to construct a device that operates in a cycle and produces no other effect than the performance of work and the exchange of heat with a single reservoir.

...that is, with a net positive amount of work done by the system.

With Clausius' statement we are in a position to prove Carnot's theorem:

No engine operating between two reservoirs can be more efficient than a Carnot engine operating between those same two reservoirs.

To prove this, consider a potentially fantabulous engine, $M'$ that has an efficiency greater than our Carnot engine, $M$, that is, in one cycle:

$$\eta' =\frac{W'}{Q_2'} > \frac{W}{Q_2}=\eta$$.

Since the Carnot engine is **reversible**, we can just as easily configure it as refrigerator, where the work $W'$ done by $M'$ is precisely equal to the input work $W$ into the Carnot refrigerator. In pictures...

Applying the first law to the fantabulous machine, $$|W'| = |Q_2'|-|Q_1'|.$$

For the Carnot refrigerator, $$-|W| = |Q_1|-|Q_2|\Rightarrow |W| = |Q_2|-|Q_1|$$

Since the output of $M'$ is coupled to the input of $M$, we have $|W'|=|W|$, so... $$|Q_2|-|Q_1|=|Q_2'|-|Q_1'|,$$

$$\Rightarrow |Q_2|-|Q_2'|= |Q_1|-|Q_1'|.$$

The efficiency equation, with $W'=W$ forces us to believe that... $$|Q_2| > |Q_2'|,$$

$$ \Rightarrow |Q_1| > |Q_1'|$$

Now, trouble comes because, the effect of the composite machine $M$ coupled to $M'$ is that...

  • No net work has been done: $W'=-W$.
  • An amount of heat, $|Q_1|-|Q_1'| > 0$ has been extracted from a cold reservoir, and
  • An equal amount of heat, $|Q_2|-|Q_2'|>0$ has been delivered to the hot reservoir.

This violates the Clausius statement of the 2nd law. So, if we believe that, it would appear that our fantabulous engine must be *toast*, and in fact, no engine with a greater efficiency than a Carnot engine (with efficiency $\eta_C$ is possible.

We can us this same kind of reasoning to show that

  • Any irreversible heat engine must have a lower efficiency than the Carnot engine, and
  • Any reversible heat engine must have at best the same efficiency as the Carnot engine.

How do we know if a particular cycle is reversible or not?

Wouldn't it be nice if we had a variable that could just tell us??

How about a state variable instead of just any ol' variable???

[We shall see that the entropy, $S$, will fulfill all these desires!]

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