Irreversible processes

The laws of mechanics are invariant under time reversal:



And yet, in our macroscopic world, we can usually tell when a video is running forwards or backwards...

Examples

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 Clausius, such a fantabulous engine would be impossible. And more generaly, engine with a greater efficiency than a Carnot engine (with efficiency $\eta_C$ is possible.

We can use 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?

Couldn't we please have a variable that could just tell us?? And...how about a state variable instead of just any ol' variable???

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

Image credit

CircaSassy