Thermodynamics III: second law

The first law of thermodynamics says that heat is a form of energy. There is a lot of heat about!  For instance, the amount of heat energy it would take to change the temperature of the world’s oceans by one degree is about \(6 \times 10^{24}\) joules.  That is four orders of magnitude greater than the world’s annual energy consumption!  So, if we could somehow how to figure out how to extract one degree’s worth of heat energy from the oceans, we could power the world for ten thousand years! 

Unfortunately, the second law of thermodynamics sets limits on the effectiveness of this kind of process,   Here is one way of expressing it: there is no way to convert heat energy from a single reservoir (at uniform temperature) into mechanical work, while leaving everything else unchanged.

If there were such a technique, we could use the resulting mechanical energy (by friction or otherwise) to heat something else – even something hotter than the original reservoir.  Effectively, we could make heat flow “uphill”, from a cooler body to a hotter body.  Conversely, if we could make heat flow uphill we could exploit the resulting temperature difference to extract mechanical work via the Carnot cycle (to be described in a moment). Thus an equivalent form of the second law is, There is no transformation whose end result is solely to transfer heat from a cooler to a hotter body.

How does the Carnot cycle work? It exploits the distinction between isothermal and adiabatic processes, which we analyzed for an ideal gas in last week’s post.  An isothermal transformation is one at constant temperature; an adiabatic transformation is one in which no heat is exchanged with the environment.   On the \(p,V\)-diagram, adiabatics are in general “steeper” than isothermals.  A Carnot cycle is a quadrilateral made up of two isothermal and two adiabatic arcs, as shown below.

p-V diagram of the Carnot cycle

The Carnot cycle, from

During the isothermal (12) arc, the system is in contact with a high temperature source, and is expanded, drawing heat from the source.  During the adiabatic (23)  arc, further expansion takes place, accompanied by a decrease in temperature because the system is thermally insulated.  During the isothermal (23) arc, the system is in contact with a low temperature sink and is compressed, releasing heat to the sink.  Finally, an adiabatic compression (41) heats the system and returns it to its starting point.   The overall effect of the cycle is to extract a certain amount of heat energy from the source, some of which is converted to work (the area of the cycle, \(\int pdV\) ) and some of which is deposited in the sink.  The efficiency of the Carnot cycle is the fraction of the heat energy extracted from the source which is converted into work.

Note that the Carnot cycle is reversible.   Running it back the other way, we obtain a heat pump: a device which absorbs mechanical energy and uses it to move heat “uphill”, from a low-temperature to a high-temperature reservoir.

Theorem: No heat engine can be more efficient than a Carnot cycle (operating between the same temperatures), and any reversible engine must have the exact same efficiency as a Carnot engine.

Proof.  Otherwise, by stringing together the hypothetical engine with a reversed Carnot engine, we could violate the second law of thermodynamics.

The efficiency of a reversible heat engine operating between temperatures \(t_1\) and \(t_2\) is therefore a function only of these temperatures, which it is convenient to write \(1-f(t_1,t_2)\); thus \(f\) is the ratio of the heat energy lost to the sink at temperature \(t_2\) to the heat energy absorbed from the source at temperature \(t_1\).  By stringing together reversible engines as in the previous proof, we see that \(f\) is a cocycle: that is,

\[ f(t_1,t_2)f(t_2,t_3)=f(t_1,t_3). \]

There is no cohomology involved here, so there is a function \(\theta(t)\), determined up to a multiplicative constant, of which \(f\) is the coboundary: \( f(t_1,t_2) = \theta(t_2)/\theta(t_1) \).  This \(\theta\) is called the absolute thermodynamic temperature: to put it another way, we agree to \emph{define} the temperature in such a way that \(\theta(t)=t\) (the multiplicative constant is of course just the “size” of one degree, conventionally fixed by having 100 degrees between the freezing and boiling points of water).  With this understanding, the efficiency of a Carnot engine (and therefore the upper bound for the efficiency of any engine) operating between temperatures \(t_1\) and \(t_2\) is

\[ 1 – \frac{t_2}{t_1} = \frac{t_1-t_2}{t_1}. \]

Remark: Thinking about a Carnot cycle where the system is an ideal gas, one easily checks that the thermodynamic temperature above is equal to the temperature measured by an ideal gas thermometer.



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