The efficiency of a Carnot heat engine is a universal property that directly follows from the laws of thermodynamics, and the beautiful reasoning behind it was a key triumph of physics in the 19th century. As Feynman wrote, at the beginning of his Section 44-4:

The efficiency of an ideal engine…to find this universal law…is one of the very beautiful pieces of reasoning in physics

That section of the Feynman Lectures is a must-read, to appreciate the beauty of this reasoning. However, I found that section pretty challenging and had to re-read it many times before I finally got it. So I would like to provide a simplified and more accessible explanation that I hope retains the essence.

The main question is: what is the efficiency of a Carnot heat engine? We know (see my prior post) that the Second Law of Thermodynamics tells us that for any two given temperatures, all Carnot engines connected to those two temperatures have the same efficiency. This means that if you consider the heat flow between the Carnot engine and the first temperature reservoir, and compare that to the heat flow between the Carnot engine and the second temperature reservoir, the ratio of the two heat flows must only depend on the two temperatures and not on any other details of the Carnot engine. And so the question is: how exactly does the efficiency depend on the two temperatures?

To answer this, the main idea is to define a new, fundamental type of temperature scale, the thermodynamic temperature scale, that is universally defined and independent of any specific material or measurement device. Contrast this with the mercury temperature scale that is more familiar, where we measure the expansion of liquid mercury and use that as the temperature scale, which clearly is very much dependent on the specific properties of mercury.

Here’s what we do: start with a reservoir at some specific reference temperature \(T_R\) that we pick for convenience (and of course we would need to standardize this so that different organizations can agree on the temperature scale), and define this reference temperature to be “one degree” in our thermodynamic temperature scale. Then, take a new reservoir at a different temperature \(T_{new}\) and consider a Carnot engine between those two temperatures. Let the heat flows be \(Q_{new}\) from the new temperature \(T_{new}\) and \(Q_R\) from the reference temperature \(T_R=1\).

Then, the definition of the thermodynamic temperature of the new reservoir is very simple: it is defined to be \(T_{new} = \lvert Q_{new}/Q_R \rvert T_R\), which simplifies to \(\lvert Q_{new}/Q_R \rvert\). In other words, for any Carnot engine the ratio of thermodynamic temperatures is the same as the absolute value of the ratio of heat flows. So to measure the temperature of an unknown reservoir, we connect a Carnot engine between the unknown reservoir and the reference reservoir at \(T_R=1\), we measure the heat flows \(Q_{new}\) and \(Q_R\), and the unknown temperature is then the ratio \(\lvert Q_{new}/Q_R \rvert\).

BTW, we know that for any pair of temperatures, the higher temperature will have a higher absolute value of heat flow. This is due to the emperical observation that we can withdraw heat from a hotter temperature and partially convert it to work and deposit the rest of the heat into a cooler temperature; but it’s impossible to withdraw heat from a cooler temperature and partially convert it to work while depositing the rest of the heat into a hotter temperature. This means that we will always measure a lower thermodynamic temperature less than one degree when \(T_{new}\) is cooler than \(T_R\) and vice versa, which is good!

Therefore, using the thermodynamic temperature scale, the efficiency of a Carnot engine operating between a higher temperature \(T_1\) and a lower temperature \(T_2\) is

\[\begin{aligned} \eta(T_1,T_2) &= \frac{\vert Q_1 \vert - \vert Q_2 \vert}{\vert Q_1 \vert}\\ &= 1-\frac{T_2}{T_1} \end{aligned}\]

We would like this measurement methodology to yield consistent temperature values, irrespective of how we choose to designate the reference reservoir. This means that for a specific set of unknown reservoirs, if two experimenters each independently picks a reference reservoir and uses that to measure the temperatures of all the unknown reservoirs, they should obtain two sets of temperature values that are consistent up to a scaling factor, where the scaling factor is the ratio of the two reference reservoir temperatures.

To demonstrate this, let’s take three reservoirs at temperatures \(T\), \(T_{R1}\), and \(T_{R2}\), where \(T\) is an unknown temperature and \(T_{R1}\) and \(T_{R2}\) are two different choices of reference temperatures. Consider these two cases:

  • Case 1: a single Carnot engine between \(T\) and \(T_{R1}\) with respective heat flows \(Q\) and \(Q_{R1}\)
  • Case 2: two Carnot engines \(A\) and \(B\) in series, with one engine between \(T\) and \(T_{R2}\) and another engine between \(T_{R2}\) and \(T_{R1}\).

For Case 2, we size the two Carnot engines to be consistent with Case 1, which means that engine \(A\) has heat flow \(Q\) from \(T\) and heat flow \(Q_{R2A}\) into \(T_{R2}\); and engine \(B\) has heat flow \(Q_{R2B}\) from \(T_{R2}\) and heat flow \(Q_{R1}\) from \(T_{R1}\). Then, since the ratio of heat flows for any Carnot engine between two temperatures is just the ratio of the thermodynamic temperatures, we must have

\[\lvert \frac{Q}{Q_{R2A}} \rvert = \frac{T}{T_{R2}}\]

and

\[\lvert \frac{Q_{R2B}}{Q_{R1}} \rvert = \frac{T_{R2}}{T_{R1}}\]

Therefore, multiplying the two equations together,

\[\lvert \frac{Q}{Q_{R1}} \frac{Q_{R2B}}{Q_{R2A}} \rvert = \frac{T}{T_{R1}}\]

But we know for the Carnot engine of Case 1 that \(\lvert Q/Q_{R1} \rvert = T/T_{R1}\). Therefore, we have \(\lvert Q_{R2B}/Q_{R2A}\rvert = 1\). Thus, given a Carnot engine of Case 1, we can always construct two Carnot engines \(A\) and \(B\) so that they are each sized to match the Carnot engine of Case 1 at the temperatures \(T\) and \(T_{R1}\), and so that they furthermore share a common value (up to a potential negative sign) of heat flow \(Q_{R2}\) at \(T_{R2}\).

Therefore, if Alice measures the unknown temperature using a reference reservoir at \(T_{R1}\) defined to be at 1 degree, the result in Alice’s temperature scale would be \(T_{Alice} = \lvert Q/Q_{R1} \rvert\). Similarly, Bob, using reference \(T_{R2}\) defined to be at 1 degree, would measure in Bob’s temperature scale \(T_{Bob} = \lvert Q/Q_{R2} \rvert\). And you could always convert between the two temperature scales using the scaling factor \(\lvert Q_{R1}/Q_{R2} \rvert\). Therefore, we have the consistency that we were looking for.

Finally, we won’t discuss it here, but the thermodynamic temperature scale can be shown to be the same as the temperature measured by an ideal-gas thermometer.

Key Insights

Let’s take a step back and think about what we just went through.

  • The Second Law of Thermodynamics led us to conclude that all Carnot engines operating between a given pair of temperatures must have the same efficiency. Therefore, given a pair of temperatures, there will be a unique absolute value of the ratio of heat flows for the temperature pair.
  • We are free to construct different temperature scales by mapping an existing temperature scale to a new scale, as long as the mapping is monotonically increasing. This lets us preserve our emperical observations that certain phenomena change in specific directions as temperature increases.
  • Therefore, we are able to define a thermodynamic temperature scale that is simply proportional to the absolute value of the ratio of heat flows when one of the temperatures is anchored at a reference temperature
  • Furthermore, any Carnot engine operating between two temperatures can be considered to be a combination of two Carnot engine in series, where the two Carnot engines share a third temperature/reservoir
  • Therefore, any experimenter can arbitrarily choose a reference reservoir with its temperature defined to be one degree, and the resultant temperature measurements will be consistent with any other experimenter’s choice of reference reservoir, up to a scaling constant.

Share or comment on Mastodon