Thermodynamic temperature scale and Carnot cycle

[chimay]

TL;DR Summary

This thread is about the concept of thermodynamic temperature scale and its derivation through thought experiments involving Carnot engines.

Hi all,
recently I started following the MIT course "Statistical Mechanics I: Statistical Mechanics Of Particles" by MIT (here).
In the second lesson Prof. Kardar introduces the concept of thermodynamic temperature analyzing the behavior of two Carnot engines that share a thermal reservour at temperatre $T_2$. The lecture notes can be found here.

My doubt is about Eq. I.21 and I.22 at pag. 10. It seems to me that from
$$1-\eta(T1,T2) = \frac{1-\eta(T1,T3)}{1-\eta(T2,T3)}$$
I can conclude that
$$1-\eta(T1,T2) = \frac{f(T_1)}{f(T_2)}$$
by taking $T_3$ as a reference temperature (note that $T_1>T_2>T_3$). In the previous equation $f$ is a generic function but since the definition of $T$ is arbitrary we can say
$$1-\eta(T1,T2) = \frac{T_1}{T_2}$$.

Now the problem is that from the definition of efficiency of a Carnot engine
$$1-\eta(T1,T2) = \frac{Q_2}{Q_1}$$

and equating the last two equations it results
$$\frac{Q_2}{Q_1}= \frac{T_1}{T_2}$$
that is clearly wrong (see Eq. I.22).

Where is my mistake here?

Thank you in advance for your reply.

 

[Hill]

$\frac {f(T_1)} {f(T_2)} < 1, T_1>T_2$. Thus, you cannot replace $f(T)$ with $T$, but you can replace it with $\frac 1 T$. Then, everything works just fine.

 

[chimay]

Hi Hill,
thank you for your reply. It makes sense now.