Feynman Lectures. Derivation of efficiency law from logic

Question

Question is about Feynman Lectures. Lecture 44. Ch.44-4 The efficiency of an ideal engine (feynmanlectures.caltech.edu/I_44.html).

In this chapter Feynman derives the formula $\frac{Q_1}{T_1} = \frac{Q_2}{T_2}$ ¹⁾ by two ways: first, by using the properties of the ideal gas, and second, by logic. But trying to do it by the 2nd way (from the logic), he did not do it. Here is a quote:

Now we shall see how this universal law could also be obtained by logical argument, without knowing the properties of any specific substances, as follows. Suppose that we have three engines and three temperatures, let us say $T_1$, $T_2$, and $T_3$. Let one engine absorb heat $Q_1$ from the temperature $T_1$ and do a certain amount of work $W_{13}$, and let it deliver heat $Q_3$ to the temperature $T_3$ (Fig. 44–8). Let another engine run backwards between $T_2$ and $T_3$. Suppose that we let the second engine be of such a size that it will absorb the same heat $Q_3$, and deliver the heat $Q_2$. We will have to put a certain amount of work, $W_{32}$, into it—negative because the engine is running backwards. When the first machine goes through a cycle, it absorbs heat $Q_1$ and delivers $Q_3$ at the temperature $T_3$; then the second machine takes the same heat $Q_3$ out of the reservoir at the temperature $T_3$ and delivers it into the reservoir at temperature $T_2$. Therefore the net result of the two machines in tandem is to take the heat $Q_1$ from $T_1$ and deliver $Q_2$ at $T_2$. The two machines are thus equivalent to a third one, which absorbs $Q_1$ at $T_1$, does work $W_{12}$, and delivers heat $Q_2$ at $T_2$, because $W_{12}=W_{13}−W_{32}$, as one can immediately show from the first law, as follows:
$W_{13} - W_{32} = (Q_1 - Q_3) - (Q_2 - Q_3) = Q_1 - Q_2 = W_{12}.$
We can now obtain the laws which relate the efficiencies of the engines because there clearly must be some kind of relationship between the efficiencies of engines running between the temperatures $T_1$ and $T_3$, and between $T_2$ and $T_3$, and between $T_1$ and $T_2$.

First, I don't understand why W₃₂ is negative and why is calculated as Q₂−Q₃ and not Q₃−Q₂.
Mathematics must be correct in all cases, even if we don't know is the work done on or by the machine 2.

Second. How to calculate efficiencies? I try next:
$\eta _{\text{machine 1}}=\tfrac{W_{13}}{Q_1}$

$\eta _{\text{machine 2}}=\tfrac{W_{32}}{Q_2}$

$\eta _{\text{machine 3}}=\tfrac{W_{12}}{Q_1}$
$W_{12} = Q_1-Q_2$
$W_{13} = Q_1-Q_3$
$W_{32} = Q_2-Q_3$
$(W_{13} - W_{32}= W_{12})$
So we have 6 equations and 9 unknowns. The system is unsolvable. Besides the system does not contain temperature, and we cannot use $PV\sim T$, because it's the ideal gas equation (and in 2nd way we shouldn't use properties of any substance).

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¹⁾ Here $Q_1$ is the heat absorbed from the heat pad at temperature $T_1$, $Q_2$ is the heat delivered to the heat pad at temperature $T_2$.

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ADDITIONAL QUESTION
Feynman derives the formula $W_{13} - W_{32}= W_{12}$ from the assumption that the engine 3 will absorb heat $Q_1$ and deliver $Q_2$. How he knows that? Could it be that the engine 3 absorbs $Q_1$, but delivers unknown $Q_?$. Hence it follows that 3rd engine's work is $Q_1 - Q_?$, and so equation $W_{13} - W_{32}= W_{12}$ is wrong.

Answer 1

First: Feynman is just being a little incautious with the sign of W_32. He states that it's sign is negative, but then uses W_13 - W_32. Since W_13 is regarded as positive, that would result in an even larger result when you subtract them, but actually you want a smaller result-- he is saying that the work you get from the machine 1-->3-->2 is less than the work you'd get from just 1-->3. So what he has done is explicitly inserted the minus sign and then treated W_32 like a positive number. The more formally correct thing to do is say W_13 + W_32 = W_12, which is just the correct algebra of the subscripts. So there's nothing wrong in his argument, but the equations cannot be taken literally as written, you have to fix the signs manually. This is a common problem in physics, sign language can be ambiguous-- the classic example is when someone says "I lost -5 dollars", it's very unclear if they gained 5 dollars, or if they are simply doubly emphasizing the fact that it was a loss and not a gain, essentially by stressing that their change in money was -5 dollars, and that constitutes a "loss." We are usually more careful, but Feynman lapsed a little there.

Second: Actually, you have 7 equations there. It looks like there are 9 unknowns, but actually there are only 7, because you don't actually care what Q_1 is, so you could call it 1 and measure all other energies relative to that. So immediately you are down to 8 unknowns. Also, you don't care what the efficiencies actually are, you just want a relationship between them. That relationship will ultimately involve ratios between the efficiences, rather than the efficiences themselves. So that's again one less variable, and now you are down to 7 equation in 7 unknowns, and you can solve it. The actual efficiencies will depend on the actual T, but again you don't really care what the Ts are, you only care about the ratios between them.