(1)We have seen that the Carnot cycle, a reversible heat engine with an ideal gas working fluid, has an efficiency,
Is this the best we can do? Does there exist another cycle which has an efficiency greater than the efficiency of the Carnot cycle? We will now show that if such a cycle exists, with an efficiency greater than a Carnot cycle, then both of our word statements of the second law will be violated. That is, if a cycle (a heat engine working between the two heat reservoirs at TU and TL) exists with an efficiency greater than the efficiency of a Carnot cycle, then we can see heat spontaneously moving from a low temperature to a higher temperature with no other effect, and we will be able to convert heat quantitatively into work with no other effect.
Let us call the cycle with higher efficiency the "better" cycle and just say about it that it has an efficiency, e', which is greater than e.
We will set up the two cycles such that the "better" cycle drives the
Carnot cycle. That is, the "better" cycle will be operated as a heat engine
and it will drive the Carnot cycle which then operates as a heat pump.
We know that ΔU = 0 for each cycle independently
and for the sum of the two cycles. Then,
or(2)
We can adjust each cycle to alter the various q's and w's by changing where the adiabatic curves intersect the isotherms. Changing the positions of the adiabatic lines changes the area enclosed by the cycle and also changes the heats absorbed and released on the isothermal lines.(3)
As we have said, in both of our two experiments the "better" cycle runs
clockwise - as an engine, and the Carnot cycle runs counterclockwise as
a heat pump. The engine drives the heat pump. Some or all of the work produced
by the "better" cycle is used to run the (Carnot cycle) heat pump.
Experiment 1
Adjust the parameters of the two cycles such that w' + w = 0. That is, the work produced by the "better" cycle is entirely used up to drive the Carnot heat pump. This also means that there is no net work either on the system or on the surroundings. Then,
w = − w' ,which we can use in the expressions for the efficiencies,
In Equation 4b use the fact that w = − w' on the left-hand-side. For the right-hand-side, use the fact that qU is negative and move the negative side to the denominator so that both numerator and denominator are positive on the right. Then,(4a, b)
In Equation (4c) w is positive so we can divide both sides by w without affecting the inequality.(4a, b, c)
since both q'U and − qU are positive. Then,(5,a b)
If you were to stand back and take a look at the overall effect of this experiment, including both cycles, you would see that, according to Equation (6), heat is being released into the heat bath at the upper temperature. This is heat that must have been absorbed from the heat bath at the lower temperature because of the first law and as indicated in Equation (3). However, we also know that there is no net work being done on the two systems. The work of the "better" cycle is entirely used up driving the Carnot cycle. It appears, looking at the overall effect, that heat is spontaneously disappearing from the lower temperature heat bath and appearing in the upper temperature heat bath and there is no other effect. This violates our first word statement of the second law.(6)
Experiment 2
This time adjust the parameters of the experiment so that heat given to the lower temperature heat bath by the "better" cycle is exactly balanced by the heat absorbed at the lower temperature by the Carnot cycle. That is, set
This time we find that(7)
Invert (8d) to get(8a, b, c, d)
Recall that, by construction of Experiment 2, we have from Equation (7),(9a,b)
Equation (9b) becomes(10)
The net effect is that heat has been absorbed at the upper temperature and work has been done on the surroundings, but there was no net heat transferred to the lower temperature heat bath. This violates the second word statement of the second law. If we regard the two cycles as one large heat engine, then that engine has no "radiator." It would be nice if such an arrangement would work, and from time to time people propose schemes which are designed to make it work, but it appears that nature does not allow it.(11a, b, c)
Conclusion
The initial assumption that we can find another cycle, with an efficiency, e', better than the Carnot cycle efficiency, leads to a contradiction of both word statements of the second law. We are forced to conclude that no cycle can have an efficiency greater than the efficiency of a Carnot cycle. That is
Equation (12) is useful and has a number of interesting and important consequences, but it does not yet provide us with the most significant result of the second law. On the next page we will use Equation (12) to define a new state function, S, called "entropy," and write the second law in a mathematical form which will give us enormous new calculating power.. (12)
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