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(1)We have seen that there are two word statements of the second law of thermodynamics. Both statements are just statements of universal observation. That is, no one has ever observed a violation of these statements and no one expects that a violation ever will be observed. As we did with the first law, we elevate these statements to the status of a "law" and assume that they are universally valid. Then we derive the consequences of this law, which can be checked out by experiment. So far, within the domain of its validity, no one has ever observed a violation of the second law, and its consequences are consistent with experimental observation.
Recall that the word statements of the second law are:
1. Heat does not move spontaneously from a cold body to a hot body with no other effect.The first statement is pretty obvious. Heat flows from hot bodies to cold bodies not the other way around. It would be very startling if all of the heat in your pencil spontaneously flowed into the eraser end so that the eraser melted and caught fire. The second statement is probably not so obvious unless you are an engineer, but it is one of the reasons that automobiles have radiators. In practice it means that you can't convert heat into work without dissipating some of the heat into a heat bath at a lower temperature than your heat source.2. You can not convert heat quantitatively into work with no other effect.
We demonstrated on the previous page that both of these statements lead to the conclusion that no heat engine can have an efficiency greater than a Carnot cycle.
(Recall that the efficiency of a cycle operating as a heat engine between two heat baths, one at an upper temperature, TU, and the other at a lower temperature, TL, can be written in several ways. For the Carnot cycle we can write,
Likewise, for some other cycle, which we will indicate by putting a prime on the heats and work - still operating between TU and TL - we can write,
We are now considering the cycles to be operating as heat engines (as opposed to refrigerators, or heat pumps) - going around the cycle clockwise, so that w and w' are negative, qU and q'U are positive, and qL and q'L are negative. (We will need to know the signs of these quantities in order to carry out the algebra of inequality signs.). (2)
Since we know from the second law that no cycle can have an efficiency greater than a Carnot cycle, we can write,
(We might ask what circumstances might cause the efficiency of an engine to decrease. The most obvious answer is that irreversibility, for example, friction, would cause the efficiency to go down. If there is any irreversibility in the cycle its efficiency will be degraded.)(3)
Using our various expressions for the efficiency, Equations 1 and 2, we can rewrite equation 3 as
or,, (4)
Multiply both sides of Equation 5 by − 1, which reverses the inequality,. (5)
In Equation 6 we have purposely associated the minus sign with the q'L which makes both the numerators and denominators of both sides positive. With all quantities in this equation positive we can cross-multiply at will without worrying about the direction of the inequality. Cross-multiplying we get. (6)
or, (7)
The equal sign holds when the cycle is fully reversible and "greater than" sign holds if there is any irreversibility in the cycle. In the case where the cycle is fully reversibly, then,. (8)
This latter equation makes it look like,. (9)
(Recall that two of the "legs" of the cycle were adiabatic so that there is no reversible heat on those legs.) We can generalize Equation 10 to an arbitrary closed path in p, V space by noting that we can fill an arbitrary closed path with a large number of small Carnot cycles (an infinite number in the limit of small cycles).. (10)
When we sum over all the cycles inside the heavy line the inside "lines" of the cycles (for example the blue filled Carnot cycle in the center) will cancel because each inside line is traversed once clockwise and once counterclockwise. Therefore, only the outside lines (the heavy lines) remain. If we index the (small) cycles making up the complete cycle, by an index, k, where k ranges over all the cells inside the heavy line, this gives,
In the limit where we approximate our arbitrary closed path by making the isotherms and adiabatic lines closer and closer together to create an infinite number of cycles, the summation in Equation 11 becomes,. (11)
The equal sign holds when the entire path is reversibly,. (12)
This equation says that the integral of reversible heat over temperature around a closed path is zero. Equation 13 is independent of the shape of the path - as long as it is a closed loop - which implies that,. (13)
is the differential of a state function. We will call this state function entropy and give it the symbol, S. Equation 14 becomes(14)
Combining Equations (12), (13), and (15) we get,(15)
Since the closed path is arbitrary, Equation (16) must be valid for any possible closed path. The only way this can be true is if the equation is true in its differential form. That is,(16)
which can be rewritten as simply(17)
or, the best way to remember it,(18)
with the understanding that the equal sign holds when the process is reversible and the greater-than sign holds when the process is irreversible.(19)
Equation (19) is the second law of thermodynamics in equation form.
Equation (18) would work just as well, but most of us would prefer (19)
because it is in the form of a definition of dS.
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