We now come to discuss two statements of the second law of thermodynamics which we shall see gives us a upper bound to the efficiency of any imaginable engine used to do work.
We start with a few experimental facts that were known by Carnot who did the original work on this topic.
Facts
1. We can get work from an engine if it is working between two heat sources of different temperatures. Clearly, we would like our engine to leave our heat sources unchanged and we would like the engine to return to its original state after a while in order to begin another cycle. These two requirements can be satisfied if our engine uses processes that are reversible.
2.It is possible for no work to be done when heat moves from a hot body to a cold body while the system is returning to equilibrium. Therefore any return to thermal equilibrium where no work has be done must counted as loss. We therefore want the engine to work between heat sources that are close in temperature as possible. This way we can reduce any in-efficiency as much as possible because, remember that inefficiency is defined as \( \nu = \frac{W}{Q_1}\) where W is the work done and \( Q_1 \) is the heat got from the heat source as opposed to the sink(reservoir at the colder temperature). Now one obvious way of getting the best efficiency namely, 1 is to make \( W= Q_1 \) but is this possible? The answer is a resounding no. This leads us to two statements that are the second law of thermodynamics which are actually equivalent. The second law of thermodynamics says that the efficiency of an engine can never be one and that in fact there is an upper bound that is less than 1.
Second Law of Thermodynamics
1. It is impossible to construct an engine whose sole purpose is to extract heat from a source and convert all of it into work.
2. It is impossible to construct a device that operating in a cycle has the sole purpose of transferring heat from a cold reservoir to a hot reservoir with no other effect.
As will later be proved these statements are indeed equivalent but they both infer the existence of an amount of heat, \( Q_2 \) which leaks from the engine to the cold reservoir so that not all the original heat \( Q_1\) is converted into work. Therefore the amount of work done can never equal to \(Q_1 \) but must be equal to \( Q_1 -Q_2 \). Plugging this into the equation for efficiency we get that
\( \nu = 1 - \frac{Q_2}{Q_1}\).
Let us now prove that there exists an upper-bound \( \nu_c \). This will be proved by contradiction. We shall assume that the upper-bound can be violated and show that this would involve a violation of the second law.
Theorem: There exists an efficient engine,C, with efficiency,\( \nu_c \). This will extract heat ,\( Q_{c_1} \) from the heat source and leak heat \( Q_{c_2} \)to the cold reservoir. This engine C has the property that \( \nu_c \) is the upper-bound for the efficiency of any system.
Proof
Let us assume there is a more efficient engine,E, than our brilliant engine and let us further assume that they both do the same work difference. So we have that
\(\frac{W}{Q_1}=\nu > \nu_c \) therefore it must be that \( Q_{c_1}> Q_1\) since we are assuming that \( W=W_c \). Now comes the key idea or trick.
Let us imagine a composite system composite system composed of these two engines, E and C except that the engine that is working at efficiency \( \nu_c\), C, is working backwards (it is a refrigerator) and the hypothetical engine , E,that has more efficiency than our upper-bound is doing work that is being put as an input to run the refrigerator.
Recall, that an engine working backwards requires us to put in work so that energy from a cold reservoir can be put into a hotter reservoir. In other words,the engine that is violating our bound on efficiency is running the refrigerator.
To make everything very explicit, E is extracting \( Q_1 \) from a hot reservoir and doing work, W, and leaking \( Q_2\) into the cold reservoir. Engine C is using this same work W, to extract \( Q_{c_2}\) from the cold reservoir and placing into into \( Q_{c_1}\) into the hot reservoir.
This means that the composite system is extracting from the cold reservoir positive \(Q_{c_1} -W - (Q_1- W)= Q_{c_1} -Q_1 \) and placing the same amount of work into the hot reservoir and the reservoirs are unchanged by the amount of heat added or extracted and stay at the same temperature. But this entails a violation of the second formulation of the second law of thermodynamics. This device extracts energy from the cold reservoir and put all of it into the hot reservoir with no work done.
So what has gone wrong? Well, we got that \( Q_{c_1}> Q_1\) which followed from the fact that E was more efficient than C. So this assumption has to be wrong.What we have at this stage is that there is an upper bound to the efficiency that must be less than one. The engine that produces this upper-bound famously goes by the name Carnot's engine and the cycle that the engine goes through in-order to produce this efficiency is called a Carnot cycle.
Note that we have not mentioned anything about entropy or disorder. We shall see that as a consequence of the formulation of the second law we shall prove the existence of a state function whose change in values from one equilibrium state to another is never negative. This we shall call entropy.
The second statement of the second law is what Clausius had in mind when he discovered or defined the notion of entropy. It must be emphasized that entropy will then be given the notion of disorder but we can not do that now as we do not have yet the concept of micro-states or atoms as was true in Clausius' time and also by the fact that thermal physics only cares about properties of macro-systems in equilibrium and says nothing about micro-states.
We start with a few experimental facts that were known by Carnot who did the original work on this topic.
Facts
1. We can get work from an engine if it is working between two heat sources of different temperatures. Clearly, we would like our engine to leave our heat sources unchanged and we would like the engine to return to its original state after a while in order to begin another cycle. These two requirements can be satisfied if our engine uses processes that are reversible.
2.It is possible for no work to be done when heat moves from a hot body to a cold body while the system is returning to equilibrium. Therefore any return to thermal equilibrium where no work has be done must counted as loss. We therefore want the engine to work between heat sources that are close in temperature as possible. This way we can reduce any in-efficiency as much as possible because, remember that inefficiency is defined as \( \nu = \frac{W}{Q_1}\) where W is the work done and \( Q_1 \) is the heat got from the heat source as opposed to the sink(reservoir at the colder temperature). Now one obvious way of getting the best efficiency namely, 1 is to make \( W= Q_1 \) but is this possible? The answer is a resounding no. This leads us to two statements that are the second law of thermodynamics which are actually equivalent. The second law of thermodynamics says that the efficiency of an engine can never be one and that in fact there is an upper bound that is less than 1.
Second Law of Thermodynamics
1. It is impossible to construct an engine whose sole purpose is to extract heat from a source and convert all of it into work.
2. It is impossible to construct a device that operating in a cycle has the sole purpose of transferring heat from a cold reservoir to a hot reservoir with no other effect.
As will later be proved these statements are indeed equivalent but they both infer the existence of an amount of heat, \( Q_2 \) which leaks from the engine to the cold reservoir so that not all the original heat \( Q_1\) is converted into work. Therefore the amount of work done can never equal to \(Q_1 \) but must be equal to \( Q_1 -Q_2 \). Plugging this into the equation for efficiency we get that
\( \nu = 1 - \frac{Q_2}{Q_1}\).
Let us now prove that there exists an upper-bound \( \nu_c \). This will be proved by contradiction. We shall assume that the upper-bound can be violated and show that this would involve a violation of the second law.
Theorem: There exists an efficient engine,C, with efficiency,\( \nu_c \). This will extract heat ,\( Q_{c_1} \) from the heat source and leak heat \( Q_{c_2} \)to the cold reservoir. This engine C has the property that \( \nu_c \) is the upper-bound for the efficiency of any system.
Proof
Let us assume there is a more efficient engine,E, than our brilliant engine and let us further assume that they both do the same work difference. So we have that
\(\frac{W}{Q_1}=\nu > \nu_c \) therefore it must be that \( Q_{c_1}> Q_1\) since we are assuming that \( W=W_c \). Now comes the key idea or trick.
Let us imagine a composite system composite system composed of these two engines, E and C except that the engine that is working at efficiency \( \nu_c\), C, is working backwards (it is a refrigerator) and the hypothetical engine , E,that has more efficiency than our upper-bound is doing work that is being put as an input to run the refrigerator.
Recall, that an engine working backwards requires us to put in work so that energy from a cold reservoir can be put into a hotter reservoir. In other words,the engine that is violating our bound on efficiency is running the refrigerator.
To make everything very explicit, E is extracting \( Q_1 \) from a hot reservoir and doing work, W, and leaking \( Q_2\) into the cold reservoir. Engine C is using this same work W, to extract \( Q_{c_2}\) from the cold reservoir and placing into into \( Q_{c_1}\) into the hot reservoir.
This means that the composite system is extracting from the cold reservoir positive \(Q_{c_1} -W - (Q_1- W)= Q_{c_1} -Q_1 \) and placing the same amount of work into the hot reservoir and the reservoirs are unchanged by the amount of heat added or extracted and stay at the same temperature. But this entails a violation of the second formulation of the second law of thermodynamics. This device extracts energy from the cold reservoir and put all of it into the hot reservoir with no work done.
So what has gone wrong? Well, we got that \( Q_{c_1}> Q_1\) which followed from the fact that E was more efficient than C. So this assumption has to be wrong.What we have at this stage is that there is an upper bound to the efficiency that must be less than one. The engine that produces this upper-bound famously goes by the name Carnot's engine and the cycle that the engine goes through in-order to produce this efficiency is called a Carnot cycle.
Note that we have not mentioned anything about entropy or disorder. We shall see that as a consequence of the formulation of the second law we shall prove the existence of a state function whose change in values from one equilibrium state to another is never negative. This we shall call entropy.
The second statement of the second law is what Clausius had in mind when he discovered or defined the notion of entropy. It must be emphasized that entropy will then be given the notion of disorder but we can not do that now as we do not have yet the concept of micro-states or atoms as was true in Clausius' time and also by the fact that thermal physics only cares about properties of macro-systems in equilibrium and says nothing about micro-states.
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