We know prove the equivalence of the two statements of the second law of thermodynamics. Recall that they are the following:
1. No engine or process exists such that its sole purpose is to turn all the heat it extracts from a reservoir into work. It must release some energy into a colder reservoir.
2. No engine or process exists whole sole purpose is to transfer heat from a cold reservoir to a hot reservoir with no other effects.
We shall prove their equivalence but assuming the violation of one of them and showing that implies the violation of the second.
Proof
Let us assume the violation of the first statement. This means that the engine E get Q from a reservoir and produces work, W, such that Q=W. Now imagine a composite system with this engine E and a second engine C. Let the work done by engine E be put in the engine C so that C can get heat \( Q_2\) from a cold reservoir and supply \(Q_2 +W \) to the hot reservoir. So in terms of the composite system (E and C) we got Q from the hot reservoir and used engine C to put back \( Q_2 +W = Q_2 +Q_1\). Therefore we transferred \( Q_2\) from cold reservoir to the hot reservoir with no work supplied i.e from the view of the composite system, the process consisting of the engine E and engine C, the second statement of the second law was violated.
For the second part of the proof, we assume that the second statement was violated and the goal will be to show that this implies the violation of the first statement. This time assume that engine E takes \( Q_2\) from a cold reservoir and places it all in the hot reservoir with any work supplied and with no other effects. Now imagine another engine C getting \( Q_1\) from the hot reservoir doing work, W ,and leaking \( Q_2\) into the cold reservoir. Thus the amount of work done by engine C is \( W = Q_1 -Q_2 \). Again let us look at these two engines as one system. From this point of view . Thus the total amount of heat got from the hot reservoir was \( Q_1 -Q_2\) and we used all of it as work. This violates the first statement.
Hence the two statements are equivalent.
Note: The first statement does not forbid us from putting work, W, into a system and the engine turning all of this into heat, Q, which is then dumped into a reservoir. This process is what will be used to arrive at the concept of entropy.
1. No engine or process exists such that its sole purpose is to turn all the heat it extracts from a reservoir into work. It must release some energy into a colder reservoir.
2. No engine or process exists whole sole purpose is to transfer heat from a cold reservoir to a hot reservoir with no other effects.
We shall prove their equivalence but assuming the violation of one of them and showing that implies the violation of the second.
Proof
Let us assume the violation of the first statement. This means that the engine E get Q from a reservoir and produces work, W, such that Q=W. Now imagine a composite system with this engine E and a second engine C. Let the work done by engine E be put in the engine C so that C can get heat \( Q_2\) from a cold reservoir and supply \(Q_2 +W \) to the hot reservoir. So in terms of the composite system (E and C) we got Q from the hot reservoir and used engine C to put back \( Q_2 +W = Q_2 +Q_1\). Therefore we transferred \( Q_2\) from cold reservoir to the hot reservoir with no work supplied i.e from the view of the composite system, the process consisting of the engine E and engine C, the second statement of the second law was violated.
For the second part of the proof, we assume that the second statement was violated and the goal will be to show that this implies the violation of the first statement. This time assume that engine E takes \( Q_2\) from a cold reservoir and places it all in the hot reservoir with any work supplied and with no other effects. Now imagine another engine C getting \( Q_1\) from the hot reservoir doing work, W ,and leaking \( Q_2\) into the cold reservoir. Thus the amount of work done by engine C is \( W = Q_1 -Q_2 \). Again let us look at these two engines as one system. From this point of view . Thus the total amount of heat got from the hot reservoir was \( Q_1 -Q_2\) and we used all of it as work. This violates the first statement.
Hence the two statements are equivalent.
Note: The first statement does not forbid us from putting work, W, into a system and the engine turning all of this into heat, Q, which is then dumped into a reservoir. This process is what will be used to arrive at the concept of entropy.
No comments:
Post a Comment