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What is the Clausius Statement of the Second Law of Thermodynamics?
Refrigerator: A mechanical device that transfer heat from a colder to a hotter place is called a refrigerator and the working substance of a refrigerator is called a refrigerant.
Suppose, a cold body, say an ice-box is at temperature of T2 [Fig.]. The surrounding temperature T1 in most cases is greater than T2. So the temperature of the cold body begins to increase due to receiving heat from the surroundings. Now if any mechanical device removes heat at the same rate at which the cold body receives heat, the temperature of the body will remain constant i.e., the body will remain in the same cold state. This mechanical arrangement is called a refrigerator, denoted by R in Fig.
Generally, with the help of some external work, the refrigerator transfers heat from the lower temperature T2 to the higher temperature T1 of the surroundings. Our house-hold refrigerator is a familiar example of this machine. Like a heat engine, a refrigerator also works in a cyclic process. At the end of each cycle, the working substance returns to its initial state, and then the next cycle starts.
Coefficient of performance of a refrigerator: The aim of a refrigerator is to extract heat from a cold body at the expense of some external mechanical work. The external supplied work is known as the input and the heat extracted from the cold body is called the output.
Suppose, in each complete cycle,
heat received by the refrigerator from the colder body at temperature T2 = Q2 [Fig.];
heat delivered by the refrigerator to the surroundings at higher temperature T1 = Q1;
external work done = W
From the principle of conservation of energy, we can write,
W + Q2 = Q1
or, W = Q1 – Q2
From the definition of coefficient of performance (e) of a refrigerator we have,
e \(=\frac{\text { output }}{\text { input }}\) = \(\frac{Q_2}{W}\) = \(\frac{Q_2}{Q_1-Q_2}\) …… (2)
Ideal refrigerator: Obviously, less the amount of external work supplied to a refrigerator to run it, the better is its performance. If any refrigerator can transfer heat from low temperature to a higer temperature without any help of external work [Fig.], then it is called an ideal refrigerator. For example, if any household refrigerator could run without any assistance of electricity, then that would be an ideal refrigerator. For this ideal refrigerator,
W = 0, i.e., Q1 – Q2 = 0 or, Q1 = Q2
So, coefficient of performance of an ideal refrigerator,
e = \(\frac{Q_2}{W}\) = \(\frac{Q_2}{0}\) → ∞
i.e., the coefficient of performance of an ideal refrigerator is infinite.
Clausius statement of the second law of thermodynamics: The statement, given in section 1.12 is as follows.
No self-acting machine can transfer heat from a lower to a higher temperature. On the basis of the discussions about refrigerators, this statement can be expressed in the form of an easy alternative: An Ideal refrigerator does not exist in nature.
Discussions:
i) On comparing Fig’s, it is apparent that the working principles of a heat engine and a refrigerator are exactly opposite to each other. But it does not mean that these two machines are used for opposite purposes. heat engine is a machine for conversion of heat into work, but a refrigerator is not a machine for conversion of work into heat. On the other hand, the refrigerator is a machine which transfers heat from a low temperature to a higher temperature, but heat engines are never used to transfer heat from a higher to a lower temperature.
ii) Again, on comparing Fig’s, it is found that if it is possible to conduct the different operations of a heat engine (i.e., heat intake from higher temperature, transformation of heat into work and rejection of heat at low temperature) in the reverse direction, then it will act as a refrigerator. Obviously, the necessary condition for this is that each operation should have to be a reversible process, i.e., the engine should have to be a reversible heat engine. In that case, the refrigerator run in the opposite direction will be a reversible refrigerator.
iii) A reversible process is nothing but an ideal process. In nature this process does not exist-all real processes are irreversible. So, actually, there is no opportunity to use a heat engine as a refrigerator, and vice versa.