There are two important statements of second law.
They are:
1. Kelvin Planck’s Statement,
2. Clausius Statement,
1. Kelvin Planck’s Statement:
“It is impossible for a heat engine to produce network in a cycle, if it exchanges heat with a single reservoir” or
“Heat engine cannot produce work output, if it exchanges heat with a single reservoir”.
By practical experience, it is noted that total heat supplied cannot be converted into useful work W i.e., Q1 ≠ W. A heat engine can never be 100% efficient. But W < Q1 therefore there has to be heat rejection i.e., Q2 > 0.
Therefore for a heat engine to produce network in a cycle, it has to exchange heat with two reservoirs.
But if, Q2 = 0, then Q1 = W or ηTh = 100%, the heat engine will produce network in a complete cycle by exchanging heat with only one reservoir, thus it violates Kelvin Planck’s statement. Such a heat engine is called a Perpetual Motion Machine of the second kind. (PMM-2) Ref. Fig. 3.8. A PMM-2 is impossible and it is just a conceptual engine. All the attempts made so far to make such a machine have failed. Thus they show the validity of Kelvin Planck’s, statement.
2. Clausius Statement:
We know that, heat always flow from a hot body to a cold body. The reverse process never occurs by itself.
Clausius statement is given as, “It is impossible to construct a device, which operating in a cycle will produce no effect other than transfer of heat from a low temperature body to a high temperature body”. Or
“Heat cannot flow by itself from a cold body to a hot body, in order to achieve this some work must be expended”. For example- Refrigerator—in this case heat is removed from the cold body A and transferred to the atmosphere but by consuming work input W.
Equivalence of Kelvin Planck and Clausius Statements:
The Kelvin Plank and Clausius statement as follow:
(i) Kelvin Planck’s Statement:
“Far a heat engine to produce Wnet it has to exchange heat with two reservoirs”.
(ii) Clausius Statement:
“Heat cannot flow by itself from a cold body to a hot body. In order to achieve this some work must be expended”.
At first sight these two statements will appear to be two different statements and are unconnected. But it can be easily shown that, these are the two parallel statements of second law and are equivalent in all respects.
The equivalence of the two statements will be proved if we prove that violation of one statement results into the violation of the other.
(a) Violation of Clausius Statement Leads into the Violation of Kelvin Planck’s Statement:
Consider a cyclic heat pump P, which transfers heat from a LTR at t2 to HTR at t1 without consuming work input (i.e., W = 0), thus violating Clausius statement.
Now, let us assume a cyclic heat engine E which also operates between the same two reservoirs at t1 and t2 respectively. The rate of working of the heat engine is such that it draws an amount of heat Q1 from HTR equal to that discharged by the heat pump. Then the HTR may be eliminated and heat Q1 discharged by the heat pump may be directly fed to the heat engine. So, the heat pump P and heat engine E acting together will form a heat engine, operating in cycles and produce network by exchanging heat with one reservoir. This violates the Kelvin Planck’s statement.
(b) Violation of Kelvin Planck Statement Leads into the Violation of Clausius Statement:
Let us consider a PMM-2 (E), which produces Wnet in a cycle by exchanging heat with only one reservoir at t1 and thus violates Kelvin Planck’s statement.
Now, let us assume a cycle heat pump P extracting heat Q2 from LTR at t2 and supplying heat to HTR at t1 by consuming work W equal to that PMM-2 supplies in a complete cycle. So, E and P together will form a heat pump and producing the complete effect of transferring heat from LTR to HTR, without any external aid.
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