Carnot theorem

Carnot theorem

1. PANDIT DEENDAYAL PETROLUEM UNIVERSITY CARNOT’S THEOREM, COROLLARY OF CARNOT’S THEOREM AND THERMODYNAMIC TEMPERATURE SCALE - MIHIR LAD (17MSE007) 1

2. CARNOT’S THEOREM “ All the heat engines operating between a given constant temperature source and given constant temperature sink , NONE has higher efficiency than a reversible engine” 2

3. Let’s take two Heat engines: E A and E B . E A  any heat engine E B a REVERSIBLE heat engine. At constant source temperature T 1 and constant sink temperature T 2 . 3

4. Let’s assume  A   B Q A = Q B = Q 1 Since,  A   B W A  W B . Since E B is a reversible heat engine, the magnitudes of heat and work transfer quantities will remain same but their directions will be reversed. Since W A  W B , some part of W A , which is equal to W B ,may be fed to (E B ) rev. . 4

5. Since, Q A = Q B = Q 1, the heat discharged by (E B ) rev may be supplied to E A . So source may be eliminated. 5

6. The net result is that E A and (E B ) rev together constitute a heat engine which, operating in a cycle, produces net work (W A - W B ), while exchanging heat with a single heat reservoir at T 2 . This violates the Kelvin-Planck statement of second law. Hence, our assumption is wrong. So,  B   A 6

7. COROLLARY OF CARNOT’S THEOREM 7 “The efficiency of all the reversible heat engines operating between the same temperature levels is the same .” So, it is independent of the nature or amount of the working substance undergoing the cycle.

8. ABSOLUTE THERMODYNAMIC TEMPERATURE SCALE The efficiency of any heat engine cycle receiving heat Q 1 and rejecting heat Q 2 is given by By the second law of thermodynamics ,it is necessary to have a temperature difference (T 1 -T 2 ) to obtain work from any cycle. We know that efficiency is independent of working substance. And depends on temperature T 1 and T 2 .  rev =f( T 1 ,T 2 ) 8

9.  If some functional relationship is assigned between T 1 , T 2 and (Q 2 / Q 1 ),the equation becomes the definition of a temperature scale. Let’s consider two reversible engines,E 1 receiving heat from the source at T 1 ,and rejecting heat at T 2 to E 2 ,which rejects heat to the sink at T 3 . 9

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11. E 1 and E 2 together constitute another heat engine E 3 operating between T 1 and T 3 . 11 Ratio Q 1 / Q 2 depends only on T 1 and T 2 , Independent of T 3 .

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13. Kelvin defined the simplest possible way to define the Absolute Thermodynamic Temperature T is to let (T)=T. 13 Also known as Kelvin scale. It is independent of the characteristics of any particular substance .

14. Also, the triple point of water is taken as standard reference point . 14

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