# CU-Boulder AREN 2110 - CLAUSIUS INEQUALITY AND ENTROPY (18 pages)

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## CLAUSIUS INEQUALITY AND ENTROPY

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Lecture Notes

- Pages:
- 18
- School:
- University of Colorado at Boulder
- Course:
- Aren 2110 - Thermodynamics

**Unformatted text preview: **

CLAUSIUS INEQUALITY AND ENTROPY with a little history thrown in CARNOT PROPOSED THAT HEAT MUST ALWAYS BE WASTED IN ORDER FOR A HEAT ENGINE TO PRODUCE NET WORK BUT HE DID NOT QUANTIFY HOW MUCH HEAT HAD TO BE WASTED IN 1850 RUDOLF CLAUSIUS CONFIRMED THE EXISTING THEORIES 1 ENERGY IS CONSERVED 1ST LAW WAS QUANTIFIED AS Q W U 2 HEAT FLOWS NATURALLY FROM HOT TO COLD NOT QUANTIFIED AND ADDED AN IMPORTANT CONTRIBUTION CLAUSIUS KNEW THAT SOME HEAT HAD TO BE REJECTED FROM A REVERSIBLE CARNOT HEAT ENGINE QL AS CARNOT PROPOSED AND HE KNEW THAT QL Wnet QH 1ST LAW BUT NO PRINCIPLE FIXED THE ABSOLUTE AMOUNT OF REJECTED HEAT CLAUSIUS THEN OBSERVED THAT FOR REVERSIBLE HEAT ENGINES THE RATIO OF THE HEAT INPUT TO THE REJECTED HEAT WAS CONSISTENTLY EQUAL TO THE RATIO OF THE ABSOLUTE TEMPERATURES OF THE HIGH AND LOW TEMPERATURE RESERVOIRS Q H TH QL TL 1 1 If and only if the temperature of the high TH and low TL temperature reservoirs are always expressed in degrees Kelvin REARRANGING 1 FOR A REVERSIBLE HEAT ENGINE QH QL TH TL 2 CLAUSIUS THEN EXAMINED IRREVERSIBLE PROCESSES AND FOUND THAT THE RELATION 2 DID NOT HOLD HE REASONED FOR EXAMPLE IF 10 JOULES 0F HEAT FLOW FROM A HOT OBJECT AT 350K INTO A COOL ROOM AT 300K THEN THE HEAT TRANSFER TERM FOR HEAT LEAVING THE OBJECT AND HEAT TRANSFERRED INTO THE ROOM IS THE SAME AND Q Q 10 10 or TH TL 350 300 3 That is for irreversible processes the ratio of heat over absolute temperature increases in the direction of natural heat flow COMBINING 2 AND 3 FOR A HEAT ENGINE WITH ONE HEAT INPUT AND ONE HEAT OUTPUT QH QL 0 TH TL 4 OR FOR ANY SEQUENCE OF PROCESSES WITH DISCRETE HEAT TRANSFER TERMS Q K T 0 K 5 2 WHERE Q T K THE RATIO OF THE AMOUNT OF HEAT TRANSFERRED IN A PROCESS TO THE TEMPERATURE OF THE SURROUNDINGS WHERE HEAT IS TRANSFERRED CALCULATING THE RATIO AS THE INTEGRAL OF A Q CONTINUOUS FUNCTION OF T CLAUSIUS PRINCIPLE FOR A REVERSIBLE OR IRREVERSIBLE CYCLE IS Q 0 T 6 FOR REVERSIBLE CYCLES Q 0 T and Q 0 K T K 7 FOR IRREVERSIBLE CYCLES Q 0 and T

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