Consider the heat engine shown below. The cyclic integral of dQ is greater than zero. Does this violate the Clausius Inequality? Explain. 









Read : 
This problem is
designed to make you think very carefully about how to use a cyclic integral. The key is that the temperature is not
the same for both of the heat transfer interactions in this cycle. 










Diagram: 
See the problem
statement. 










Given: 
T_{H} 
400 
K 



Q_{H} 
450 
kJ 

T_{C} 
300 
K 



Q_{C} 
350 
kJ 







W 
100 
kJ 



















Find: 
Does the cycle violate the Clausius Inequality? 










Assumptions: 
None. 











Equations
/ Data / Solve: 


















The Clausius Inequality is: 



Eqn 1 











Because the cycle only exchanges heat with the hot and cold thermal reservoirs, the integrals can be simplified: 












Eqn 2 




Eqn 3 












100 
kJ 



0.04167 
kJ/K 












This cycle is irreversible because the cyclic integral in the Clausius Inequality is less
than zero. 











It is true that the cyclic integral of dQ > 0.
But the Clausius Inequality is still
satisfied. 











Confusion about the cyclic integrals sometimes arises if you mistakenly pull T out of the cyclic integral. 


















Eqn 4 











You could only pull T out of the cyclic integral if ALL
of the heat exchange
across the system boundary went to
or from reservoirs that were ALL at the SAME temperature. 











That is almost never going to happen. It is definitely not the case in this problem as the hot and cold reservoirs are at 400 K and 300 K,
respectively. 










Verify: 
None. 


















Answers : 
Yes, the cycle does indeed satisfy the Clausius
Inequality. 









