Confusion about relation of entropy with temperature.

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SUMMARY

The discussion clarifies the relationship between entropy and temperature, specifically addressing scenarios where entropy can remain constant despite temperature changes. It highlights the equation ##\delta Q = T dS##, emphasizing its application in adiabatic processes where ##\delta Q = 0##. The entropy change for a closed system of an ideal gas is defined by the equation ##\Delta S = c_p \cdot Ln\frac{T_2}{T_1} + R \cdot Ln\frac{P_2}{P_1}##. The conversation concludes that while entropy change is influenced by heat transfer and internal irreversibilities, real processes must adhere to the second law of thermodynamics, which states that the total entropy change must exceed the entropy transferred with heat.

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  • Familiarity with the concepts of entropy and heat transfer
  • Knowledge of ideal gas behavior and equations
  • Basic grasp of thermodynamic processes, particularly adiabatic processes
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Engineer1
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Why can sometimes entropy remain constant with increase of temperature and vice versa?Entropy implies transfer of heat and heat must increase with temperature.I am unable to intuitively understand.
 
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With ##\delta Q = T dS## you can see that this happens for an adiabatic (##\delta Q = 0##) change of state, e.g. an expansion. (reversible, because of the ##dS=0##, so an idealization).
 
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You can see why by considering the entropy change for a closed system of an ideal gas (assuming constant specific heats).

##\Delta S = c_p \cdot Ln\frac{T_2}{T_1} + R \cdot Ln\frac{P_2}{P_1}##

So if we ##\Delta S = 0## do you see anything that would require ##T_1 = T_2##? Recall that entropy change may occur due to heat transfer, and internal irreversibilities. The second law requires that for all real processes the entropy change will be larger than the entropy that is transferred along with heat.
 
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