Confusion about relation of entropy with temperature.

[Engineer1]

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.

 

[BvU]

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).

 

[mfig]

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.