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When all other thermodynamic variables pertaining to a system are kept constant, how does the entropy $S$ of a system change with the increase in temperature? Does it strictly increase with the increase in temperature? Whatever is the behaviour i.e., whether entropy increase or decrease with the increase in temperature be worked out from the thermodynamic relations?

One way to find the answer is to find the sign of the derivative $\Big(\frac{\partial S}{\partial T}\Big)_{V,N}$. If it's positive, then I can say for sure that $S$ increases with $T$.

Please do not answer by citing examples of the temperature dependence of entropy of some specific systems.

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If it is OK to take the following as a starting point, then I will proceed further: $$dU=TdS-PdV+\mu dN$$ If I use this equation to determine the partial derivative of U with respect to T at constant V and N, I obtain:$$\left(\frac{\partial U}{\partial T}\right)_{V,N}=T\left(\frac{\partial S}{\partial T}\right)_{V,N}$$But, the left hand side of this equation is equal to $NC_V$. Therefore, $$\left(\frac{\partial S}{\partial T}\right)_{V,N}=N\frac{C_V}{T}$$The sign of this is always positive.

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