What determines the shape of the temperature-entropy graph?

AI Thread Summary
The shape of the temperature-entropy graph is influenced by the inherent properties of the material, particularly its heat capacity. The graph is expected to be monotonically increasing, reflecting the relationship between temperature and entropy. The first derivative, representing the slope, indicates temperature, while the second derivative relates to heat capacity. Variations in material properties will affect these derivatives and, consequently, the graph's shape. Understanding these relationships is crucial for analyzing thermodynamic behavior.
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in my thermo class when we were formalizing the definition of temperature (\frac{1}{T}=\frac{∂S}{∂U}), we drew out all the combinations of various slopes and concavities of the ∂S/∂U graphs.

http://imgur.com/cR4V8K8

The shape of this graph i figure should be a reflection of the inherent nature of the material. so, my question is, what properties of the material determines the shape of this graph?
 
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I think the graph should be monotonically increasing since the temperature is the slope. Furthermore, the second derivative should also be considered since it relates to the heat capacity.
 
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