Question

Entropy & heat capacity: The definitions for the entropy and heat capacity are pretty similar. So, what is actually the difference between dS=dq/T and cp = dq/dT? Apply a simple and plausible example to illustrate the different nature of S and cp.

Answer 1

Heat capacity  of an object is the proportionality constant between the heat Q that the object absorbs or loses & the resulting temperature change dT of the object. Entropy change is the amount of energy dispersed reversibly at a specific temperature. But they have the same unit joule/kelvin like work & energy. Entropy and heat capacity are different as one concern with temperature change and other only at a specific temperature.

Entropy by itself is not useful and cannot even be measured. Heat capacity, on the other hand, can have its absolute value determined experimentally, and it won't depend on a reference value like entropy does. Its absolute value is immediately useful.

The heat capacity is a material-dependent object that - as you say - measures the difference in temperature when energy is absorbed by the material. The entropy change on the other hand is a material-independent quantity. The entropy change is proportional to the heat transfer in a reversible process. However, most processes are irreversible, so the quantity is process dependent.