I have seen the equation $S(T_2)=S(T_1)+C_p\ln(T_2/T_1)$ where $C_p$ is the molar heat capacity at a constant pressure. I understand that this assumes that the temperature range is sufficiently small that the constant pressure heat capacity does not vary significantly over it.
My question pertains to the derivation of this expression. One of the first steps is substituting $q=C_pdT$ into $dS=\frac{q_{rev}}{T}$. I am not 100% sure as to why these enthalpies are equated. Is it because the heat exchanged by a system at a constant pressure is always reversible heat, or is this not true?
If the above were true, it would explain to me why I never see an expression involving constant volume heat capacity. If the above is not true, could someone also explain why suh a form does not exist?
Finally, is the equation $S(T_2)=S(T_1)+C_p\ln(T_2/T_1)$ only applicable where the pressures are the same at the initial and final temperatures? On the one hand, the derivation assumes a constant pressure in the infintesimal changes from the initial to final temperature (and of course uses a constant pressure heat capacity!); however on the other hand entropy is a state function. So I would think that if one could find how entropy varies wit pressure at a constant temperature, and the change from the final to the initial state involved both a temperature and pressure change, then I would think that the above expression can be used to find the change in entropy due to the temperature change only, and then the additional factor from the pressure could be used?
Thank you in advance.
EDIT: For the second part, from having a quick think about the reversible heat enxchange in and isothermal process, I think $S(T_2,p_2)=S(T_2,p_1)+R\ \ln(p_1/p_2)$. So then one could construct a Hess-like cycle to go from one temperature and pressure/volume, to another temperature and pressure/volume. The use of the cycle and step-wise calulcation works because entropy is a state function I think?
Hopefully I answered the second part of my query (I would appreciate if someone could verify that it is correct), although I am still not sure about why the heats used are the same in the derivation, and whether the expression really does refer to the same pressure at the initial and final temperatures, even though the process in getting from these two temperatures can involve pressure changes (again, thinking about $S$ being a state function, so the process itself when changing states does not matter).