Is there a name for this sort of thermo relationship? taking physical chemistry at the moment.
My textbook does not go over the derivation of the relationship below:
$$H_{vap\;/\;sub\;/\;cond}(T')-H_{vap\;/\;sub\;/\;cond}(T)= \int_{T}^{T'}\Delta C_{p,m}\; dT$$
Where $T$ is the enthalpy value at standard conditions, and $T'$ is $(T + dT)$, such that we may find the "new" enthalpy value at that temperature.
Can I use a similar process to determine a new $\Delta S _{vap}$ or $\Delta G_{vap}$, provided C(T) and the $S$ or $G$ value at standard conditions? Is there a name I can use to look into this topic more deeply?
 A: The relation arises from integrating the general partial-derivative expansion
$$dH=\left(\frac{dH}{dT}\right)_PdT+\left(\frac{dH}{dP}\right)_TdP,$$
or—replacing the partial derivatives with the corresponding material properties—
$$dH=C_P\,dT+V(1-αT)dP,$$
with constant-pressure heat capacity $C_P$, temperature $T$, bulk modulus $K$, thermal expansion coefficient $\alpha$, and volume $V$, for the specific cases of an ideal gas, for which $\alpha = 1/T$, or constant pressure ($dP=0$), thus giving $dH=C_P\,dT$ and then $\Delta H=\int C_P\,dT$.
To calculate $\Delta S$, for instance, you'd express $dS$ in the variables you wish to use, e.g.,
$$dS=\left(\frac{dS}{dT}\right)_PdT+\left(\frac{dS}{dP}\right)_TdP.$$
Then you'd figure out what material properties those partial derivatives refer to, simplify, and integrate over the temperature range of interest. Here, $C_P\equiv T\left(\frac{\partial S}{\partial T}\right)_P$, so at constant pressure $dS=\frac{C_P}{T}dT$ and then $\Delta S=\int\frac{C_P}{T}dT$.
The same general strategy would be applied to calculate $\Delta G$.
