Is there a name for this sort of thermo relationship? H_(vap sub cond)(T')-H_(vap sub cond)(T)= int_(T)^(T')Delta C_(p,m)dT

Is there a name for this sort of thermo relationship?
${H}_{vap\phantom{\rule{thickmathspace}{0ex}}/\phantom{\rule{thickmathspace}{0ex}}sub\phantom{\rule{thickmathspace}{0ex}}/\phantom{\rule{thickmathspace}{0ex}}cond}\left({T}^{\prime }\right)-{H}_{vap\phantom{\rule{thickmathspace}{0ex}}/\phantom{\rule{thickmathspace}{0ex}}sub\phantom{\rule{thickmathspace}{0ex}}/\phantom{\rule{thickmathspace}{0ex}}cond}\left(T\right)={\int }_{T}^{{T}^{\prime }}\mathrm{\Delta }{C}_{p,m}\phantom{\rule{thickmathspace}{0ex}}dT$
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

hottchevymanzm
The relation arises from integrating the general partial-derivative expansion
$dH={\left(\frac{dH}{dT}\right)}_{P}dT+{\left(\frac{dH}{dP}\right)}_{T}dP,$
or replacing the partial derivatives with the corresponding material properties
$dH={C}_{P}\phantom{\rule{thinmathspace}{0ex}}dT+V\left(1-\alpha T\right)dP,$
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 $dH={C}_{P}\phantom{\rule{thinmathspace}{0ex}}dT$ and then $\mathrm{\Delta }H=\int {C}_{P}\phantom{\rule{thinmathspace}{0ex}}dT$
To calculate $\mathrm{\Delta }S$ for instance, you'd express dS in the variables you wish to use, e.g.,
$dS={\left(\frac{dS}{dT}\right)}_{P}dT+{\left(\frac{dS}{dP}\right)}_{T}dP.$
Then you'd figure out what material properties those partial derivatives refer to, simplify, and integrate over the temperature range of interest. Here, ${C}_{P}\equiv T{\left(\frac{\mathrm{\partial }S}{\mathrm{\partial }T}\right)}_{P}$, so at constant pressure $dS=\frac{{C}_{P}}{T}dT$ and then $\mathrm{\Delta }S=\int \frac{{C}_{P}}{T}dT$
The same general strategy would be applied to calculate $\mathrm{\Delta }G$