I am currently reading this bookDemirel's Nonequilibrium Thermodynamics in order to learn more about entropy and nonequlibrium thermodynamics
However I and stumped on how the authors do this step here as shown in the left side of the image
Background: I am trying toauthor's do some research on modelling relationship between entropy and chemical potentials for isomerisation reactions, which requires computing partial derivatives where the variables are chemical potentials of particles in a system. Thus I have to make sure I have used the maths correctly (especially integrating multivariate expressions expressed as differentials), thus this question is the firstfollowing step to it
EDIT: Zoom in of the problem
IsThe pressure $P$ and temperature $T$ define the result fromvalues at each point of the text obtained fromsystem and are therefore called direct integrationintensive properties, or it just looks like it is integrated because of the consequence of Euler Homogeneous Function Theorem?
That issome of which can be expressed as derivatives of extensive properties, is thissuch as temperature \begin{align} X=\sum_i X_in_i &&&& [1] \end{align}
$$T=\left(\frac{\partial U}{\partial S}\right)_{V,N}$$ where $U$ is actually not this
\begin{align} dX=\sum_i X_idn_i &&&& [2] \end{align}
integrated
But just because of Euler Homogeneous Function Theorem, ifthe energy and $[2]$$S$ is true, thenthe entropy. If $[1]$ must also be true$X$ denotes any extensive property (not necessarily a thermodynamic propert) of a phase, we may derive intensive properties denoted by the$X_i$ and called as partial properties $X_i=\left(\partial X/\partial n_i\right)_{T,p,n_i}$ ($i\neq j$). For any partial property we have $dX=\sum_i\left(\partial X/\partial n_i\right)dn_i=\sum_iX_idn_i$ at constant $T$ and $P$. The Euler theorem? shows that $X=\sum_iX_in_i$
Is the result from the text obtained from direct integration, or it just looks like it is integrated because of the consequence of Euler Homogeneous Function Theorem?
That is, is this relation $$ X=\sum_i X_in_i \tag{1} $$ the integration of this relation $$ dX=\sum_i X_idn_i? \tag{2} $$ But just because of Euler Homogeneous Function Theorem, if (2) is true, then (1) must also be true by the theorem?
