Fundamental equations for open systems (dni ⁄= 0)

3.32 Fundamental equations for open systems (\(dn_i \neq 0\))

For systems containing only one component we already introduced the chemical potential \(\mu\) (cf. e.g. Eq (3.33)). To later on allow for the description of chemical reactions we now will generalize this concept to systems with many components \(i\); the fundamental equations now are

\begin{equation*} \begin{split} dH = &\;\; \left( \frac{\partial H}{\partial S }\right)_{p,n_i} dS + \left( \frac{\partial H}{\partial p }\right)_{S,n_i} dp + \sum_i \left( \frac{\partial H}{\partial n_i }\right)_{S,p,n_{i \neq j}} dn_i\\ dU = &\;\; \left( \frac{\partial U}{\partial S }\right)_{V,n_i} dS + \left( \frac{\partial U}{\partial V }\right)_{S,n_i} dV + \sum_i \left( \frac{\partial U}{\partial n_i }\right)_{S,V,n_{i \neq j}} dn_i\\ dG = &\;\; \left( \frac{\partial G}{\partial T }\right)_{p,n_i} dT + \left( \frac{\partial G}{\partial p }\right)_{T,n_i} dp + \sum_i \left( \frac{\partial G}{\partial n_i }\right)_{T,p,n_{i \neq j}} dn_i\\ dF = &\;\; \left( \frac{\partial F}{\partial T }\right)_{V,n_i} dT + \left( \frac{\partial F}{\partial V }\right)_{T,n_i} dV + \sum_i \left( \frac{\partial F}{\partial n_i }\right)_{T,V,n_{i \neq j}} dn_i\\ \end{split} \label{eq:fund_eq_mu} \end{equation*}

(3.83)

so one finds

\begin{equation*} \mu_i = \left( \frac{\partial H}{\partial n_i }\right)_{S,p,n_{i \neq j}} = \left( \frac{\partial U}{\partial n_i }\right)_{S,V,n_{i \neq j}} = \left( \frac{\partial G}{\partial n_i }\right)_{T,p,n_{i \neq j}} = \left( \frac{\partial F}{\partial n_i }\right)_{T,V,n_{i \neq j}} \label{eq:mu_for_fund} \end{equation*}

(3.84)

So as an example \(\mu_i\) gives the change of \(G\) when component \(i\) is added to the system at constant \(T\), \(p\), and constant number of moles of all of the other species, thus \(\mu\) represents the chemical non-expansion work. Thus for a pure phase we find \(\mu = G / n\).