We will now discuss an astonishingly general relation which holds for all extensive properties \(\Phi(p,T,n_i)\) of a mixture which can be written as
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| \begin{equation*} \Phi = \sum_i n_i \varphi_i \quad \mbox{with} \quad \varphi_i = \left( \frac{\partial \Phi}{\partial n_i}\right)_{p,T,n_{j \neq i}} \label{eq:Phi_Gibbs_Duhem} \end{equation*} | (5.17) |
As for any function with the same coordinates as the Gibbs potential for constant \(p\) and \(T\) the total derivative is
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| \begin{equation*} d\Phi = \sum_i \left( \frac{\partial \Phi}{\partial n_i}\right)_{p,T,n_{j \neq i}} dn_i = \sum_i \varphi_i dn_i \label{eq:d_Phi_1} \end{equation*} | (5.18) |
On the other hand for a special function defined by Eq. (5.17) the total derivative can be written as
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| \begin{equation*} d\Phi = \sum_i \varphi_i dn_i + \sum_i n_i d\varphi_i \label{eq:d_Phi_2} \end{equation*} | (5.19) |
Comparing Eq. (5.18) and Eq. (5.19) we find
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| \begin{equation*} 0 = \sum_i n_i d\varphi_i \label{eq:Gibbs_Duhem} \end{equation*} | (5.20) |
which is the Gibbs-Duhem equation. So in a system with \(N\) components, only \(N-1\) partial molar quantities are independent, e.g. the chemical potential of one component in a solution cannot be varied independently of the chemical potentials of the other components. Integration of Eq. (5.20) allows to calculate \(\Delta \phi_i\) relations:
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| \begin{equation*} \int d \varphi_1 = - \frac{n_2}{n_1} \int d\varphi_2 \label{eq:int_Gibbs_Duhem} \end{equation*} | (5.21) |
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© J. Carstensen (TD Kin I)