Real solutions
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For a component in an ideal or dilute ideal solution, the chemical potential is given by: \begin{equation} \mu_{i}^{id}=\mu_{i}^{0}+RTlnx_i \end{equation} Solving for $ x_i$
\begin{equation} x_i=e^{[(\mu_i-\mu_{i}^{0})/RT]} \end{equation}
We define the activity of component i in a real solution as:
\begin{equation} a_i=e^{[(\mu_i-\mu_{i}^{0})/RT]} \end{equation} Activity plays the same role in real solutions as mole fraction in ideal ones.
Therefore, the chemical potential of a component in any solution (ideal or real) is given by:
\begin{equation} \mu_{i}=\mu_{i}^{0}+RTlna_i \end{equation}
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Convention I. The mole fractions of the components vary over a wide range (cannot distinguish between solvent and solutes).
To define the normal state, we drop the term $RTlnx_i\gamma_i$ in the equation, $\mu_i=\mu_{I,i}^{^0}+RTlnx_i\gamma_{I,i}$. To do this, we choose a state with, $\gamma_{I,i}\rightarrow 1$ and $x_i\rightarrow 1$, so that $lnx_i\gamma_{I,i}\rightarrow 0$.
The normal state $\mu_{I,i}^{0}=\mu_{i}^{\ast}$ is defined as component i being pure at the temperature and pressure of the solution.
Note that $\mu_i=\left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_{j\neq i}}$ does not depend on the choice of the normal state. However, the activity and the activity coefficient do depend.
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Excess functions represent the difference between the thermodynamic function of a solution and said function in a hypothetical ideal solution of the same composition. \begin{equation} G^{E}=G-G^{id}=G-G^{id}+G^{\ast}-G^{\ast}=G-G^{\ast}-(G^{id} -G^{\ast})=\Delta G_{mez}-\Delta G_{mez}^{id} \end{equation} Analogously: \begin{equation} S^{E}=\Delta S_{ mix}-\Delta S_{mez}^{id} \end{equation} \begin{equation} H^{E}=\Delta H_{mez}-\cancel{\Delta H_{mez}^{id}} \end{equation} \begin{equation} V^{E}=\Delta V_{mez}-\cancel{\Delta V_{mez}^{id}} \end{equation}
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The above formalism is useless if we cannot determine the activity coefficients.
Convention I
\begin{equation} \mu_i=\mu_{I,i}^{0}+RTln\gamma_{I,i}x_i\;\;\;\rightarrow\;\;\;\mu_i=\mu_{I ,i}^{0}+RTlna_{I,i} \end{equation} Raoult's Law, $P_i=x_iP_{i}^{\ast}$, can be generalized to real solutions simply by changing the fraction molar times the activity, $P_i=a_{I,i}P_{i}^{\ast}$ Solving for the activity, we obtain an equation that allows us to calculate it. \begin{equation} a_{I,i}=\frac{P_i}{P_{i}^{\ast}} \end{equation} The partial pressure of component i is calculated using Dalton's Law, $P_i= x_{iv}P_T$, and it is necessary to experimentally measure the total pressure, $P_T$, and the composition of the vapor over the solution, $x_{iv}$ \begin{equation} \gamma_{I,i}=\frac{ P_i}{x_{i,l}} \end{equation} This last equation allows us to calculate the activity coefficient, once the activity and composition of the liquid phase are known.
Read more: Determination of Activities and Activity Coefficients
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Nonvolatile solute activity coefficients can be determined from vapor pressure data using the Gibbs-Duhem equation. We start from the equation that gives us G for a solution as the sum of the product of the moles of each component times its chemical potential.
\begin{equation} G=\sum_{i}n_i\mu_i \end{equation} Differentiating:
\begin{equation} dG=\sum_{i}n_id\mu_i-\sum_{i}\mu_idn_i \end{equation}
Writing the Gibbs equation for dG
\begin{equation} dG=-SdT+VdP+\sum_{i}\mu_idn_i \end{equation}
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The activity coefficients of nonvolatile solutes cannot be determined by measuring the partial pressure of solute since it is too small.
Therefore, the vapor pressure on the solution (solvent pressure) $P_A$ is measured and with it the activity coefficient $\gamma_A$ is calculated based on the composition of the solution. Using the Gibbs-Duhem equation, the activity coefficient of the solvent is related to that of the solute $\gamma_B$.
We write the Gibbs-Duhem equation
\begin{equation} \sum_{i}n_id\mu_i=0 \end{equation}
We develop the equation for two components A and B.
\begin{equation} n_Ad\mu_A+n_Bd\mu_B= 0 \end{equation}
Dividing by the total moles: $n_A+n_B$
\begin{equation} x_Ad\mu_A+x_Bd\mu_B=0 \end{equation}
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Starting from the chemical potential of a solute according to the convention II
\begin{equation} \mu_{i}=\mu_{II,i}^{0}+RTln\gamma_{II,i}x_i \end{equation}
Molality of component i is given by, $m_i=\frac{n_i}{n_AM_A}$ where $M_A$ is the molecular weight of the solvent. Since the solvent is very abundant we can approximate the moles of A by the totals and suppose that, $x_i=\frac{n_i}{n_A}$. Substituting this mole fraction into the chemical potential equation
\begin{equation} \mu_i=\mu_{II,i}^{0}+RTln\left(\gamma_{II,i}m_ix_AM_A\frac{m^{0} }{m^{0}}\right) \end{equation}
In this last equation we multiply and divide by $m^0$ in order to separate the Neperian into two dimensionless addends.
\begin{equation} \mu_i=\underbrace{\mu_{II,i}^{0}+RTln(M_Am^0)}_{\mu_{m,i}^{0}}+RTln(\underbrace{ x_A\gamma_{II,i}}_{\gamma_{m,i}}m_i/m^0) \end{equation}
Read more: Activity coefficients on the molality and molar concentration scales