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Real solutions

Activity and Activity Coefficients

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Written by: Germán Fernández
Category: real solutions
Published: 10 October 2012
Hits: 725

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}

Read more: Activity and Activity Coefficients

Normal States in Components of Real Solutions

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Written by: Germán Fernández
Category: real solutions
Published: 10 October 2012
Hits: 584

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.

Read more: Normal States in Components of Real Solutions

Excess Functions

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Written by: Germán Fernández
Category: real solutions
Published: 10 October 2012
Hits: 635

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}

Determination of Activities and Activity Coefficients

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Written by: Germán Fernández
Category: real solutions
Published: 10 October 2012
Hits: 2391

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

Gibbs-Duhem equation

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Written by: Germán Fernández
Category: real solutions
Published: 10 October 2012
Hits: 804

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}

Read more: Gibbs-Duhem equation

Activity coefficients of nonvolatile solutes

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Written by: Germán Fernández
Category: real solutions
Published: 10 October 2012
Hits: 665

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}

Read more: Activity coefficients of nonvolatile solutes

Activity coefficients on the molality and molar concentration scales

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Written by: Germán Fernández
Category: real solutions
Published: 10 October 2012
Hits: 784

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

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