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# 物理代写|热力学代写Thermodynamics代考|What Are Excess Functions?

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## 物理代写|热力学代写Thermodynamics代考|What Are Excess Functions?

In practice, no real mixture is ideal, but mixtures formed from similar chemical substances do behave in large measure as ideal. When mixtures are not ideal, it is usual to discuss their behavior in terms of the excess molar functions $X_{\mathrm{m}}^{\mathrm{E}}$ defined by
$$X_{\mathrm{m}}^{\mathrm{E}}=\Delta_{\mathrm{mix}} X_{\mathrm{m}}-\Delta_{\mathrm{mix}} X_{\mathrm{m}}^{\mathrm{id}}$$
so that the excess property is expressed with respect to the ideal mixture denoted by id and defined by Equations 4.113 to 4.116 .
Thus, for a binary mixture, the excess molar Gibbs energy is
\begin{aligned} G_{\mathrm{m}}^{\mathrm{E}}\left(T, p, x_{\mathrm{A}}\right)= & \Delta_{\mathrm{mix}} G_{\mathrm{m}}\left(T, p, x_{\mathrm{A}}\right)-R T\left{x_{\mathrm{A}} \ln x_{\mathrm{A}}+x_{\mathrm{B}} \ln x_{\mathrm{B}}\right} \ = & x_{\mathrm{A}} \mu_{\mathrm{A}}\left(T, p, x_{\mathrm{B}}\right)+x_{\mathrm{B}} \mu_{\mathrm{B}}\left(T, p, x_{\mathrm{A}}\right)-\left[x_{\mathrm{A}} \mu_{\mathrm{A}}^(T, p)+x_{\mathrm{B}} \mu_{\mathrm{B}}^(T, p)\right] \ & -R T\left{x_{\mathrm{A}} \ln x_{\mathrm{A}}+x_{\mathrm{B}} \ln x_{\mathrm{B}}\right} \end{aligned}
using the definitions of $G_m(T, p, x)$ from Equation 4.106 and $\Delta_{\text {mix }} G_m(T, p, x)$ from Equation 4.107. This can be written as
$$\mathrm{G}{\mathrm{m}}^{\mathrm{E}}=x{\mathrm{A}} \mu_{\mathrm{A}}^{\mathrm{E}}+x_{\mathrm{B}} \mu_{\mathrm{B}}^{\mathrm{E}}$$
where
$$\mu_{\mathrm{A}}^{\mathrm{E}}\left(T, p, x_{\mathrm{A}}\right)=\mu_{\mathrm{A}}\left(T, p, x_{\mathrm{A}}\right)-\mu_{\mathrm{A}}^e(T, p)-R T \ln x_{\mathrm{A}}$$
and
$$\mu_{\mathrm{B}}^{\mathrm{E}}\left(T, p, x_{\mathrm{B}}\right)=\mu_{\mathrm{B}}\left(T, p, x_{\mathrm{B}}\right)-\mu_{\mathrm{B}}^*(T, p)-R T \ln x_{\mathrm{B}} .$$

## 物理代写|热力学代写Thermodynamics代考|What Are Activity Coefficients?

The chemical potential of substance $\mathrm{A}$ in a binary liquid mixture can be obtained from its value at saturation from the equation
$$\mu_{\mathrm{A}, 1}\left(T, p, x_{\mathrm{A}}\right)=\mu_{\mathrm{A}, 1}\left(T, p^{\text {sat }}, x_{\mathrm{A}}\right)+\int_{p^{\mathrm{st}}}^p V_{\mathrm{A}, 1}\left(T, p, x_{\mathrm{A}}\right) \mathrm{d} p,$$
which follows from its definition in Equations 3.25 and 3.73 .
At equilibrium (saturation), when the pressure in the system is the saturation vapor pressure of the mixture $p^{\text {sat }}$, the liquid and gas phase chemical potentials of substance A are equal so that
$$\mu_{\mathrm{A}, 1}\left(T, p^{\mathrm{sat}}, x_{\mathrm{A}}\right)=\mu_{\mathrm{A}, \mathrm{g}}\left(T, p^{\mathrm{sat}}, y_{\mathrm{A}}\right)$$
In view of Equation 4.61, Equation 4.126 can be written as
$$\mu_{\mathrm{A}, \mathrm{l}}\left(T, p^{\mathrm{sat}}, x_{\mathrm{A}}\right)=\mu_{\mathrm{A}, \mathrm{g}}^{\ominus}(T)+R T \ln \left{\frac{y_{\mathrm{A}} p^{\mathrm{sat}}}{p^{\ominus}}\right}+\int_0^{p^{\mathrm{st}}}\left{V_{\mathrm{A}, \mathrm{g}}\left(T, p, x_{\mathrm{A}}\right)-\frac{R T}{p}\right} \mathrm{d} p .$$
Equation 4.125 is then
\begin{aligned} \mu_{\mathrm{A}, 1}\left(T, p, x_{\mathrm{A}}\right)= & \mu_{\mathrm{A}, \mathrm{g}}^{\ominus}(T)+R T \ln \left{\frac{y_{\mathrm{A}} p^{\mathrm{sat}}}{p^{\ominus}}\right}+\int_0^{p^{\mathrm{stt}}}\left{V_{\mathrm{A}, \mathrm{g}}\left(T, p, y_{\mathrm{A}}\right)-\frac{R T}{p}\right} \mathrm{d} p \ & +\int_{p^{\mathrm{stt}}}^p V_{\mathrm{A}, \mathrm{l}}\left(T, p, x_{\mathrm{A}}\right) \mathrm{d} p \end{aligned}
For substance B, the equivalent equation is
\begin{aligned} \mu_{\mathrm{B}, 1}\left(T, p, x_{\mathrm{B}}\right)= & \mu_{\mathrm{B}, \mathrm{g}}^{\ominus}(T)+R T \ln \left{\frac{y_{\mathrm{B}} p^{\mathrm{sat}}}{p^{\ominus}}\right}+\int_0^{p^{\mathrm{st}}}\left{V_{\mathrm{B}, \mathrm{g}}\left(T, p, y_{\mathrm{B}}\right)-\frac{R T}{p}\right} \mathrm{d} p \ & +\int_{p^{\mathrm{st}}}^p V_{\mathrm{B}, 1}\left(T, p, x_{\mathrm{B}}\right) \mathrm{d} p . \end{aligned}

## 物理代写|热力学代写Thermodynamics代考|What Are Excess Functions?

$$X_{\mathrm{m}}^{\mathrm{E}}=\Delta_{\operatorname{mix}} X_{\mathrm{m}}-\Delta_{\operatorname{mix}} X_{\mathrm{m}}^{\mathrm{id}}$$

$$\mathrm{Gm}^{\mathrm{E}}=x \mathrm{~A} \mu_{\mathrm{A}}^{\mathrm{E}}+x_{\mathrm{B}} \mu_{\mathrm{B}}^{\mathrm{E}}$$

$$\mu_{\mathrm{A}}^{\mathrm{E}}\left(T, p, x_{\mathrm{A}}\right)=\mu_{\mathrm{A}}\left(T, p, x_{\mathrm{A}}\right)-\mu_{\mathrm{A}}^e(T, p)-R T \ln x_{\mathrm{A}}$$

$$\mu_{\mathrm{B}}^{\mathrm{E}}\left(T, p, x_{\mathrm{B}}\right)=\mu_{\mathrm{B}}\left(T, p, x_{\mathrm{B}}\right)-\mu_{\mathrm{B}}^*(T, p)-R T \ln x_{\mathrm{B}}$$

## 物理代写|热力学代写Thermodynamics代考|What Are Activity Coefficients?

$$\mu_{\mathrm{A}, 1}\left(T, p, x_{\mathrm{A}}\right)=\mu_{\mathrm{A}, 1}\left(T, p^{\mathrm{sat}}, x_{\mathrm{A}}\right)+\int_{p^{\mathrm{st}}}^p V_{\mathrm{A}, 1}\left(T, p, x_{\mathrm{A}}\right) \mathrm{d} p,$$

$$\mu_{\mathrm{A}, 1}\left(T, p^{\mathrm{sat}}, x_{\mathrm{A}}\right)=\mu_{\mathrm{A}, \mathrm{g}}\left(T, p^{\mathrm{sat}}, y_{\mathrm{A}}\right)$$

$\backslash$ begin ${$ aligned $} \backslash m m_{-}{\backslash \operatorname{mathrm}{A}, 1} \backslash$ left $\left(T, p_1 x_{-}{\backslash \operatorname{mathrm}{A}} \backslash r i g h t\right)=\& \backslash m u_{-}{\backslash \operatorname{mathrm}{A}, \mid \operatorname{mathrm}{g}} \wedge{\backslash \operatorname{lominus}}(T)+R T \backslash \ln \backslash \ln \backslash \operatorname{left}{\backslash 1$

\begin{aligned} \mu_{\mathrm{B}, 1}\left(T, p, x_{\mathrm{B}}\right)= & \mu_{\mathrm{B}, \mathrm{g}}^{\ominus}(T)+R T \ln \left{\frac{y_{\mathrm{B}} p^{\mathrm{sat}}}{p^{\ominus}}\right}+\int_0^{p^{\mathrm{st}}}\left{V_{\mathrm{B}, \mathrm{g}}\left(T, p, y_{\mathrm{B}}\right)-\frac{R T}{p}\right} \mathrm{d} p \ & +\int_{p^{\mathrm{st}}}^p V_{\mathrm{B}, 1}\left(T, p, x_{\mathrm{B}}\right) \mathrm{d} p . \end{aligned}

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