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物理代写|热力学代写Thermodynamics代考|PHY360 Solutions, Phase-Separated Systems

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物理代写|热力学代写Thermodynamics代考|Solutions, Phase-Separated Systems

The chemical potential of a substance A in a solution can be calculated by focusing on the gas phase above the solution. At equilibrium, the chemical potential of $\mathrm{A}$ in the solution is equal to the chemical potential of $\mathrm{A}$ in the gas phase, so that the first task is to calculate the chemical potential of $\mathrm{A}$ in the gas phase.

The chemical potential $\mu$ is dependent on the pressure $(p)$ at constant temperature according to Eq. (6.2), i.e.:
$$\left(\frac{\partial \mu}{\partial p}\right)T=V{\mathrm{m}}$$
For a gas which obeys the ideal gas law, the following $\mu-p$ relationship is obtained:
\begin{aligned} &d \mu=V_{\mathrm{m}} d p ; V_{\mathrm{m}}=\frac{R T}{p} \Rightarrow d \mu=R T \frac{d p}{p} \ &\mu=\mu_0+R T \ln \left(\frac{p}{p_0}\right) \end{aligned}
where $\mu_0$ is the chemical potential of the ideal gas at a reference pressure $\left(p_0\right)$. Figure $7.1$ shows (i) a closed system with a pure liquid $\mathrm{A}$ and a gas phase in equilibrium with the liquid phase (the pressure of component $\mathrm{A}$ is $p_{\mathrm{A}}{ }^$ ) and (ii) a similar closed system but with a binary solution with the molar composition $x_{\mathrm{A}}$. The chemical potential of the pure liquid $\mathrm{A}\left(\mu_{\mathrm{A}}{ }^\right)$ is according to Eq. (7.2) given by:
$$\mu_{\mathrm{A}}^=\mu_{\mathrm{A}, 0}+R T \ln \left(\frac{p_{\mathrm{A}}^}{p_{\mathrm{A}, 0}}\right) \Rightarrow \mu_{\mathrm{A}, 0}=\mu_{\mathrm{A}}^-R T \ln \left(\frac{p_{\mathrm{A}}^}{p_{\mathrm{A}, 0}}\right)$$
and the chemical potential of $\mathrm{A}$ in the solution is given by:

$$\mu_{\mathrm{A}}=\mu_{\mathrm{A}, 0}+R T \ln \left(\frac{p_{\mathrm{A}}}{p_{\mathrm{A}, 0}}\right)$$
The following expression is obtained by combining Eqs. (7.3) and (7.4):
$$\mu_{\mathrm{A}}=\mu_{\mathrm{A}}^-R T \ln \left(\frac{p_{\mathrm{A}}^}{p_{\mathrm{A}, 0}}\right)+R T \ln \left(\frac{p_{\mathrm{A}}}{p_{\mathrm{A}, 0}}\right) \Rightarrow \mu_{\mathrm{A}}=\mu_{\mathrm{A}}^+R T \ln \left(\frac{p_{\mathrm{A}}}{p_{\mathrm{A}}^}\right)$$

物理代写|热力学代写Thermodynamics代考|Chemical Equilibrium

A central part of chemical thermodynamics is concerned with chemical reactions and chemical equilibrium. This is often treated in a separate course without the strictness typical of thermodynamics. In this final chapter, we shall deal with this topic using Gibbs free energy and related state functions to describe chemical equilibrium.
Let us start with a very simple reaction: $A(g) \leftrightarrows B(g)$. The changes in the number of moles of $\mathrm{A}\left(n_{\mathrm{A}}\right)$ and $\mathrm{B}\left(n_{\mathrm{B}}\right)$ with the extent of the forward reaction, which denoted the reaction coordinate $(\xi)$, are given by:
$$d n_{\mathrm{A}}=-d \xi ; d n_{\mathrm{B}}=d \xi$$
The differential change in Gibbs free energy at constant $T$ and constant $p$ is given by:
$$d G=-\mu_{\mathrm{A}} d \xi+\mu_{\mathrm{B}} d \xi=\left(\mu_{\mathrm{B}}-\mu_{\mathrm{A}}\right) d \xi$$
which means that:
$$\left(\frac{\partial G}{\partial \xi}\right){p, T}=\mu{\mathrm{B}}-\mu_{\mathrm{A}}$$
This derivative, which is also a difference, is abbreviated $\Delta G_{\mathrm{r}}$ :
$$\Delta G_{\mathrm{r}}=\left(\frac{\partial G}{\partial \xi}\right){p, T}=\mu{\mathrm{B}}-\mu_{\mathrm{A}}$$
and, according to Chap. 7 (Eq. (7.8)), it can be expressed as:

$$\Delta G_{\mathrm{r}}=\mu_{\mathrm{B}}-\mu_{\mathrm{A}}=\left(\mu_{\mathrm{B}}^-\mu_{\mathrm{A}}^\right)+R T \ln \left(\frac{a_{\mathrm{B}}}{a_{\mathrm{A}}}\right)$$
where the activity ratio, $a_{\mathrm{B}} / a_{\mathrm{A}}$, is denoted the reaction quotient $(Q)$. Figure $8.1$ shows a generic $G-\xi$ diagram. The curvature, which originates from the logarithmic term in Eq. (8.5), is the reason for the presence of a minimum point, chemical equilibrium in the $G-\xi$ diagram.

On the left-hand side of the minimum, $\Delta G_{\mathrm{r}}<0$, which indicates that the forward reaction $(\mathrm{A} \rightarrow \mathrm{B})$ dominates. This continues until the point where $\Delta G_{\mathrm{r}}=0$. If, on the other hand, the system is on the right-hand side of the minimum, $\Delta G_{\mathrm{r}}<0$ and the backward reaction ( $\mathrm{B} \rightarrow \mathrm{A}$ ) is dominant. Again, the system is striving to reach the minimum $G$-value ( $\Delta G_{\mathrm{r}}=0$ ). The condition for equilibrium can be expressed as:
$$\Delta G_{\mathrm{r}}=0$$

物理代写|热力学代写THERMODYNAMICS代考|SOLUTIONS, PHASE-SEPARATED SYSTEMS

$$\left(\frac{\partial \mu}{\partial p}\right) T=V \mathrm{~m}$$

$$d \mu=V_{\mathrm{m}} d p ; V_{\mathrm{m}}=\frac{R T}{p} \Rightarrow d \mu=R T \frac{d p}{p} \quad \mu=\mu_0+R T \ln \left(\frac{p}{p_0}\right)$$

$$\mu_{\mathrm{A}}=\mu_{\mathrm{A}, 0}+R T \ln \left(\frac{p_{\mathrm{A}}}{p_{\mathrm{A}, 0}}\right)$$

物理代写|热力学代写THERMODYNAMICS代考|CHEMICAL EQUILIBRIUM

$$d n_{\mathrm{A}}=-d \xi ; d n_{\mathrm{B}}=d \xi$$

$$d G=-\mu_{\mathrm{A}} d \xi+\mu_{\mathrm{B}} d \xi=\left(\mu_{\mathrm{B}}-\mu_{\mathrm{A}}\right) d \xi$$

$$\left(\frac{\partial G}{\partial \xi}\right) p, T=\mu \mathrm{B}-\mu_{\mathrm{A}}$$

$$\Delta G_{\mathrm{r}}=\left(\frac{\partial G}{\partial \xi}\right) p, T=\mu \mathrm{B}-\mu_{\mathrm{A}}$$

$$\Delta G_{\mathrm{r}}=0$$

Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。