# 数学代写|信息论代写information theory代考|ELEG630

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## 数学代写|信息论代写information theory代考|Liquid Water

Water is known to be a structured liquid. However, there is no agreement on how to define the structure of water, see Ben-Naim [9, 10]. Look at Table 2.1, the entropy of vaporization of water is larger than the value expected from Trouton’s rule. Also, we see that the entropy of vaporization of heavy water $\left(\mathrm{D}_2 \mathrm{O}\right)$ is slightly larger than water $\left(\mathrm{H}_2 \mathrm{O}\right)$. This is consistent with the common view that heavy water is a more structured liquid than water. However, we can see in Table 2.1 that ethanol has almost the same entropy of vaporization as heavy water though it is difficult to claim that ethanol is more structured than either water or heavy water.

Tables of standard entropy are available for many liquids as well as for water and heavy water. It is not easy to compare values of standard entropies of different substances with different degrees of freedom such as vibration, rotation and electronic.

In this chapter we have interpreted the entropy values of a simple liquid in terms of the MI associated with the correlation functions which in turn is associated with the strength of the molecular interactions.

It is usually assumes that $\mathrm{H}_2 \mathrm{O}$ and $\mathrm{D}_2 \mathrm{O}$ have approximately the same internal degrees of freedom. It follows that the higher the entropy of vaporization of $\mathrm{D}_2 \mathrm{O}$ compared with $\mathrm{H}_2 \mathrm{O}$ is due to stronger intermolecular interactions. In this case the main part of the interactions is due to hydrogen bonding, see Ben-Naim [9, 10].
Another measure of the “structure” or the extent of intermolecular interactions in the liquid is the entropy of solvation. The solvation process is depicted in Fig. 2.10. A single solute molecule $(s)$ is transferred from a fixed position in an ideal gas phase into a fixed position in an ideal gas phase. Figure 2.11 shows some values of the self-solvation entropy of $\mathrm{H}_2 \mathrm{O}$ and $\mathrm{D}_2 \mathrm{O}$ at several temperatures (self-solvation is the process of solvation of a molecule in its own liquid). In all cases we see that $\Delta S^*$ of $\mathrm{D}_2 \mathrm{O}$ is more negative than the corresponding value of $\mathrm{H}_2 \mathrm{O}$.

It is tradition to interpret these values in terms of structural effects (or ordering). Within our interpretation of entropy as a special case of SMI we view the difference in the values of $\Delta S^*$ in $\mathrm{H}_2 \mathrm{O}$ and $\mathrm{D}_2 \mathrm{O}$ due to the stronger interaction between $\mathrm{D}_2 \mathrm{O}$ molecules compared with $\mathrm{H}_2 \mathrm{O}$ molecules.

In Appendix, we derive a relationship between the entropy of solvation of a solute $s$ in a solvent in terms of difference in SMI. In the next section we also discuss the solvation entropy of inert gases in water. Here however, we discuss the solvation entropy of $\mathrm{H}_2 \mathrm{O}$ in pure $\mathrm{H}_2 \mathrm{O}$, i.e. the “solute” is also a water molecule. This is sometimes called self-solvation, i.e. in Fig. 2.10 instead of a solute $s$ inserted in water, we insert a water molecule into pure water. If we do this process at constant temperature $T$ and volume $V$, the solvation entropy energy is given by:
$$\Delta S_w^*=\left(k_B \ln 2\right)\left[\operatorname{SMI}\left(N \mid R_s\right)-\operatorname{SMI}(N)\right]$$

## 数学代写|信息论代写information theory代考|Aqueous Solutions of Inert Gases

The thermodynamics of aqueous solutions of inert gases involves a few, very exciting and mysterious problems. We shall discuss in this section only one aspect of these systems; the solvation entropy of an inert solute, say, argon in water.

The solubilities of inert solutes such as argon, neon, methane and the like are very small. In the early 1930 s and 1940 s the data available on the solubility of these solutes in water was very inaccurate. It was known that the solubility of these solutes in water is much smaller than in other organic liquid.

The entropy of solvation (previously referred to as the entropy of solution) of these solutes could be obtained only from very accurate data on the solubility and its dependence on temperature. For more details, see Ben-Naim [11, 12]. It was known that the entropy of solvation of inert solutes in water is large and negative compared with the solvation entropy of the same solutes in typical organic solvents. A few examples are shown in Table 2.2.

In 1945, Frank and Evans [13] published a very influential article on the thermodynamics of solvation of inert solutes in water and in other liquids. They noted that the entropy of solvation of these solutes is much larger and negative in water as compared with the entropy of solvation of the same solutes in other liquids. To explain these findings, the authors conjectured that when an inert solute dissolves in water it forms, or builds some kind of structure, which the called “icebergs,” around it. This idea was revolutionary at that time. It has captured the imagination of many scientists for more than half a century. How can an inert solute, weakly interacting with water molecules, form an “iceberg”? Frank and Evans did not offer any proof that an inert solute builds up iceberg around it, nor did they provide any explanation as to why inert solute should form icebergs. All they did was to interpret the negative change in entropy in terms of increasing the order, or equivalently increasing the structure of water. Yet, this idea was not only accepted by, but used by many scientists to explain the entropy and the enthalpy of solvation of the non-polar solute in water.
The truth is that Frank and Evans did not contribute anything to understanding the entropy of solvation of inert gases in water. The last statement might be shocking to many chemists who believe that Frank and Evans actually solved the problem. Unfortunately, they did not. Entropy at that time was viewed (and still is) as a measure of the extent of order or disorder in the system. “Structure” is another word for order. Therefore, Frank and Evans suggestion was nothing but the translation of the experiment fact about the negative entropy of solvation into the language of order-disorder. Thus, negative $\Delta S_s^*$ is equivalent to more order, or more structure, or picturesquely formation of icebergs. For more details, see Ben-Naim [9].

# 信息论代写

## 数学代写|信息论代写information theory代考|Liquid Water

$$\Delta S_w^*=\left(k_B \ln 2\right)\left[\operatorname{SMI}\left(N \mid R_s\right)-\operatorname{SMI}(N)\right]$$

## 数学代写|信息论代写information theory代考|Aqueous Solutions of Inert Gases

1945年，Frank和Evans[13]发表了一篇非常有影响力的关于惰性溶质在水和其他液体中的溶剂化热力学的文章。他们注意到，与相同溶质在其他液体中的溶剂化熵相比，这些溶质在水中的溶剂化熵要大得多，而且是负的。为了解释这些发现，作者推测，当惰性溶质溶解在水中时，它会在其周围形成或形成某种结构，即所谓的“冰山”。这个想法在当时是革命性的。半个多世纪以来，它俘获了许多科学家的想象力。一个与水分子弱相互作用的惰性溶质是如何形成“冰山”的?弗兰克和埃文斯没有提供任何证据证明惰性溶质在其周围形成冰山，也没有提供任何解释为什么惰性溶质会形成冰山。他们所做的就是把熵的负变化解释为序的增加，或者等价地说，水的结构的增加。然而，这个想法不仅被许多科学家所接受，而且还被许多科学家用来解释水中非极性溶质的熵和溶剂化焓。

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## 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。

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