## 物理代写|热力学代写thermodynamics代考|How to Determine the Equilibrium Conditions

As we have already known that finding thermodynamic equilibrium conditions requires finding the maximum of the entropy function or the minimum of the energy functions. Such a process involves a function of several variables which may be related to each other by one or more constraint equations. For simple systems, these mathematical operations are straight forward as we have demonstrated in the previous chapter. However, for non-uniform systems, the total entropy and the total energy of such a system often involves integration of local properties, and the mathematical analysis to find the maximum of the entropy function or the minimum of the energy functions is relatively complex. Therefore, we need to introduce a method of undetermined Lagrange multipliers.

For example, let us consider minimizing a function $\mathrm{z}=x^2+y^2$ where the variables $x$ and $y$ are subject to a constraint equation: $x y-1=0$. An intuitive approach may be to solve the constraint equation first to get:
$$y=1 / x$$
Then substituting $y$ with this equation into $\mathrm{z}=x^2+y^2$ yields:
$$\mathrm{z}=x^2+1 / x^2 .$$
Differentiating this equation and setting it to zero (i.e., condition required for minimum) gives:
$$\frac{d z}{d x}=2 x-\frac{2}{x^3}=0$$
Solving this equation, one can find the minimum positions are $(+1,+1)$ and $(-1$, $-1)$

However, the approach used in the above example may not be practical for some more complicated equations. A more general approach is the method of undetermined Lagrange multipliers, as outlined below.

## 物理代写|热力学代写thermodynamics代考|Introduction to Interfaces and Three-Phase Contact Lines

Up to now, all the thermodynamics theories we have discussed are the theories for simple three dimensional (3D) bulk phase systems. The size of such a system is measured by volume, and the mechanical work mode for these systems is $P d V$ type of work. One of the conditions for simple systems is that effects of the boundaries of the bulk phases on the equilibrium states are not considered. However, in a broad spectrum of applied science and engineering applications, the bulk phase boundaries, such as an interface between a liquid droplet and its vapor phase, play important roles in determination of the equilibrium states of the system. Examples where surface or interface effects are important include (bubble, droplet or ice) nucleation processes, bubble flotation processes used in mineral and oil processing, two-phase (liquid-gas or water-oil) transport phenomena in a porous medium, etc.

The boundaries between immiscible bulk phases include interfaces and threephase contact lines. An interface or a surface is the boundary between two immiscible bulk phases. A three-phase contact line is the mutual boundary of three immiscible bulk phases or three surfaces.

When do we have to consider the effects of these boundaries? Generally, when the dimension of the system is small enough and the energy associated with the surfaces and lines is comparable to the energy associated with the volume of the bulk phase, the effects of the surfaces and three-phase contact lines must be considered.

It should be realized that the systems with important surface or line effects are not the “simple” systems as we defined before. In this chapter, we will show how to apply the general thermodynamic theory we learned before to the two-dimensional (2D) surfaces and the one-dimensional (1D) lines. The objectives in this chapter are to learn.
(1) How to apply the thermodynamic theory to model systems involving surfaces and lines, and
(2) How these boundaries will affect the equilibrium conditions.

# 热力学代考

## 物理代写|热力学代写thermodynamics代考|How to Determine the Equilibrium Conditions

$$y=1 / x$$

$$\mathrm{z}=x^2+1 / x^2$$

$$\frac{d z}{d x}=2 x-\frac{2}{x^3}=0$$

## 物理代写|热力学代写thermodynamics代考|Introduction to Interfaces and Three-Phase Contact Lines

(1) 如何将热力学理论应用于涉及面和线的模型系统，以及
(2) 这些边界将如何影响平衡条件。

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