# 最小能量原理

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The **principle of minimum energy** is essentially a restatement of the second law of thermodynamics. It states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. External parameters generally means the volume, but may include other parameters which are specified externally, such as a constant magnetic field.

The principle of minimum energy is essentially a restatement of the second law of thermodynamics. It states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. External parameters generally means the volume, but may include other parameters which are specified externally, such as a constant magnetic field.

最小能量原理实质上是热力学第二定律的重述。它指出，对于外部参数和熵不变的封闭系统，内部能量将减少，并在平衡时达到最小值。外部参数通常指体积，但也可能包括外部指定的其他参数，如恒定磁场。

In contrast, for isolated systems (and fixed external parameters), the second law states that the entropy will increase to a maximum value at equilibrium. An isolated system has a fixed total energy and mass. A closed system, on the other hand, is a system which is connected to another, and cannot exchange matter (i.e. particles), but other forms of energy (e.g. heat), with the other system. If, rather than an isolated system, we have a closed system, in which the entropy rather than the energy remains constant, then it follows from the first and second laws of thermodynamics that the energy of that system will drop to a minimum value at equilibrium, transferring its energy to the other system. To restate:

In contrast, for isolated systems (and fixed external parameters), the second law states that the entropy will increase to a maximum value at equilibrium. An isolated system has a fixed total energy and mass. A closed system, on the other hand, is a system which is connected to another, and cannot exchange matter (i.e. particles), but other forms of energy (e.g. heat), with the other system. If, rather than an isolated system, we have a closed system, in which the entropy rather than the energy remains constant, then it follows from the first and second laws of thermodynamics that the energy of that system will drop to a minimum value at equilibrium, transferring its energy to the other system. To restate:

相比之下，对于孤立系统(和固定的外部参数) ，第二定律指出，熵将增加到平衡时的最大值。孤立系统的总能量和总质量是固定的。另一方面，一个封闭的系统是一个与另一个系统相连接的系统，它不能交换物质(即:。粒子) ，但其他形式的能量(例如:。热) ，与其他系统。如果，而不是一个孤立的系统，我们有一个封闭的系统，其中熵而不是能量保持不变，那么它从第一个和第二个热力学定律，该系统的能量将下降到平衡的最小值，将其能量转移到其他系统。重申:

- The maximum entropy principle: For a closed system with fixed internal
**energy**(i.e. an isolated system), the**entropy**is maximized at equilibrium. - The minimum energy principle: For a closed system with fixed
**entropy**, the total**energy**is minimized at equilibrium.

- The maximum entropy principle: For a closed system with fixed internal energy (i.e. an isolated system), the entropy is maximized at equilibrium.
- The minimum energy principle: For a closed system with fixed entropy, the total energy is minimized at equilibrium.

- 最大熵原理: 对于具有固定内能的封闭系统(即:。一个孤立的系统) ，熵在平衡时最大化。
- 最小能量原理: 对于具有固定熵的封闭系统，总能量在平衡时最小。

## Mathematical explanation

## Mathematical explanation

# = 数学解释 =

The total energy of the system is [math]\displaystyle{ U(S,X_1,X_2,\dots) }[/math] where *S* is entropy, and the [math]\displaystyle{ X_i }[/math] are the other extensive parameters of the system (e.g. volume, particle number, etc.). The entropy of the system may likewise be written as a function of the other extensive parameters as [math]\displaystyle{ S(U,X_1,X_2,...) }[/math]. Suppose that *X* is one of the [math]\displaystyle{ X_i }[/math] which varies as a system approaches equilibrium, and that it is the only such parameter which is varying. The principle of maximum entropy may then be stated as:

The total energy of the system is U(S,X_1,X_2,\dots) where S is entropy, and the X_i are the other extensive parameters of the system (e.g. volume, particle number, etc.). The entropy of the system may likewise be written as a function of the other extensive parameters as S(U,X_1,X_2,...). Suppose that X is one of the X_i which varies as a system approaches equilibrium, and that it is the only such parameter which is varying. The principle of maximum entropy may then be stated as:

系统的总能量为 u (s，x _ 1，x _ 2，点) ，其中 s 是熵，x _ i 是系统的其他扩展参数(例如:。体积、粒子数等)。系统的熵同样可以写成其他广义参数的函数，如 s (u，x _ 1，x _ 2，...)。假设 x 是 x _ i 中的一个，随着系统趋于平衡而变化，这是唯一一个变化的参数。最大熵原理可以这样写:

- [math]\displaystyle{ \left(\frac{\partial S}{\partial X}\right)_U=0 }[/math] and [math]\displaystyle{ \left(\frac{\partial ^2S}{\partial X^2}\right)_U \lt 0 }[/math] at equilibrium.

- \left(\frac{\partial S}{\partial X}\right)_U=0 and \left(\frac{\partial ^2S}{\partial X^2}\right)_U < 0 at equilibrium.

- 左(frac { partial s }{ partial x } right) _ u = 0，左(frac { partial ^ 2S }{ partial x ^ 2} right) _ u < 0。

The first condition states that entropy is at an extremum, and the second condition states that entropy is at a maximum. Note that for the partial derivatives, all extensive parameters are assumed constant except for the variables contained in the partial derivative, but only *U*, *S*, or *X* are shown. It follows from the properties of an exact differential (see equation 8 in the exact differential article) and from the energy/entropy equation of state that, for a closed system:

The first condition states that entropy is at an extremum, and the second condition states that entropy is at a maximum. Note that for the partial derivatives, all extensive parameters are assumed constant except for the variables contained in the partial derivative, but only U, S, or X are shown. It follows from the properties of an exact differential (see equation 8 in the exact differential article) and from the energy/entropy equation of state that, for a closed system:

第一个条件表明熵处于极值状态，第二个条件表明熵处于极值状态。注意，对于偏导数，除了偏导数中包含的变量外，所有的扩展参数都假定为常数，但是只显示 u、 s 或 x。它来自于一个精确微分的性质(见精确微分的文章中的方程8)和能量/熵的状态方程，对于一个封闭的系统:

- [math]\displaystyle{ \left(\frac{\partial U}{\partial X}\right)_S = -\,\frac{\left(\frac{\partial S}{\partial X}\right)_U}{\left(\frac{\partial S}{\partial U}\right)_X} =-T\left(\frac{\partial S}{\partial X}\right)_U = 0 }[/math]

- \left(\frac{\partial U}{\partial X}\right)_S = -\,\frac{\left(\frac{\partial S}{\partial X}\right)_U}{\left(\frac{\partial S}{\partial U}\right)_X}

=-T\left(\frac{\partial S}{\partial X}\right)_U = 0

- 左(frac { partial u }{ partial x } right) _ s =-，frac { left (frac { partial s }{ partial x } right) _ u }{ left (frac { partial u } right) _ x } =-t left (frac { partial s }{ partial x } right) _ u = 0

It is seen that the energy is at an extremum at equilibrium. By similar but somewhat more lengthy argument it can be shown that

It is seen that the energy is at an extremum at equilibrium. By similar but somewhat more lengthy argument it can be shown that

可以看出，能量处于极值的平衡状态。通过类似但稍微长一点的论证，我们可以证明

- [math]\displaystyle{ \left(\frac{\partial^2U}{\partial X^2}\right)_S=-T\left(\frac{\partial^2S}{\partial X^2}\right)_U }[/math]

- \left(\frac{\partial^2U}{\partial X^2}\right)_S=-T\left(\frac{\partial^2S}{\partial X^2}\right)_U

- left (frac { partial ^ 2U }{ partial x ^ 2} right) _ s =-t left (frac { partial ^ 2S }{ partial x ^ 2} right) _ u

which is greater than zero, showing that the energy is, in fact, at a minimum.

which is greater than zero, showing that the energy is, in fact, at a minimum.

大于零，表明能量实际上是最小的。

## Examples

## Examples

# = 实例 =

Consider, for one, the familiar example of a marble on the edge of a bowl. If we consider the marble and bowl to be an isolated system, then when the marble drops, the potential energy will be converted to the kinetic energy of motion of the marble. Frictional forces will convert this kinetic energy to heat, and at equilibrium, the marble will be at rest at the bottom of the bowl, and the marble and the bowl will be at a slightly higher temperature. The total energy of the marble-bowl system will be unchanged. What was previously the potential energy of the marble, will now reside in the increased heat energy of the marble-bowl system. This will be an application of the maximum entropy principle as set forth in the principle of minimum potential energy, since due to the heating effects, the entropy has increased to the maximum value possible given the fixed energy of the system.

Consider, for one, the familiar example of a marble on the edge of a bowl. If we consider the marble and bowl to be an isolated system, then when the marble drops, the potential energy will be converted to the kinetic energy of motion of the marble. Frictional forces will convert this kinetic energy to heat, and at equilibrium, the marble will be at rest at the bottom of the bowl, and the marble and the bowl will be at a slightly higher temperature. The total energy of the marble-bowl system will be unchanged. What was previously the potential energy of the marble, will now reside in the increased heat energy of the marble-bowl system. This will be an application of the maximum entropy principle as set forth in the principle of minimum potential energy, since due to the heating effects, the entropy has increased to the maximum value possible given the fixed energy of the system.

比如，我们熟悉的一个例子是碗边的大理石。如果我们把弹珠和碗看作是一个孤立的系统，那么当弹珠下落时，势能就转化为弹珠运动的动能。摩擦力将这种动能转化为热能，达到平衡时，大理石将在碗底处于静止状态，大理石和碗的温度将略高一些。大理石碗系统的总能量不变。以前大理石的势能，现在将存在于大理石碗系统增加的热能中。这将是最小势能原理中所阐述的最大熵原理的应用，因为由于热效应，熵增加到给定系统固定能量的可能的最大值。

If, on the other hand, the marble is lowered very slowly to the bottom of the bowl, so slowly that no heating effects occur (i.e. reversibly), then the entropy of the marble and bowl will remain constant, and the potential energy of the marble will be transferred as energy to the surroundings. The surroundings will maximize its entropy given its newly acquired energy, which is equivalent to the energy having been transferred as heat. Since the potential energy of the system is now at a minimum with no increase in the energy due to heat of either the marble or the bowl, the total energy of the system is at a minimum. This is an application of the minimum energy principle.

If, on the other hand, the marble is lowered very slowly to the bottom of the bowl, so slowly that no heating effects occur (i.e. reversibly), then the entropy of the marble and bowl will remain constant, and the potential energy of the marble will be transferred as energy to the surroundings. The surroundings will maximize its entropy given its newly acquired energy, which is equivalent to the energy having been transferred as heat. Since the potential energy of the system is now at a minimum with no increase in the energy due to heat of either the marble or the bowl, the total energy of the system is at a minimum. This is an application of the minimum energy principle.

另一方面，如果大理石缓慢地下降到碗底部，以至于不会产生加热效果(即。可逆) ，那么大理石和碗的熵将保持不变，大理石的势能将作为能量转移到周围。周围的环境会最大化它的熵，给予它新获得的能量，这相当于作为热量传递的能量。由于系统的势能现在处于最小值，而且不会因大理石或碗的热量而增加能量，因此系统的总能量处于最小值。这是最小能量原理的应用。

Alternatively, suppose we have a cylinder containing an ideal gas, with cross sectional area *A* and a variable height *x*. Suppose that a weight of mass *m* has been placed on top of the cylinder. It presses down on the top of the cylinder with a force of *mg* where *g* is the acceleration due to gravity.

Alternatively, suppose we have a cylinder containing an ideal gas, with cross sectional area A and a variable height x. Suppose that a weight of mass m has been placed on top of the cylinder. It presses down on the top of the cylinder with a force of mg where g is the acceleration due to gravity.

或者，假设我们有一个圆柱体，里面有一个理想气体，横截面积为 a，高度为 x。假设质量为 m 的物体被放置在圆柱体的顶部。它以毫克的力向圆柱体顶部施压，其中 g 是重力作用下的加速度。

Suppose that *x* is smaller than its equilibrium value. The upward force of the gas is greater than the downward force of the weight, and if allowed to freely move, the gas in the cylinder would push the weight upward rapidly, and there would be frictional forces that would convert the energy to heat. If we specify that an external agent presses down on the weight so as to very slowly (reversibly) allow the weight to move upward to its equilibrium position, then there will be no heat generated and the entropy of the system will remain constant while energy is transferred as work to the external agent. The total energy of the system at any value of *x* is given by the internal energy of the gas plus the potential energy of the weight:

Suppose that x is smaller than its equilibrium value. The upward force of the gas is greater than the downward force of the weight, and if allowed to freely move, the gas in the cylinder would push the weight upward rapidly, and there would be frictional forces that would convert the energy to heat. If we specify that an external agent presses down on the weight so as to very slowly (reversibly) allow the weight to move upward to its equilibrium position, then there will be no heat generated and the entropy of the system will remain constant while energy is transferred as work to the external agent. The total energy of the system at any value of x is given by the internal energy of the gas plus the potential energy of the weight:

假设 x 小于它的平衡值。气体向上的力大于重量向下的力，如果允许气体自由运动，气缸中的气体会将重量迅速向上推，并且会有摩擦力将能量转化为热量。如果我们指定一个外部介质压在这个重量上，以至于非常缓慢地(可逆地)允许这个重量向上移动到它的平衡位置，那么就不会产生热量，当能量作为功转移到外部介质时，系统的熵将保持不变。系统的总能量在 x 的任意值都是由气体的内能加上重量的势能得到的:

- [math]\displaystyle{ U=TS-PAx+\mu N+mgx\, }[/math]

- U=TS-PAx+\mu N+mgx\,

- u = TS-PAx + mu n + mgx,

where *T* is temperature, *S* is entropy, *P* is pressure, μ is the chemical potential, *N* is the number of particles in the gas, and the volume has been written as *V=Ax*. Since the system is closed, the particle number *N* is constant and a small change in the energy of the system would be given by:

where T is temperature, S is entropy, P is pressure, μ is the chemical potential, N is the number of particles in the gas, and the volume has been written as V=Ax. Since the system is closed, the particle number N is constant and a small change in the energy of the system would be given by:

其中 t 是温度，s 是熵，p 是压强，μ 是化学势，n 是气体中粒子的数量，体积被写成 v = Ax。由于系统是封闭的，粒子数 n 是常数，系统能量的一个小变化可以通过以下方式给出:

- [math]\displaystyle{ dU = T\,dS-PA\,dx+mg\,dx }[/math]

- dU = T\,dS-PA\,dx+mg\,dx

- dU = t，dS-PA，dx + mg，dx

Since the entropy is constant, we may say that *dS*=0 at equilibrium and by the principle of minimum energy, we may say that *dU*=0 at equilibrium, yielding the equilibrium condition:

Since the entropy is constant, we may say that dS=0 at equilibrium and by the principle of minimum energy, we may say that dU=0 at equilibrium, yielding the equilibrium condition:

由于熵是常数，我们可以说平衡态下的 dS = 0，根据最小能量原理，我们可以说平衡态下的 dU = 0，产生了平衡条件:

- [math]\displaystyle{ 0=-PA+mg\, }[/math]

- 0=-PA+mg\,

- 0 =-PA + mg,

which simply states that the upward gas pressure force (*PA*) on the upper face of the cylinder is equal to the downward force of the mass due to gravitation (*mg*).

which simply states that the upward gas pressure force (PA) on the upper face of the cylinder is equal to the downward force of the mass due to gravitation (mg).

它简单地说明了气体压力(PA)在圆筒上面的上升力等于质量的下降力由于重力(毫克)。

## Thermodynamic potentials

## Thermodynamic potentials

# = 热力学势 =

The principle of minimum energy can be generalized to apply to constraints other than fixed entropy. For other constraints, other state functions with dimensions of energy will be minimized. These state functions are known as thermodynamic potentials. Thermodynamic potentials are at first glance just simple algebraic combinations of the energy terms in the expression for the internal energy. For a simple, multicomponent system, the internal energy may be written:

The principle of minimum energy can be generalized to apply to constraints other than fixed entropy. For other constraints, other state functions with dimensions of energy will be minimized. These state functions are known as thermodynamic potentials. Thermodynamic potentials are at first glance just simple algebraic combinations of the energy terms in the expression for the internal energy. For a simple, multicomponent system, the internal energy may be written:

最小能量原理可以推广到除固定熵之外的约束条件。对于其他约束条件，其他具有能量维数的状态函数将被最小化。这些状态函数称为热力学势。热力学势初看起来只是内能表达式中能量项的简单代数组合。对于一个简单的多组分系统，内能可以写成:

- [math]\displaystyle{ U(S,V,\{N_j\})=TS-PV+\sum_j\mu_jN_j\, }[/math]

- U(S,V,\{N_j\})=TS-PV+\sum_j\mu_jN_j\,

- U(S,V,\{N_j\})=TS-PV+\sum_j\mu_jN_j\,

where the intensive parameters (T, P, μ_{j}) are functions of the internal energy's natural variables [math]\displaystyle{ (S,V,\{N_j\}) }[/math] via the equations of state. As an example of another thermodynamic potential, the Helmholtz free energy is written:

where the intensive parameters (T, P, μj) are functions of the internal energy's natural variables (S,V,\{N_j\}) via the equations of state. As an example of another thermodynamic potential, the Helmholtz free energy is written:

其中强度参数(t，p，μj)是通过状态方程得到的内能自然变量(s，v，{ n _ j })的函数。作为另一个热动力位能的例子，《亥姆霍兹自由能写道:

- [math]\displaystyle{ A(T,V,\{N_j\})=U-TS\, }[/math]

- A(T,V,\{N_j\})=U-TS\,

- a (t，v，{ n _ j }) = U-TS,

where temperature has replaced entropy as a natural variable. In order to understand the value of the thermodynamic potentials, it is necessary to view them in a different light. They may in fact be seen as (negative) Legendre transforms of the internal energy, in which certain of the extensive parameters are replaced by the derivative of internal energy with respect to that variable (i.e. the conjugate to that variable). For example, the Helmholtz free energy may be written:

where temperature has replaced entropy as a natural variable. In order to understand the value of the thermodynamic potentials, it is necessary to view them in a different light. They may in fact be seen as (negative) Legendre transforms of the internal energy, in which certain of the extensive parameters are replaced by the derivative of internal energy with respect to that variable (i.e. the conjugate to that variable). For example, the Helmholtz free energy may be written:

温度已经取代熵成为一个自然变量。为了理解热力学势的值，有必要从不同的角度来看待它们。它们实际上可以被看作是内能的(负的) Legendre 变换，其中某些广泛的参数被相对于该变量的内能的导数所取代。该变量的变形)。例如，亥姆霍兹自由能可以这样写:

- [math]\displaystyle{ A(T,V,\{N_j\})=\underset{S}\mathrm{min}(U(S,V,\{N_j\})-TS)\, }[/math]

- A(T,V,\{N_j\})=\underset{S}\mathrm{min}(U(S,V,\{N_j\})-TS)\,

- a (t，v，{ n _ j }) = 下集{ s } mathrm { min }(u (s，v，{ n _ j })-TS) ,

and the minimum will occur when the variable *T* becomes equal to the temperature since

and the minimum will occur when the variable T becomes equal to the temperature since

当变量 t 等于自年以来的温度时，最小值就会出现

- [math]\displaystyle{ T=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_j\}} }[/math]

- T=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_j\}}

- t = left (frac { partial u }{ partial s } right) _ { v，{ n _ j }

The Helmholtz free energy is a useful quantity when studying thermodynamic transformations in which the temperature is held constant. Although the reduction in the number of variables is a useful simplification, the main advantage comes from the fact that the Helmholtz free energy is minimized at equilibrium with respect to any unconstrained internal variables for a closed system at constant temperature and volume. This follows directly from the principle of minimum energy which states that at constant entropy, the internal energy is minimized. This can be stated as:

The Helmholtz free energy is a useful quantity when studying thermodynamic transformations in which the temperature is held constant. Although the reduction in the number of variables is a useful simplification, the main advantage comes from the fact that the Helmholtz free energy is minimized at equilibrium with respect to any unconstrained internal variables for a closed system at constant temperature and volume. This follows directly from the principle of minimum energy which states that at constant entropy, the internal energy is minimized. This can be stated as:

在研究温度保持不变的热力学变换时，亥姆霍兹自由能是一个有用的量。虽然减少变量的数量是一个有用的简化，主要的优势来自于这样一个事实，即对于一个封闭系统在恒定温度和体积下的任何无约束的内部变量，亥姆霍兹自由能在平衡状态下是最小的。这直接遵循最小能量原理，即在恒定熵下，内能最小化。这可以表述为:

- [math]\displaystyle{ U_0(S_0)=\underset{x}\mathrm{min}(U(S_0,x))\, }[/math]

- U_0(S_0)=\underset{x}\mathrm{min}(U(S_0,x))\,

- u _ 0(s _ 0) = 底集{ x } mathrm { min }(u (s _ 0，x)) ,

where [math]\displaystyle{ U_0 }[/math] and [math]\displaystyle{ S_0 }[/math] are the value of the internal energy and the (fixed) entropy at equilibrium. The volume and particle number variables have been replaced by *x* which stands for any internal unconstrained variables.

As a concrete example of unconstrained internal variables, we might have a chemical reaction in which there are two types of particle, an *A* atom and an *A _{2}* molecule. If [math]\displaystyle{ N_1 }[/math] and [math]\displaystyle{ N_2 }[/math] are the respective particle numbers for these particles, then the internal constraint is that the total number of

*A*atoms [math]\displaystyle{ N_A }[/math] is conserved:

- [math]\displaystyle{ N_A=N_1+2N_2\, }[/math]

where U_0 and S_0 are the value of the internal energy and the (fixed) entropy at equilibrium. The volume and particle number variables have been replaced by x which stands for any internal unconstrained variables.

As a concrete example of unconstrained internal variables, we might have a chemical reaction in which there are two types of particle, an A atom and an A2 molecule. If N_1 and N_2 are the respective particle numbers for these particles, then the internal constraint is that the total number of A atoms N_A is conserved:

- N_A=N_1+2N_2\,

其中 u _ 0和 s _ 0是平衡态内能和(固定)熵的值。体积和粒子数量变量已经被代表任何内部无约束变量的 x 所代替。作为无约束内变量的一个具体例子，我们可能有一个化学反应，其中有两种类型的粒子，一个 a 原子和一个 a2分子。如果 n _ 1和 n _ 2分别是这些粒子的粒子数，那么内在的约束是 a 原子的总数 n _ a 守恒: : n _ a = n _ 1 + 2 n _ 2,

we may then replace the [math]\displaystyle{ N_1 }[/math] and [math]\displaystyle{ N_2 }[/math] variables with a single variable [math]\displaystyle{ x=N_1/N_2 }[/math] and minimize with respect to this unconstrained variable. There may be any number of unconstrained variables depending on the number of atoms in the mixture. For systems with multiple sub-volumes, there may be additional volume constraints as well.

The minimization is with respect to the unconstrained variables. In the case of chemical reactions this is usually the number of particles or mole fractions, subject to the conservation of elements. At equilibrium, these will take on their equilibrium values, and the internal energy [math]\displaystyle{ U_0 }[/math] will be a function only of the chosen value of entropy [math]\displaystyle{ S_0 }[/math]. By the definition of the Legendre transform, the Helmholtz free energy will be:

we may then replace the N_1 and N_2 variables with a single variable x=N_1/N_2 and minimize with respect to this unconstrained variable. There may be any number of unconstrained variables depending on the number of atoms in the mixture. For systems with multiple sub-volumes, there may be additional volume constraints as well.

The minimization is with respect to the unconstrained variables. In the case of chemical reactions this is usually the number of particles or mole fractions, subject to the conservation of elements. At equilibrium, these will take on their equilibrium values, and the internal energy U_0 will be a function only of the chosen value of entropy S_0. By the definition of the Legendre transform, the Helmholtz free energy will be:

然后我们可以用一个单变量 x = n1/n2来代替 n1和 n2变量，并且对这个无约束变量最小化。根据混合物中原子的数量，可能存在任意数量的无约束变量。对于具有多个子卷的系统，可能还有额外的卷约束。 最小化是关于无约束变量的。在化学反应的情况下，通常是粒子或摩尔分数的数量，受到元素守恒的制约。在平衡状态下，它们将取平衡值，内能 u _ 0将只是选择的熵 s _ 0的函数。根据勒让德变换的定义，亥姆霍兹自由能变换将是:

- [math]\displaystyle{ A(T,x)=\underset{S}\mathrm{min}(U(S,x)-TS)\, }[/math]

- A(T,x)=\underset{S}\mathrm{min}(U(S,x)-TS)\,

- a (t，x) = 下集{ s } mathrm { min }(u (s，x)-TS) ,

The Helmholtz free energy at equilibrium will be:

The Helmholtz free energy at equilibrium will be:

处于均衡状态的亥姆霍兹自由能将是:

- [math]\displaystyle{ A_0(T_0)=\underset{S_0}\mathrm{min}(U_0(S_0)-T_0S_0) }[/math]

- A_0(T_0)=\underset{S_0}\mathrm{min}(U_0(S_0)-T_0S_0)

- a _ 0(t _ 0) = 低于{ s _ 0}最小值(u _ 0(s _ 0)-t _ 0s _ 0)

where [math]\displaystyle{ T_0 }[/math] is the (unknown) temperature at equilibrium. Substituting the expression for [math]\displaystyle{ U_0 }[/math]:

where T_0 is the (unknown) temperature at equilibrium. Substituting the expression for U_0:

其中 t0是(未知的)平衡温度。用 u _ 0代替表达式:

- [math]\displaystyle{ A_0=\underset{S_0}\mathrm{min}(\underset{x}\mathrm{min}(U(S_0,x))-T_0S_0) }[/math]

- A_0=\underset{S_0}\mathrm{min}(\underset{x}\mathrm{min}(U(S_0,x))-T_0S_0)

- a _ 0 = 逆集{ s _ 0} mathrm { min }(逆集{ x } mathrm { min }(u (s _ 0，x))-t _ 0s _ 0)

By exchanging the order of the extrema:

By exchanging the order of the extrema:

通过交换极端物品的顺序:

- [math]\displaystyle{ A_0=\underset{x}\mathrm{min}(\underset{S_0}\mathrm{min}(U(S_0,x)-T_0S_0)) = \underset{x}\mathrm{min}(A_0(T_0,x)) }[/math]

- A_0=\underset{x}\mathrm{min}(\underset{S_0}\mathrm{min}(U(S_0,x)-T_0S_0)) =

\underset{x}\mathrm{min}(A_0(T_0,x))

- A_0=\underset{x}\mathrm{min}(\underset{S_0}\mathrm{min}(U(S_0,x)-T_0S_0)) =

\underset{x}\mathrm{min}(A_0(T_0,x))

showing that the Helmholtz free energy is minimized at equilibrium.

showing that the Helmholtz free energy is minimized at equilibrium.

显示亥姆霍兹自由能在平衡状态下最小化。

The Enthalpy and Gibbs free energy, are similarly derived.

The Enthalpy and Gibbs free energy, are similarly derived.

焓和吉布斯自由能的计算也是类似的。

## References

- Callen, Herbert B. (1985).
*Thermodynamics and an Introduction to Thermostatistics*(2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8. OCLC 485487601.

## References

Category:Thermodynamics

分类: 热力学

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