换句话说，存在每个分子的自由能。
所以这就是吉布斯自由能的变量。
我们回到吉布斯自由能的表达式。
化学势是一摩尔物质的吉布斯自由能。
有谁记得？吉布斯自由能？
亥姆霍兹自由能的一个比较简单的形式。
对吉布斯自由能是压强，温度和粒子数。
For the Gibbs free energy it was the pressure and the temperature, the number of particles.
就得到了吉布斯自由能。
整个的自由能是它乘以。
所以我们需要知道,就是各种物质的自由能。
So we need to know the free energy of each one of the species.
自由能随半径的增大而单调地增加。
The free energy increases monotonically with increasing radius.
神的主权和人的自由能相容吗？
蓄在水闸内的水具有自由能。
我们知道吉布斯自由能，等于化学势之和，对吗？
And I know the Gibbs free energy is just the sum of the chemical potentials, right?
现在要进行微分，对除以温度之后的吉布斯自由能。
OK, now I have this derivative of the Gibbs free energy divided by the temperature.
每摩尔·吉布斯自由能，就是汽化的吉布斯自由能。
Per mole, which is just the Gibbs free energy of vaporization.
现在，化学势，就像我们看到的，等于吉布斯自由能。
Now, the chemical potential, as we saw, was the Gibbs free energy.
吉布斯自由能，注意，我们还可以这样写，重写一下。
The Gibbs free energy. Notice, we could also write, let's rewrite that.
所以我可以很直接的说，这就是我的亥姆赫兹自由能。
So I can in a very straightforward way say, OK, well, here is my Helmholtz free energy.
若没有自由能怎么办?。
液相中的化学势就是，液相中每摩尔a的吉布斯自由能。
气相中的化学势，就是气相中每摩尔a的吉布斯自由能。
And this is just the Gibbs free energy per mole of a in the gas phase.
现在我想，更进一步，看看化学势，和自由能的表达式。
Now what I'd like to do is go a little bit further and just look at expressions for the chemical potential and the free energies.
对压强求导，你知道吉布斯自由能，的变量是压强和温度？
With respect to pressure, when we know that the variables that we use for Gibbs free energy are pressure and temperature. Right?
现在我们知道怎么，算自由能，通过统计力学计。
And now we know how to calculate that from first principles, through statistical mechanics.
吉布斯自由能和化学势的变化，一切都，由这个子系统决定。
Changes in Gibbs free energy, changes in the chemical potential. Everything will be related to this partition function. This subsystem.
我们能够说的是，如果我们有一份混合物，自由能怎么变化？
What that's going to allow us to do is, if we say, OK I have a mixture of stuff, how does the free energy change?
这个方程把吉布斯自由能中的温度变化，和焓的变化联系起来。
And it relates the temperature change in the Gibbs free energy with the enthalpy change.
让我们通过计算出，混合物中自由能的变化，使问题更加清楚。
Let's see that a little more explicitly by just calculating out the free energy change of mixing.
这告诉我们的是，我们只有化学势，每个分子的亥姆霍兹自由能。
What this is telling us is that we just have a chemical potential, of Helmholtz free energy per molecule.
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