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So now we have a relationship between the ratios of these volumes that are reached during these adiabatic paths.
现在我们有了一个联系,这些绝热过程中,体积比的关系式。
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These two relations involving entropy are also useful because they'll let us see how entropy depends on volume and pressure.
这两个涉及熵的关系也非常有用,因为他们告诉我们,熵和体积，压强的关系。
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I'm going to say, quite to the contrary, the positive charge is concentrated at the center in a tiny, tiny, tiny volume.
我要说的是，完全相反，正电荷集中在中心,在一个非常非常小的体积内。
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So the probability of having an electron at the nucleus in terms of probability per volume is very, very high.
在单位体积内发现,一个电子的概率非常非常大。
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So, now you have a single molecule, very large molecule, with not just two binding sites but with ten binding sites.
所以如果你体内有一个细胞,一个体积很大的细胞,细胞表面不只有两个抗原结合位点,而有十个抗原结合位点
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Not to harp on the mathematical features of this, but cubing, AX*X*X you know, if you're starting to do AX star, X star, X, every time you want to cube some value in a program, it just feels like this is going to get a little messy looking, if nothing else.
不要总是说这个的数学特性，但是体积，你们懂的，如果你开始做，在一个程序中，每次你想算几个数值的体积，感觉它就变得,有一点凌乱的，如果没有其他的。
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In other words, if we don't have to worry about entropy or volume equilibrium is achieved when energy is at a minimum.
换句话说，如果我们不担心熵,和体积的平衡,那么能量就得是最小的。
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On the other hand, temperature, volume and pressure are variables that are much easier in the lab to keep constant.
另一方面，温度，体积和压强,在实验室中比较容易保持恒定。
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And we combine this with first law, which for the case of pressure volume changes we write as this.
结合第一和第二定律,对于压强体积功我们可以这样写。
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But remember that we need to multiply it by the volume here, the volume of some sphere we've defined.
但记住我们需要把它乘以体积，乘以一个我们定义的壳层的体积。
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A cubic kilometer of seawater, go into the ocean and imagine a box .
一千立方米的海水，想象一下一个体积为一千立方米的盒子。
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The important point here is it's not just a probability, it's a density, so we know that it's a probability divided by volume.
它不是概率,而是概率密度，所以我们知道，它是密，除以体积，我们。
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OK, now what we'd like to do is be able to calculate any of these quantities in terms of temperature, pressure, volume properties.
现在我们想要做的是能够利用,温度，压强和体积的性质,计算上面的物理量。
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It tells you what kind of molecule it is andgives you twovariables that are state variables You could have the volume and the temperature.
告诉你它是哪种分子,还给你了两个状态变量,它们可以是体积或温度。
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Because we did work at constant pressure, and so it's just volume difference times pressure.
因为是在恒压下做功，所以功就等于体积变化乘以压力。
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And the equation of state, pressure versus volume at constant temperature, is going to have some form, let's just draw it in there like that.
系统的态函数，恒温下压强比体积，变化曲线，就像这样。
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So, all I want to do now is look at the derivatives of the free energies with respect to temperature and volume and pressure.
我现在所要做的一切就是,考察自由能对,温度，体积和压强的偏导数。
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I don't need to tell you the volume here, because you've got enough information to calculate the volume.
这里我不需要告诉你体积，因为你已经获得了足够多的信息,来计算体积。
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OK, now, we're going to look at the internal energy, and we're going to pretend that it is explicitly a function of temperature and volume.
好，我们接下来看看内能，我们假设,它是温度和体积的函数。
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This is an example where the external pressure here is kept fixed as the volume changes, but it doesn't have to be kept fixed.
在我们举的这个例子中,外界压强不变，气体体积改变。
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And this volume, temperature and pressure doesn't care how you got there. It is what it is.
另一个状态,也有一组确定的体积。
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We know how the volume and temperature vary with respect to each other at constant pressure.
知道在恒定压强下,体积如何随着温度变化。
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V2 So, if one goes to two and V1 goes to V2, and it's constant temperature, just what we've specified there.
如果状态1变到状态1并且体积从V1到，同时温度保持不变，这就是我们这里要讨论的问题。
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and final points, a relationship between the temperature and volume for the initial and final points.
我们就得到了,初末态的温度,和体积间的关系。
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Then the second derivative gives the change in entropy with respect to the variable that we're differentiating, with respect to which is either pressure or volume.
二阶导数给出熵,随着变量变化的情况，这些变量包括压强或者体积。
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So this is still adiabatic. It's insulated, but now it's constant volume, OK.
这仍然是绝热的，是隔热的，但现在它的体积是恒定的。
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It's now, all we have to do is say we're going to have heat at constant volume.
我们需要做的就是,计算恒定体积下的热量。
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But it's allowed to say the internal energy is a function of temperature and volume.
但是我们也可以说内能,是温度和体积的函数。
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dw=0 Well for constant volume, dw is equal to zero.
约束是恒定体积，此时。
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You just need a few macroscopic variables that are very familiar to you, like the pressure, the temperature, the volume, the number of moles of each component, the mass of the system.
你只需要某些你非常熟悉的宏观变量,比如压强，温度，体积，每个组分的摩尔数，系统的质量。
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