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This should be particularly bothersome to you because, as you've already experienced in 5.60, There are a lot of partial derivatives.
对你们来说这可能很让人头疼，就像你们在5。60里体验过的那样，这有很多偏微分和变量。
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I can get better grades and you know, better average rather than going different uni, so.
我可以得到更好的成绩，中等偏上，比去其他大学要好。
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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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The baby is being approached, goes like this , smiles like this , and then the aversion .
有人走近这个婴儿，像这样，眉毛上挑凝视，开始微笑，像这样笑，大笑头偏向左，然后是讨厌的人，大笑头避开。
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If you can kick the ball very hard, but not very accurately, then if you try and shoot to the left or right, you're slightly more likely to miss.
如果你只会大力抽射但是缺乏精准,当你射向左路或者右路时,你有可能会射偏
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Little coffins, people were smaller then.
窄小的棺材,那时的人们身材偏小
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It gets in the way of arriving too quickly at meaning, if indeed one arrives at meaning at all.
当你想理解含义时却偏不让你迅速理解，前提是你真的理解了。
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Now when it reaches up to the right more towards the heart of that red zone, " 100 spikes per second.
现在，当它移动到,右上侧偏红色区域的时候，我们会听到嗑哩的声音，每秒钟100次，we，may，hear，“keri…
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But, of course, it's going to come from the fact that these second derivatives are also equal.
但是，结果同样是依赖于,二阶混合偏导数相等。
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So for an ideal gas then, dH/dp under 0 constant temperature, that has to be equal to zero.
所以对于理想气体,偏H偏p在恒温下，等于。
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That is, it's easy to write down straight away that dG with respect to temperature at constant pressure S is minus S.
这就是说，可以很简单的写出dG在,恒定压强下对温度的偏导数,是负。
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When you say that, it implies that the differential is given by this pair of partial derivatives.
这就意味着，内能的微分,等于偏u偏T，保持体积不变。
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We have discovered that this partial derivative that appears in the definition, the abstract definition of the differential for internal energy, is just equal to the constant volume heat capacity.
我们还发现,这个偏微分出现在了，内能的偏微分,定义式中，它也就是热容。
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Now let's take it in the other order.
我们用另一种顺序求二阶偏导数。
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And then we can take the derivative with respect to temperature, it's just R over molar volume minus b.
这样我们求,压强对温度的偏导数，结果等于R除以摩尔体积V杠减去b的差。
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p This is going to get us dH/dp constant temperature. What is this experiment?
这帮助我们理解恒温条件下的偏H偏,那么这个实验具体是什么呢？
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G We can take the derivative of G with respect to how much material there is.
我们可以取,对物质总量的偏导数。
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Can determine how entropy is going to behave as the volume changes.
这些偏导数,你就可以知道当体积变化时熵如何变化。
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These things have to be equal to each other.
这两个偏导数是相等的。
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Because these mixed second derivatives are the same thing.
因为这两个混合二阶偏导数,是相等的。
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v du/dV under constant temperature. du/dT v under constant volume. You use the Joule expansion to find these quantities.
像偏u偏v，恒温下的偏u偏,恒容下的偏u偏,你们知道怎么运用焦耳定律。
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So d/dT of dA/dV, just like this.
即对dA/dV求对温度T的偏导数。
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What is dH/dT as a function, keeping pressure constant, what is dH/dp, keeping temperature constant?
恒定时偏H偏T是什么,温度恒定时的偏H偏p又是什么呢？,好的，让我们解决第一个问题？
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So we already know that. So now we can write CpdT or differential dH as Cp dT plus dH/dp, pdp constant temperature, dp.
我们已经知道了这个，所以我们现在,可以写出H的微分式：dH等于，加上恒温时的偏H偏。
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So we have dH/dT keeping pressure constant, is du/dT keeping pressure constant.
等于偏U偏T，p恒定加上，偏pv偏T，p恒定。
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Now, for an ideal gas, du/dV under =0 constant temperature is equal to zero.
对于理想气体，温度一定,时偏U偏V等于零。
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We already did that. OK, dH/dT constant pressure is Cp. That was easy one.
我们已经做过这个计算了，好的，在恒压,状态下的偏H偏T就是Cp，这个很简单。
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dT That means that dH is also equal to dH/dT, constant pressure dT. All right, so now I've T ot more dH/dT under constant pressure.
也等于偏H偏T恒压乘以，现在我已经得到了在恒压,状态下的偏H偏。
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If T is less than T inversion, you have the opposite case, and dT/dp is greater than zero.
如果T比转变温度低,情况就相反，偏T偏p大于零。
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