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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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Little coffins, people were smaller then.
窄小的棺材,那时的人们身材偏小
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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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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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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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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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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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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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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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So, using those, now, what happens if we take the second derivative of A, the mixed derivative, partial with respect to T and the partial with respect to V.
如果我取A的二阶导数,混合导数，对T偏微分,再对V偏微分。
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Therefore, from experiments, u is only a function of temperature for an ideal gas, H and therefore from these experiments, 0 we come out with delta H dH/dp is equal to zero.
因此，从实验可以得出,对于理想气体u只是温度的态函数，因此从这些实验中我们得到Δ,偏H偏p等于。
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du/dT And we discover that du/dT at constant V T is equal to du/dT at constant V.
可以发现恒定体积下的,等于恒定体积下的偏u偏。
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OK, so now we have the other one, p dH/dp constant temperature.
好的，现在我们来研究另一个量，在恒温条件下的偏H偏。
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dA/dT dS/dV So this is negative dS/dV.
是负S，It’s，negative，S。,这个二阶偏导数是负的恒定温度下的。
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T Remember, we're trying to get delta H, p we're trying to get dH/dT constant pressure and dH/dp constant temperature. OK, these are the two things were trying to get here.
想要得到在恒压状态下的偏H偏，和在恒温状态下的偏H偏,好的，这是两个我们，在这里想要得到的东西。
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That's where that term comes from, du/dV dV/dT.
乘以偏V偏T，p恒定，这项的来源。
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The dA/dV is calculated at constant temperature.
就像这样，dA/dV是在恒定温度下的偏导数。
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With respect to n, the number of moles.
对n的偏导数，这里的n是摩尔数。
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