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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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We want to integrate. So let's take the integral of both sides, going from the initial point to the final point.
分别从初态,到末态做积分，消去微分。
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So, all you will have the opportunity to solve differential equations in your math courses here. We won't do it in this chemistry course. In later chemistry courses, you'll also get to solve differential equations.
你们在数学课中有机会,遇到解微分方程,我们在这化学课里就不解了,在今后的化学课程里，你们也会遇到解微分方程的时候。
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Not only did he formulate laws of gravitation, he also invented calculus and he also learned how to solve the differential equation for calculus.
他不仅找出了引力定律的公式,还发明了微积分,同时也得出了微分方程的解法
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This is a discontinuous payoff function, so differentiating isn't going to get you very far.
这是一个不连续的收益函数,所以不会涉及太多微分的东西
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So now I have my first of my two slopes, in terms of something that's related to my system the heat capacity of the system.
好，我们现在得到了，两个微分式中的一个,它等于与系统密切相关的一个量。
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So, these two are equal to each other as well which tells me that this derivative, Cp dH/dT constant pressure is Cp.
所以这两者也相等,这告诉我们在恒压下微分，等于。
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So here I've written for the hydrogen atom that deceptively simple form of the Schrodinger equation, where we don't actually write out the Hamiltonian operator, but you remember that's a series of second derivatives, so we have a differential equation that were actually dealing with.
这里我写出了,氢原子薛定谔方程的,最简单形式，这里我们实际上,没有写出哈密顿算符,但是请记住那你有，一系列的二次导数,所有我们实际上会处理一个微分方程。
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Now we can use it to derive differential relations for all of the thermodynamics quantities.
现在我们可以利用他们,推导所有热力学量的微分关系。
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OK, but in order to relate turning these physical knob to this quantity here, which we don't have a very good feel for, we've got to have a feel for the slopes.
热量是怎么进一步改变的,但是为了把这些物量同我们，不是很理解的焓联系起来，我们对微分已经有了一定的了解。
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And, so you propose that there is no, that this derivative is zero, and that the internal energy is given simply by this quantity.
你认为这是零，这个微分是零，内能仅仅由,这个简单的量决定。
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Well we can go look up here, looking at the differential, there are no approximations here.
好的我们可以看这儿,看这个微分方程，这里没有做近似。
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And this little slash here means an in inexact differential.
请注意这里的小横杠,说明这不是一个准确的微分。
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We can start taking derivatives of this with respect to time.
我们首先来对时间求微分
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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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So the result is we can combine all of these as a single differential, and just like we've seen before, what that suggests is that we define another new quantity given by this expression.
结果就是我们能把所有的结果,整理成一个单一的微分，就像我们前面看到的一样，这说明我们可以利用这个式子,定义一个新的物理量。
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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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The Joule experiment was a constant energy experiment, right. Here we're going to have to find a constant enthalpy experiment, and that is going to be the Joule-Thomson experiment. That's going to extract out a physical meaning to this derivative here.
非常像焦耳实验,焦耳实验是一个能量恒定的实验,我们这里要做的是，找到一个焓不变的实验,也就是焦耳-汤姆逊实验,这个实验可以把这里的微分式形象化。
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Integration is not an algorithmic process like differentiation.
积分不是微分那种演算过程
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We can measure the heat capacity at constant volume, and now we have another term, and if we can figure out how to measure it, we'll have a complete form for this differential du which will enable us to calculate du for any process.
我们能够测量恒定体积下的热容，这里我们有另一项，如果能够知道怎么测量它，问我们就有了这个完整的微分式,就能够对任何过程计算。
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You won't have to solve it in this class, you can wait till you get to 18.03 to start solving these types of differential equations, and hopefully, you'll all want the pleasure of actually solving the Schrodinger equation at some point. So, just keep taking chemistry, 18 03 you'll already have had 18.03 by that point and you'll have the opportunity to do that.
你们不用在课堂上就解它，你们可以等到得到18，03之后,再开始解这些类型的微分方程，希望你们都想得到,实际解薛定谔方程的乐趣,所以，保持来上化学课，你们在那个点将会得到,你们有机会做到的。
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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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Mathematically, it's done by taking the expression for R and differentiating everything in sight that can be differentiated as vector t.
从数学上看是这样的,写出 R 的表达式,并对所有可以进行矢量微分的部分,对时间 t 求微商
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So, let's say we start off at the distance being ten angstroms. We can plug that into this differential equation that we'll have and solve it and what we find out is that r actually goes to zero at a time that's equal to 10 to the negative 10 seconds.
也就大约是这么多,所以我们取初始值10埃,我们把它代入到,这个微分方程解它,可以发现，r在10的,负10次方秒内就衰减到零了。
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This is going to be probably a homework at some point to do this. For now, let's take it for granted. Let's take it for granted that we know how to calculate this derivative from an equation of state like this.
这可能是将来的一个课后作业,现在，请把这当成理所当然的，理所当然地认为我们,知道怎样从一个状态,方程计算这样的微分式。
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A It tells me that the partial of A with respect to T at constant V is minus S. Right?
他告诉我们,在恒定体积下对温度的微分等于负S，对吗？
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OK, and now we return to this differential.
好，让我们回到这个微分式。
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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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du/dT constant pressure is the direct derivative with respect to temperature here, which is sitting by itself under constant volume keeping this constant but there is temperature sitting right here too.
偏U偏T，p恒定是对,温度的直接微分，而它本身对体积不变,保持它不变，但是这里也有一个温度,这就是偏U偏V，T恒定。
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In other words, your job is to guess a function whose second derivative is a, and this is called integration, which is the opposite of differentiation, and integration is just guessing.
换言之,你的任务是要猜出一个二阶导数为a的函数,这就是积分,和微分恰恰相反,积分就是猜
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