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We're not going to have the constant pressure heat capacity, we're going to have the constant volume heat capacity, right.
这里出现的,不是等压热容,而是等体热容。
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It's related to the heat capacity, the constant volume of heat capacity and something you could measure.
它联系了热能，恒容热容和一些,我们能够测量的物理量。
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You're allowed Cv comes out here for this adiabatic expansion, which is not a constant volume only because this is always true for an ideal gas.
绝热过程写下,这个式子是因为它对理想气体都成立，并没有用到等容过程的条件,只用了理想气体的条件。
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Now, you know with constant volume, H now it's not going to be delta H that's U straightforward to measure, it's going to be dealt u, all right.
好，现在你们知道在体积恒定的条件下，我们得到的不是Δ,我们直接测量到的是Δ,好，但这基本上也是一样的。
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So now we have a constant volume reversible temperature change.
所以现在我们有一个,等体，可逆的温度变化。
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When I flail my arms around I generate work and heat. This is not a constant volume process.
这不是一个恒容过程,但如果我是一个系统，当我做这些的时候。
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You just change volume to pressure and basically you're looking at enthalpy under a constant -- anything that's done at a constant volume path with energy, there's the same thing happening under constant pressure path for enthalpy.
可以看到这就是把体积换成了压强,一般我们都是在一种恒定状态下,考虑焓的，任何在恒容条件下,能伴随能量变化的东西，也在恒压条件下伴随焓同样地变化,所以你可以经常。
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So what we've discovered from this relationship dq that du at constant volume is equal to dq v.
从这个关系式里我们发现,恒体积时的du等于恒体积时的。
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So the first path then, the first path, 1 constant volume constant V, so I'm going to, again, let's just worry about energy.
首先，是路径，等压过程。
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You know how pressure changes with temperature at constant volume if you know the equation of state.
如果你知道状态方程,知道在体积恒定的时压强如何随着温度变化。
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Yes, and if we have gases involved, it's pretty similar, but now what will have is something like this. We'll have a reaction vessel that's sealed, it's constant volume.
如果涉及了气体，情况也很相似，只是现在的装置是这样的,我们有一个密封的反应容器,它的体积是恒定的。
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OK, constant volume.
下一个。
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Now it's constant volume when the volume is constant.
因为体积,保持不变。
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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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We discovered that the quantity dA, under conditions of constant volume and temperature, dA TS And A is u minus TS.
我们发现在恒定的体积和温度下，亥姆赫兹自由能的变化，小于零，is，less，than，zero。,亥姆赫兹自由能A等于内能u减去。
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We're going to take a constant volume path.
其一是。
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nRT So, dp/dT, for our ideal gas, at constant volume, remember pV is nRT.
对于理想气体状态方程pV等于，所以对理想气体。
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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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And that in the case of constant volume, U in this case that's my delta u, and then I'll H add my little delta n term to get delta H.
这是在等体情形下，此时的到的是Δ,然后我可以加上Δn的项来得到Δ
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T So we know that T dS/dT at constant volume is Cv over T, T and dS/dT at constant pressure is Cp, over T.
在恒定压强下定压比热容Cp乘以dT除以,所以在恒定体积下dS/dT等于Cv除以，在恒定压强下dS/dT等于Cp除以。
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This is path C, constant volume. OK?
这条路径C，体积恒定？
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If you have a real gas and you write du is Cv dT and your path is not a constant volume path, then you are making a mistake. But for an ideal gas, you can always write this. And this turns out to be very useful to remember.
对于真实气体，如果其变化过程,不是恒容的，du=Cv*dT就不成立，但对于理想气体,这条规则永远成立,这一点非常有用，请记住。
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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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U It's u, because u is to q plus w right, heat and work, but it's adiabatic. So there's no heat, exchange with the environment, and it's constant volume, so there's no p dV work, right.
什么是零？是U，因为,等于q加w，热量和功，但这是绝热的，所以系统与环境间没有热量交换；,同时它是灯体的，所以也没有pdV形式的功。
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That is, most processes that we're concerned with, they'll happen with something held constant like pressure or temperature or maybe volume.
这句话是说我们所关注的大部分过程，发生的时候都是保持某个量为常数,比如压强，温度或者体积。
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Then we can take the derivative of that quantity, when we vary the temperature, holding the volume constant.
即恒定体积，改变温度，这里恒定温度下。
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This piston is being brought out, so we expect 0 the work to be negative, negative. And we start o V2 ut with zero volume. We end up with a volume p2 of V2, and the external pressure is constant to p2.
所以我们可以想象功是负的,开始的时候体积是,最终的容积是,外界的压力恒为。
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So I need, well the pressure is constant, but there's a change in volume.
压强不变，体积变化。
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If I'm working under conditions of constant temperature and volume, that's very useful.
如果在恒定的温度和体积下,进行一个过程，这是非常方便的。
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In other words, the order of taking the derivatives with respect to pressure and temperature doesn't matter And what this will show is that dS/dp dS/dp at constant temperature, here we saw how entropy varies with volume, this is going to show us how it varies with pressure.
换句话说,对温度和压强的求导顺序无关紧要,结果会表明,恒定温度下的，对应我们上面看到的，熵如何随着体积变化，这个式子告诉我们，熵如何随着压强变化。
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