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。
Now, suppose I took this two paths, and I took -- couple them together with one the reverse of the other.
其中一条的方向反过来,这是初态,末态,路径1,先冷却。
So, if we start instead with talking about the energy levels, we can relate these to frequency, because we already said that frequency is related to, or it's equal to the initial energy level here minus the final energy level there over Planck's constant to get us to frequency.
如果我们从讨论能级开始,我们可以联系到频率上,因为我们说过频率和能量相关,或者说等于初始能量,减去末态能量除以普朗克常数。
For example, if you started here and you did all this and you came back here, the average velocity would be zero, because you start and end at the same value of x, you get something; 0 divided by time will still be 0.
例如,如果你从这里开始运动,经过这个过程又回到这里,平均速度就是0,因为初态和末态的位移相同,你得到了什么呢,0除以时间还是0
p2 One of them is going to end up at pressure p2 p3 and the other is going to end up at pressure p3.
其中一种末态压强为2,另一种末态压强为。
So, the first obvious one is to take V1 to V2 first with p constant. So take this path here.
从初态到末态可以有无数种路径2,甚至像这样,我们只研究两种。
So there's going to be a line that's going to connect the initial point to the final point, and that line mathematically is not going to be the same as this one here.
连接绝热过程,初末态的曲线的方程,和等温线的方程,也不会相同。
p2 So they're both p2. external is p2 p2 V2 T2 and I have p2, V2, T2, on the other side.
都是2,这样末态是。
OK, so, the thing about a state function is that the function has a value for initial conditions and at final conditions.
态函数的特点是,在初始状态有一个值,在末状态也有一个值。
OK, you use the ideal gas law, etc., then you get a relationship that connects the pressure and the temperature, like here we got a relationship that connected the temperatures and the volumes together.
我们会得到,一个联系初末态,的压强和温度的,关系式,就像这个联系过程中。
This is going to be the connect, what connects the pressures and the temperature.
这就是联系初末,态压强和温度的关系式。
T2 Is the temperature T2 in this process smaller or larger than if I were to do the process reversibly with the same endpoint pressure.
这里的末态温度,与经过可逆绝热过程,到达相同压强的末态温度相比哪个比较高呢?
If it was non-reversible, I would be allowed to put an initial point and a final point, but I wouldn't be allowed to put a path between them like this, connecting them together.
如果是不可逆过程,我可以画出过程的初态点,和末态点,但是我不能再像这样,画出连接这两个点的,路径曲线来。
Define the initial state. Define the final state.
定义初态,定义末态,定义路径。
It only cares about the end points.
于初末态。
p1 V1 Initial find -- there are many ways p2 V2 V1 I can get from one state to the other.
初态的氩气状态为,末态为。
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。
My final stage is 50 degrees Celsius and one bar.
我的末态是50摄氏度,1巴。
This is very different from the functions of state like pressure and temperature.
联系初末两态的整个过程,这与压强。
How do you get from the initial state to the final state?
你是怎样,从初态过渡到末态的?
You could take a continuous path, where you have an infinite equilibrium points in between the two, a smooth path, whereyoudrop pressureand temperature simultaneosly in little increments.
你可以选择光滑的连续路径,在初末态间有,无数个平衡态的点,压强和温度同时一点,一点地下降。
OK, we're going to do this reversibly, which means we're going to slowly change the external pressure very, very slightly at a time, so that at every point we're basically in equilibrium, p2/ until the pressure reaches a new smaller pressure p2.
整个过程保持可逆,外界压强,变化得很慢,每一个瞬间,都保持平衡,直到压强减小到末态值。
This is going to end up at a different temperature by the way. You saw this last time in a slightly different way. Last time what you saw is we compared isothermal and adiabatic paths that ended up at the same final pressure, and what you saw is that therefore they ended up in different final volumes.
末态温度是不一样的,上次你们看到的,和这个有一点不一样,上次我们比较的是末态压强,相等的等温过程和绝热过程,因此它们的末态,体积是不一样的。
So in talking about the hydrogen atom, they actually have different names for different series, which means in terms of different n values that we end in.
所以就氢原子来说,它的每个系都有不同的名字,代表了不同的末态n值。
So this is going to end up at T2 a different temperature, we'll call it T2.
这个的末态有不同的温度,我们叫它。
Now these two endpoints here are different.
两条路径的末态不一样。
The sum of path number 2 and path number 3 get me to the same place, so the energy change by going through this time path, this intermediate point here back all the way to final state should be the same the red path.
而经过路径2和3可以3,到达同样的末态,因此经过路径,2和3带来的能量的变化,与路径1带来的,能量变化相同。
Let's say we start from some V1 and p1 here, so high pressure, small volume and we end up with a high volume low pressure, under constant temperature condition.
例如我们要从压强比较高,体积比较小V1,p1出发,到达低压强,大体积的末态,过程中温度不变。
So that's going to results in an expansion where the new volume new temperature new pressure and an external pressure, p2 which is p2 which is a smaller pressure.
这样经过膨胀,到达了,状态为V2,T2,p2的末态,外界压强也降低到。
What is y-y0? The final y is 0, the initial y is 15. You solve for that equation and you will find the final velocity.
-y0是什么,末态的y是0,初态的y是15 你们已经得到,然后你可以找到最终速度
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