翻译 露小西
压制 草太阳
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大家好，我是安德森 这里是AP物理精华 系列视频136——半衰期和放射性衰变 Hi. It's Mr. Andersen and this is AP Physics essentials 136. It is on half-life and radioactive decay.
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我和我女儿用摇色子的方式来模拟衰变过程 我们一共有32个色子 To model that, my daughter and I started rolling some dice. We have 32 dice in the cup.
她先摇色子 然后把所有的某一个点数的色子挑出来 And what she would do is she would roll the dice and then she would pull all of a certain number out.
在第一个演示中 她把所有点数1朝上的色子挑出来（就当这些色子衰变了） So in this first simulation she is going to pull all of the 1s out.
我们把它们放到一边 第一次的 第二次的 第三次的 第四次的 And so we put those to the side. That is generation one. And now generation two. And generation three. And generation four.
当第四次挑色子时 她已经到达了第一个半衰期 Now when you get to the fourth generation, she has hit her first half life.
什么意思呢 是因为我们一共有32个色子 而摇到第四次的时候已经有一半的色子衰变了 What does that mean? Well if there were 32 to start, that is the amount of time it takes for half of those to decay.
也就是16个色子被挑出来了 So that would be 16. So 16 have decayed right here.
她会继续摇色子并把1点朝上的挑出来 So she is going to keep doing that.
我们继续看这个模拟的结果 And so if we watch what happens in the rest of the simulation
你摇的色子越来越少 因为衰变剩下的色子越来越少 you are not pulling as many dice because there are not as many left.
最终 在摇了19次之后 所有的色子都衰变了 And eventually they all have decayed after 19 generations.
但是你可以大约地看到 这些半衰期是比较连续的 But what you can see is that the half life is consistent.
这里是16个色子的衰变点 这里是8个的 然后是4个的 然后是2个的 然后是1个的 So this was to 16 and then to 8 and then to 4 and then to 2 and then to 1.
所以放射性的原子核会发生衰变 And so radioactive nuclei will decay.
什么是衰变呢 就是原子核释放出某种粒子 可以是α，β，或γ粒子 What does that mean? They are going to kick off some kind of a particle, it could be alpha, beta or gamma.
然后它们通常会变成一个更稳定的 新的原子核 And then they are going to form a new, usually more stable nuclei.
所以在这样的放射性衰变中 总体的质量和电荷是守恒的 这在之前的视频中已经讲过 And so in this radioactive decay, mass and energy are conserved. We have talked about that in previous videos.
而我们在这里讨论的是 这种衰变发生的概率有多大 But what we are going to talk about here is what is the probability of that decay occurring?
这是随机的 所以我们永远不能知道那个原子会发生衰变 Well it is chance. So we never know when the next one is going to decay.
但是我们可以利用【大数定理】来计算半衰期 But what we can use is the law of large numbers to calculate the half-life.
半衰期是一半的原子核发生衰变所需要的时间 The half-life is the amount of time it takes for half of the radioactive nuclei to decay.
如果这代表所有的放射性原子核的话 达到这个状态就要用掉一个半衰期的时间 In other words if this represents all of the radioactive nuclei that would be the time it takes for half of them to decay.
对于剩下的这些未衰变的原子核 经历一个半衰期又会有一半发生衰变 Now we have all of these ones left. And so we would have another half life like that.
我们可以不断重复这样的过程 We could just keep going and going and going like that.
所以 就会有一个方程来计算 And so there is an equation we can use to figure out
每一个半衰期内将有多少原子核发生衰变 how many are going to decay at each generation.
用N表示初始放射性原子核的总数 △N表示发生衰变的放射性原子核数 And so delta N, which is the change in the radioactive nuclei,so N is going to be our radioactive nuclei.
N是在变少的 因为我们在失去放射性的原子核 So the change in N is minus, because we are losing those nuclei,
所以一个负值的衰变常数 乘以放射性原子核的总数N 再乘以时间 so negative decay constant, I will get to that in a second, times N which is the number of radioactive nuclei times time.
所以我们把这个方程代回到刚才的模型中 我们的时间变化量是摇1次色子 And so if we kind of work backwards for that, in this simulation here our time was advancing 1 generation after another.
所以△t取1 And so that is going to be 1 each time.
我们的衰变常数是多少呢 应该是1/6 What was our decay constant? Well it is 1 in 6.
因为每次摇色子都有1/6的概率1点朝上 也就是1/6的概率发生衰变 So it is a one-sixth probability that you are going to roll a 1 and that that one is going to decay.
所以我们可以用这个公式来模拟刚才摇色子的过程 我将用一个表格表示结果 And so we could model that. I will just use a quick little spreadsheet to do that.
刚开始的时候 我们一共有多少个色子呢 So at time 0 how many radioactive nuclei did we have?
我们的N是32 让我带你一起来看这个公式 Well our N value was 32. So let me walk you through this formula.
我们的时间变量是多少 是1 我们的N是多少 是32 What is our time going to be? It is going to be 1. What is going to be our N value? It is 32.
所以1乘以32 我们的衰变常数是多少 是1/6 So 1 times 32. And then what is our decay constant? It is one-sixth.
所以 负的1/6乘以32 所以结果是 负的5.33 And so it is going to be negative one-sixth times 32. So what is that value? It is going to be negative 5.33.
实际衰变的数量是几呢 是6个 但是这很接近-5.33 Now if we look, how many actually decayed? It is 6, but it is close, -5.33.
这基本正好是我们预测的值 而且我们仅用了很少的色子就做到了 This is what it would predict to be and this is what we actually got with a really small number of dice.
下一步应该怎么做呢 我们应该用32 Now how do you do the next one? Well what we have to do is we have to take that 32,
减去5.33 然就得到了下一次的N值 these ones decayed, so we are going to take 32 minus 5.33 and that is my N value for the next generation.
然后接下来 我再用一次这个公式 And so now what do I do? I go back to this formula again.
时间t等于1 新的N值是26.67 我的衰变常数不变 So it is going to be t, which is 1 times my new N which is going to be 26.67 times my decay constant, which remains constant,
所以我将得到的结果是负的4.45 这个结果与刚才的模拟结果也很接近 and so what I am going to get is negative 4.45. Now how does that match up? It is pretty close to this.
然后 我们继续做一次减法 然后再用一次这个公式 一直重复这样做 So then we subtract that value like that and we could just do this over and over again.
最后 在右边我们就可以得到每一次发生衰变的色子数 So right here on the right is going to be what we would predict to occur.
而这是在刚才的模拟中实际衰变的色子数 And this is what actually occurred in this little simulation.
这里的色子数远远小于一个实际样品中放射性原子核的数目 And the number of dice are so small compared to the number of nuclei in a sample.
但是看一下我统计的数据 绿线代表实际模拟得到的数据，而红线代表预测数据 But if you look at my data, the green line represents the actual data that we find. The red line represents the predicted.
我们可以看到 二者很快变得很接近 And so you can see that it matches up pretty quickly.
如果我们改变衰变常数会怎样的 比如我们把1/6改成1/2会怎样呢 What would happen if we changed the decay constant. What if we changed it from one-sixth to one-half.
在模拟中应该怎样做呢 把只挑1点向上的色子改成将1,3,5点向上的都挑出来 So how would you do that in the modeling? Instead of pulling 1s out, she is going to pull the 1s, 3s and 5s out.
现象会如何呢 整个过程变得很快 So what is going to happen? Well it is going to happen more quickly.
所以随着一次衰变的数目的增多 所有色子衰变所需要的次数会变少 So more of them are going to decay at each point. And so it is going to take less time for all of them to decay.
换言之 我们的半衰期明显的变短了 In other words our half-life has gotten much much shorter.
所以半衰期应该是可以计算的 And so you should be able to calculate half-life.
对于第一次的模拟 半衰期是多少呢 你可以看到大概是4次 So how long would it be for half of them decay? You can see it is going to be somewhere around 4 generations.
所以当你拿到这样一条衰变曲线的时候 第一件事情就是 And so when you look at a curve, the first thing you want to do is
找出所有原子核衰变到原来的50%所需要的时间 figure out how long does it take to go from 100 radioactive nuclei to 50 percent of that.
如果我从这张图上看的话 50%的原子核衰变用的时间是1年 And so if I look across here it took 1 year for 50 percent of them to decay.
然后我们再看下一次 你会发现这是个常数 1年 1年 1年 So let's watch the next generation. So now we should go to 25. You can see it is consistent. 1 year. 1 year. 1 year.
所以你可以说 这个理想模型的半衰期是1年 And so you could say for this perfect model it is going to be a half-life of 1 year.
但是在考试中 你更可能来到这样的问题 But on a test you are more likely to get something like this.
计算碳14的半衰期 Calculate the half-life decay of carbon 14.
如果你有这样一张图的话 我们从100%看到50% So if you are given this curve right here, we go from 100 to 50,
在50%的地方 做一条这样的辅助线 so you could count across like that. So this is 50 percent of the carbon 14.
然后读出下面对应的数值 如果这里是10,000年的话 我们的半衰期是多少 And then you just read the time on the bottom. So if this right here is 10,000 years, what is our radioactive half-life?
大约是6,000年 It is around 6,000 years.
我们继续看下去 在25%对应的地方 你可以看到 大约是12,000年 But let's keep going. So now let's go down to 25 percent and you can see it is around 12,000.
所以半衰期是很稳定的 And so it is pretty consistent over time.
每一种衰变都将放射出一种粒子 Now each form of radioactive decay is going to kick off a different particle.
我们先来看α粒子 α粒子是由2个质子和2个中子组成的 Let's start with alpha particles. An alpha particle is 2 protons and 2 neutrons.
我们来看一下铀238的α衰变 首先我们来确认质量是否守恒 So if we look at the alpha decay of uranium 238, let's make sure that the mass is conserved.
左边的质量是238 右边的质量是238 所以两边质量相当 And so mass on the left side is 238, mass on the right side is 238. So we are fine there.
我们再来确认电荷守恒 左边有92个正电荷 Let's make sure charge is conserved. 92 positive charge on the left.
右边也有92个正电荷 所以电荷也是守恒的 92 positive charge on the right. So that is conserved as well.
那铀238的半衰期是多少呢 44.7亿年 Now what is the half-life of uranium 238 decay? It is 4.47 billion years.
一半的铀238衰变就要花这么这么长的时间 It takes a huge amount of time for just half of the uranium 238 to decay.
但是世界上有那么那么多的原子核我们可以测量 这也是我们用来测定地球年龄的方法 But there are so many nuclei that we can actually measure this. And this is how we determine the age of the earth.
我们来看β衰变 Let's look at beta decay.
β衰变放射时 一个中子转变成一个质子 探后释放一个电子和一个电子中微子 Beta decay we converted a neutron to a proton. We also give off an electron and an electron antineutrino.
所以看上面 质量是守恒的 这边14 这边也是14 So if we look across the top, mass is conserved, 14 and 14.
电荷也是守恒的 左边我们又6个正电荷 右边我们又7个正电荷和一个负电荷 Charge is conserved. On the left side we have 6 positive charges. On the right side we have 7 positive, 1 negative.
所以右边也是6个正电荷 And so we have 6 positive charges on the right as well.
碳14衰变的半衰期是多少呢 What is the half-life of carbon 14 decay?
是5730年 我们刚才已经提到过了 It is going to be 5730 years. So we had shown that just a few slides ago.
它的半衰期大约是6000年 我们可以利用这一点来判断生物体化石的年龄 It was around 6000 for half-life. And we can use this to date living material.
然后我们再来看γ衰变 记得吗 在γ衰变时会有γ射线放出 And then we could look at gamma decay. Remember in gamma decay we are just giving off these gamma rays.
从带电的钡137 变成不带电的钡137 We are going from barium 137 that is charged to barium 137 that is not charged.
这个过程电荷是守恒的 质量也是守恒的 And so we are conserving charge and conserving mass.
半衰期是多少呢 2.6分钟 非常非常短 What is the half-life? 2.6 minutes. So it is really really short.
所以半衰期和衰变常数是紧密相关的 And so half-life is going to change depending on that decay constant.
通过本节课 你应该学会从这样的图中找到半衰期 But you should be able to take a graph like this. Figure out the half-life.
祝宝宝学习进步哟 And I hope that was helpful.