Ideal Gases
A-Level Physics Topic 15 13:48 English narration · English + 中文 subtitles burned in
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Hold up a single glass of water.
举起一杯普通的水。
It looks simple enough.
它看起来平平无奇。
But inside it are more molecules than you could ever count.
但它内部的分子多到你永远数不清。
One mole of water — just eighteen grams, a couple of sips — contains six hundred thousand billion billion molecules.
一摩尔的水——只有十八克，几口而已——就含有六十万亿亿个分子。
If you could count a million every second, it would take you longer than the entire age of the universe.
如果你能每秒数一百万个，你要数的时间比整个宇宙的年龄还长。
This staggering number has a name: the Avogadro constant.
这个惊人的数字有个名字： 阿伏伽德罗常量。
And it is how physicists count the uncountable.
物理学家正是用它来清点那些数不清的东西。
Gases are simple, if you know the trick: count the molecules.
气体其实很简单，只要你知道诀窍：数分子。
Today: the mole, the ideal gas equation, and the kinetic theory that connects heat to motion.
今天：摩尔、理想气体方程， 以及把热与运动联系起来的分子动理论。
Let's begin.
让我们开始吧。
Amount of substance is one of the seven base quantities in physics, measured in a unit called the mole.
物质的量是物理学七个基本量之一，用一个叫做摩尔的单位来量度。
One mole of anything contains the same fixed number of particles — the Avogadro constant, about six times ten to the twenty-three.
任何东西的一摩尔， 都含有相同的固定数目的粒子——阿伏伽德罗常量，约为六乘以十的二十三次方。
A mole of carbon, a mole of water, a mole of electrons: different masses, but always the very same count of particles.
一摩尔碳、一摩尔水、一摩尔电子：质量各不相同，但粒子的数目永远完全相同。
Now, a gas.
现在来看气体。
Squeeze it into half the space, and its pressure doubles.
把它挤进一半的空间，它的压强就加倍。
Heat it up, and it pushes harder.
给它加热，它就推得更用力。
For many gases, pressure, volume, and temperature follow one neat rule: pressure times volume is proportional to the thermodynamic temperature.
对许多气体来说，压强、体积和温度遵循一条简洁的规则：压强乘以体积，与热力学温度成正比。
A gas that obeys this rule perfectly is called an ideal gas.
完美遵守这条规则的气体，就叫做理想气体。
Watch how squeezing the volume raises the pressure, in exact step.
看压缩体积如何精确地同步抬高压强。
Before the gas laws, three small relations that questions lean on.
在讲气体定律之前，先看三个题目常常要用到的小关系式。
First, a warning: a "particle" means whatever you are counting — atoms for a monatomic element like helium, but molecules for oxygen or water.
第一是一句提醒： "粒子"指的是你正在数的东西——对氦这样的单原子元素来说是原子， 但对氧气或水来说是分子。
Always say what you are counting, because the numbers differ by a factor of two.
一定要说清你在数什么，因为两者的数目差了一倍。
Second, for n moles the number of particles is N equals n times N A.
第二，对 n 摩尔物质，粒子数是 N 等于 n 乘以 N A。
Third, the molar mass is the mass of one mole, in kilograms or grams per mole, so the mass of n moles is n times the molar mass.
第三，摩尔质量是一摩尔的质量， 单位是千克每摩尔或克每摩尔，所以 n 摩尔的质量就是 n 乘以摩尔质量。
And dividing the molar mass by the Avogadro constant gives the mass of a single particle — which is the step most people miss when a question gives them a molar mass but wants the mass of one molecule.
而用摩尔质量除以阿伏伽德罗常量，就得到单个粒子的质量—— 当题目给出摩尔质量却要求一个分子的质量时，这一步正是多数人漏掉的。
A cylinder of volume nought point nought two cubic metres holds gas at twenty-seven degrees Celsius and a pressure of two times ten to the fifth pascals.
一个体积为零点零二立方米的气瓶，装着二十七摄氏度、压强为二乘以十的五次方帕的气体。
How many moles are there?
里面有多少摩尔？
The very first move, before anything else: convert to kelvin.
在做任何别的事之前，第一步：换算成开尔文。
Twenty-seven plus two hundred and seventy-three is three hundred kelvin.
二十七加二百七十三等于三百开。
Using Celsius here is the single commonest way to lose this mark, because p V is proportional to T only in kelvin.
在这里用摄氏度，是丢掉这一分最常见的方式， 因为只有当 T 用开尔文时，p V 才与 T 成正比。
Now rearrange the equation of state: n equals p V over R T.
现在把状态方程移项： n 等于 p V 除以 R T。
Substituting gives about one point six moles.
代入后得到约一点六摩尔。
For a fixed amount of gas changing from one state to another, p one V one over T one equals p two V two over T two — and the three named laws are just this with one quantity held fixed.
对于一定量的气体从一个状态变到另一个状态，p 一 V 一除以 T 一等于 p 二 V 二除以 T 二—— 而那三条以人名命名的定律，不过是把其中一个量固定住的特例。
At constant temperature, p one V one equals p two V two: that is Boyle's law, and on a pressure-volume graph it draws a hyperbola, with a higher temperature giving a curve further from the origin.
温度不变时， p 一 V 一等于 p 二 V 二：这就是玻意耳定律，在压强与体积的图上画出一条双曲线， 温度越高，曲线离原点越远。
At constant pressure, V over T is constant: Charles's law.
压强不变时，V 除以 T 是常数：这是查理定律。
At constant volume, p over T is constant: the pressure law.
体积不变时，p 除以 T 是常数：这是气体压强定律。
You do not need to memorise three separate equations — just cover up whichever quantity is held fixed.
你不必背三个不同的方程—— 只要把被固定住的那个量遮起来就行。
A fixed mass of gas at three hundred kelvin occupies nought point five cubic metres.
一定质量的气体在三百开时占据零点五立方米。
It is heated to four hundred and fifty kelvin at constant pressure.
在压强不变的条件下把它加热到四百五十开。
Find the new volume.
求新的体积。
Because it is at constant pressure, V over T stays constant, so V two over T two equals V one over T one.
因为压强不变，V 除以 T 保持为常数，所以 V 二除以 T 二等于 V 一除以 T 一。
Rearranged, that means you simply multiply by the ratio of the temperatures: V two equals V one times T two over T one.
移项之后就是：直接乘以温度之比，V 二等于 V 一乘以 T 二除以 T 一。
Nought point five times four-fifty over three hundred gives nought point seven five cubic metres.
零点五乘以四百五十除以三百，得到零点七五立方米。
Notice the temperatures were already in kelvin — if they had been in Celsius, the ratio would have been meaningless.
注意这里的温度本来就是开尔文——要是用摄氏度，这个比值就毫无意义了。
This graph is worth dwelling on, because it is where the kelvin scale comes from.
这张图值得多看一会儿，因为开尔文温标正是从这里来的。
Measure a gas's volume against temperature in Celsius at constant pressure, and the points lie on a straight line.
在压强不变的条件下， 测量气体体积随摄氏温度的变化，这些点落在一条直线上。
Extend that line backwards, as the dashed part shows, and it reaches zero volume at about minus two hundred and seventy-three degrees Celsius.
像图中虚线那样把这条直线向后延长， 它会在大约零下二百七十三摄氏度处到达零体积。
Every gas gives the same intercept, whatever it is made of.
无论气体是什么， 每一种气体给出的截距都相同。
That intercept is absolute zero, and defining it as zero kelvin is exactly what turns the proportionality into an exact law.
那个截距就是绝对零度，而把它定义为零开尔文， 正是把这个正比关系变成一条精确定律的原因。
In practice, of course, every real gas liquefies long before it gets there.
当然在实际中， 每一种真实气体早在到达那里之前就已经液化了。
We write this as the equation of state.
我们把它写成状态方程。
Pressure times volume equals the number of moles, times the gas constant, times the temperature.
压强乘以体积，等于摩尔数，乘以气体常量，再乘以温度。
Or, counting individual molecules instead of moles: pressure times volume equals the number of molecules, times the Boltzmann constant, times the temperature.
或者，数单个分子而不是摩尔：压强乘以体积，等于分子数，乘以玻尔兹曼常量，再乘以温度。
The two constants are linked: the Boltzmann constant is just the gas constant, divided by the Avogadro number.
这两个常量是相联系的：玻尔兹曼常量就是气体常量除以阿伏伽德罗常量。
But where does gas pressure actually come from?
但气体的压强究竟从何而来？
From the molecules themselves.
来自分子本身。
The kinetic theory pictures a gas as countless tiny particles, in constant random motion, flying in straight lines until they collide.
分子动理论把气体想象成无数微小的粒子， 处于不停的随机运动中，沿直线飞行，直到相互碰撞。
We assume they are so small we can ignore their size, and that their collisions lose no energy.
我们假设它们小到可以忽略其大小， 而且它们的碰撞不损失能量。
Every time one strikes a wall, it gives a tiny push.
每当一个粒子撞到器壁，它就给一个微小的推力。
Billions of these pushes, together, are what we feel as pressure.
亿万个这样的推力加在一起，就是我们感受到的压强。
The kinetic theory explains a gas's large-scale behaviour from the random motion of its molecules, and it rests on six assumptions you should be able to list.
分子动理论用分子的无规则运动来解释气体的宏观行为，它建立在六条假设之上， 你应当能把它们列出来。
One: a large number of identical molecules in continuous random motion.
第一，大量全同的分子处于持续的无规则运动之中。
Two: the molecules' own volume is negligible compared with the container.
第二，与容器相比，分子自身的体积可以忽略。
Three: each collision lasts a negligible time compared with the time between collisions.
第三，与两次碰撞之间的时间相比， 每次碰撞持续的时间可以忽略。
Four: there are no intermolecular forces except during a collision, so molecules travel in straight lines in between.
第四，除碰撞瞬间外分子间没有作用力， 所以分子在碰撞之间沿直线运动。
Five: all collisions are elastic — no kinetic energy is lost, which is why a gas does not spontaneously cool.
第五，所有碰撞都是弹性的——不损失动能， 这正是气体不会自发变冷的原因。
Six: Newton's laws apply.
第六，牛顿定律适用。
And know when they fail: at very high pressure the molecular volume stops being negligible, and at very low temperature the intermolecular forces stop being ignorable.
还要知道它们何时失效：在极高压强下分子体积不再可以忽略， 在极低温度下分子间作用力也不再可以忽略。
Here is the derivation in five steps.
下面用五步完成这个推导。
Take a cube of side L holding N molecules each of mass m, and follow one moving along the x-axis with speed u one.
取一个边长为 L 的立方体，里面有 N 个分子，每个质量为 m， 追踪其中一个沿 x 轴以速率 u 一运动的分子。
Step one, one collision with the right wall: the velocity reverses, so the change in momentum is minus two m u one, and by Newton's third law the wall receives an impulse of plus two m u one.
第一步，与右壁的一次碰撞： 速度反向，所以动量变化是负二 m u 一，而根据牛顿第三定律， 壁受到的冲量是正二 m u 一。
Step two, the time between hits on that same wall: the molecule must travel two L, there and back, so delta t is two L over u one.
第二步，两次撞击同一面壁之间的时间： 分子必须走过二 L，来回一趟，所以 delta t 等于二 L 除以 u 一。
Step three, the average force from this one molecule is the momentum change divided by that time, which gives m u one squared over L.
第三步，这一个分子产生的平均力等于动量变化除以那个时间，得到 m u 一平方除以 L。
Step four, add over all the molecules: F equals N m over L, times the mean square of the x-velocity.
第四步，对所有分子求和：F 等于 N m 除以 L，再乘以 x 方向速度的均方。
Step five, pressure is force over area, so p equals N m times the mean square x-velocity, divided by the volume.
第五步，压强等于力除以面积，所以 p 等于 N m 乘以 x 方向速度的均方，再除以体积。
That result used only the x-direction, so we need one more step to reach the real gas.
刚才的结果只用到了 x 方向，所以要到达真实气体还需要再走一步。
Because the motion is completely random, no direction is special: on average the molecules move just as fast along y and along z as along x.
由于运动是完全无规则的，没有哪个方向是特殊的：平均而言， 分子沿 y 和沿 z 运动得和沿 x 一样快。
Since the speed squared is the sum of the three components squared, each component must contribute one third of the mean-square speed.
既然速率的平方等于三个分量平方之和， 每一个分量就必定贡献均方速率的三分之一。
Substitute that in and you get the result the syllabus wants: p V equals one third N m times the mean-square speed.
把它代进去， 你就得到考纲要求的结果：p V 等于三分之一 N m 乘以均方速率。
That equation is the whole bridge between the microscopic picture and the measurable pressure.
这个方程就是从微观图像通向可测量的压强之间的整座桥梁。
Root-mean-square speed is built in four steps, and the name tells you them backwards.
均方根速率是分四步构造出来的，而这个名字正好倒着告诉了你这四步。
Take the speeds.
取各个速率。
Square each one.
把每一个平方。
Take the mean of those squares — that is the mean-square speed.
对这些平方值取平均——那就是均方速率。
Then take the square root.
然后开平方根。
The order matters: squaring first, then averaging, is not the same as averaging first.
顺序很重要：先平方再取平均，和先取平均再平方是不一样的。
And because squaring weights the fast molecules more heavily, the r.m.s. speed always comes out slightly larger than the plain mean speed.
而且因为平方会给快速分子更大的权重，均方根速率总是略大于普通的平均速率。
It is the natural measure here because it is exactly the quantity that appears in the energy and pressure equations.
它之所以是这里最自然的量度，正是因为它恰好就是出现在能量和压强方程里的那个量。
No molecule actually travels at the r.m.s. speed — the molecules have a whole distribution of speeds, shown here.
其实没有哪个分子真的以均方根速率运动——分子的速率有一整片分布，就是这里画出的样子。
At a lower temperature the curve is tall and narrow: most molecules are near the peak and few are very fast.
温度较低时，曲线又高又窄：大多数分子集中在峰值附近，很少有跑得特别快的。
Raise the temperature and the curve broadens and shifts to the right, so the average rises and, importantly, the tail of very fast molecules grows much larger.
提高温度，曲线变宽并向右移动，于是平均值上升，而且很重要的是， 速率极高的那条尾巴长大了很多。
That growing tail is why reaction rates climb so steeply with temperature.
正是这条不断长大的尾巴， 使得反应速率随温度上升得如此陡峭。
The r.m.s. speed is marked on the curve, sitting a little above the peak.
均方根速率标在曲线上，位置略高于峰值。
Follow the collisions carefully, and out comes a remarkable equation.
仔细追踪这些碰撞，就会得出一个了不起的方程。
Pressure times volume equals one third, times the total mass of gas, times the mean-square speed of the molecules — the average of the speed squared.
压强乘以体积，等于三分之一， 乘以气体的总质量，再乘以分子的均方速率——也就是速率平方的平均值。
It links what we can measure — pressure and volume — directly to what we cannot see: the frantic motion of molecules inside.
它把我们能测量的——压强和体积——直接和我们看不见的东西联系起来：内部分子的疯狂运动。
Take the square root of that mean-square speed, and you get the root-mean-square speed — a fair average of how fast the molecules move.
对那个均方速率开平方，你就得到均方根速率——分子运动快慢的一个恰当平均。
And now, the punchline.
现在，来到高潮。
Compare our two equations for pressure, and everything cancels down to something beautiful.
比较我们关于压强的两个方程，一切都约化成一个漂亮的结果。
The average kinetic energy of a single molecule is three halves, times the Boltzmann constant, times the temperature.
单个分子的平均动能，等于二分之三，乘以玻尔兹曼常量，再乘以温度。
Temperature is molecular kinetic energy.
温度就是分子的动能。
That is what heat really is.
这才是热真正的本质。
Now the most important result in the topic, and it comes out of putting two expressions for p V side by side.
现在来看本章最重要的结果，它来自把 p V 的两个表达式并排放在一起。
From the equation of state, p V equals N k T.
由状态方程，p V 等于 N k T。
From the kinetic theory, p V equals one third N m times the mean-square speed.
由分子动理论，p V 等于三分之一 N m 乘以均方速率。
Set them equal.
把两者等同起来。
Cancel the N from both sides.
两边把 N 约掉。
Then multiply through by three halves, and the right-hand side becomes a half m times the mean-square speed — which IS the average translational kinetic energy of one molecule.
再同乘二分之三，右边就变成二分之一 m 乘以均方速率—— 而那正是一个分子的平均平动动能。
So the average kinetic energy equals three halves k T.
于是平均动能等于二分之三 k T。
Read what that says: it depends only on the thermodynamic temperature.
读一读这句话说了什么：它只取决于热力学温度。
Not on the pressure, not on the volume, and not on which gas it is.
与压强无关，与体积无关， 也与是哪一种气体无关。
Find the root-mean-square speed of oxygen molecules at three hundred kelvin, given the mass of one oxygen molecule as five point three times ten to the minus twenty-six kilograms.
求三百开时氧分子的均方根速率，已知一个氧分子的质量为五点三乘以十的负二十六次方千克。
Start from a half m mean-square speed equals three halves k T.
从二分之一 m 乘以均方速率等于二分之三 k T 出发。
Rearranging, the mean-square speed is three k T over m.
移项后，均方速率等于三 k T 除以 m。
Substitute the numbers, then take the square root, and you get about four hundred and eighty metres per second.
代入数字，再开平方根，得到约四百八十米每秒。
Sanity-check that: it is faster than sound in air, which is exactly right — sound travels through air by molecular motion, so it cannot be faster than the molecules themselves.
做个合理性检验： 它比空气中的声速还快，而这恰恰是对的——声音是靠分子运动在空气中传播的， 所以它不可能比分子本身还快。
Three consequences that questions turn on.
有三个推论是考题的着力点。
First, doubling the absolute temperature doubles the average kinetic energy, so the mean-square speed doubles — but the r.m.s. speed grows only by root two, because of that square root.
第一，把绝对温度加倍，平均动能就加倍， 于是均方速率加倍——但均方根速率只增大根号二倍，因为外面还有一个平方根。
Second, for two gases at the same temperature the lighter gas moves faster: same average kinetic energy, smaller mass, so larger mean-square speed.
第二，两种气体处于同一温度时，较轻的气体运动得更快：平均动能相同，质量更小， 所以均方速率更大。
Hydrogen molecules really are moving faster than oxygen molecules in this room.
这个房间里的氢分子，确实比氧分子跑得快。
Third, the internal energy of an ideal gas is just the total kinetic energy, three halves N k T, equivalently three halves n R T — so it too is proportional to temperature alone.
第三，理想气体的内能就是全部动能，等于二分之三 N k T， 也就是二分之三 n R T——所以它同样只与温度成正比。
And a trap: squeezing it at fixed temperature raises the pressure but does not change the average kinetic energy at all.
还有一个陷阱： 在温度不变时压缩气体会提高压强，却完全不改变平均动能。
There are simply more wall collisions per second, each one just as energetic.
只不过是每秒撞击器壁的次数变多了，而每一次撞击的能量还是一样。
Three marks to secure.
三个要拿稳的分。
First, always use the temperature in kelvin in the gas equations — never Celsius.
第一，在气体方程中一律用开尔文温度——绝不用摄氏度。
Second, know both forms: pressure-volume equals n-R-T, and equals N-k-T.
第二，两种形式都要会：压强乘体积等于 nRT，也等于 NkT。
Third, the average kinetic energy of a molecule is three-halves k T, and depends only on temperature.
第三，一个分子的平均动能是二分之三 kT，只取决于温度。
Master these, and gases are yours.
掌握这些，气体就是你的了。